Nov 8, 2010 - B.3 Forced Air Convection Thermal Switch (FACTS). ...... Figure A.6: Illustration of MightySat II.1 (MightySat II DataSheet, 2005). ...... 3 (( 4900 (.
Graduate School ETD Form 9 (Revised 12/07)
PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Derek William Hengeveld Entitled Development of a System Design Methodology for Robust Thermal Control Subsystems to Support Responsive Space
For the degree of Doctor of Philosophy
Is approved by the final examining committee: Prof. James E. Braun Chair
Prof. Eckhard A. Groll
Prof. Suresh V. Garimella
Dr. Jeffry S. Welsh
To the best of my knowledge and as understood by the student in the Research Integrity and Copyright Disclaimer (Graduate School Form 20), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy on Integrity in Research” and the use of copyrighted material.
Prof. James E. Braun Approved by Major Professor(s): ____________________________________ Prof. Eckhard A. Groll ____________________________________
Approved by: David C. Anderson Head of the Graduate Program
11/08/2010 Date
Graduate School Form 20 (Revised 1/10)
PURDUE UNIVERSITY GRADUATE SCHOOL Research Integrity and Copyright Disclaimer
Title of Thesis/Dissertation: Development of a System Design Methodology for Robust Thermal Control Subsystems to Support Responsive Space Doctor of Philosophy For the degree of ________________________________________________________________
I certify that in the preparation of this thesis, I have observed the provisions of Purdue University Teaching, Research, and Outreach Policy on Research Misconduct (VIII.3.1), October 1, 2008.* Further, I certify that this work is free of plagiarism and all materials appearing in this thesis/dissertation have been properly quoted and attributed. I certify that all copyrighted material incorporated into this thesis/dissertation is in compliance with the United States’ copyright law and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and save harmless Purdue University from any and all claims that may be asserted or that may arise from any copyright violation.
Derek William Hengeveld ______________________________________ Printed Name and Signature of Candidate
11/08/2010 ______________________________________ Date (month/day/year)
*Located at http://www.purdue.edu/policies/pages/teach_res_outreach/viii_3_1.html
DEVELOPMENT OF A SYSTEM DESIGN METHODOLOGY FOR ROBUST THERMAL CONTROL SUBSYSTEMS TO SUPPORT RESPONSIVE SPACE
A Dissertation Submitted to the Faculty of Purdue University by Derek William Hengeveld
In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
December 2010 Purdue University West Lafayette, Indiana
ii
ACKNOWLEDGEMENTS
I would like to thank my advisors Professor James E. Braun and Professor Eckhard A. Groll for their encouragement, creativity, knowledge, advice, and friendship. The opportunity to work with them has been an invaluable experience. Additionally, I want to extend my appreciation to my advisory committee members Professor Suresh V. Garimella and Dr. Jeffry S. Welsh. A special thanks to Mr. Andrew D. Williams for his guidance and input. I would like to acknowledge the financial support provided by the Air Force Research Laboratory (AFRL), the Winkelman Davidson Fellowship, the Carrier ASHRAE Fellowship, an ASHRAE Grant-In-Aid Award, an AIAA Open Topic Graduate Award, and the AFRL Space Scholars program. Many thanks to the support staff at both the Air Force Research Laboratory and Ray W. Herrick Laboratories. Thank you to the many friends I made at both Purdue and AFRL. Thank you to my family and friends for their patience, belief, and encouragement. To Dr. Kurt Bassett and Mr. Mike Twedt, thank you for your great friendship and support over myriad years. To my parents Les and Linda Hengeveld, your love, support, and encouragement have always been instrumental in my achievements. To my sister Lesley and her husband Jack, your support and encouragement is appreciated. Finally, I would like to extend a special thank you to Lori Ziegler for her enduring love, support, patience, and encouragement.
iii
TABLE OF CONTENTS
Page LIST OF TABLES.............................................................................................................. x LIST OF FIGURES ........................................................................................................ xvii NOMENCLATURE ....................................................................................................... xxx ABSTRACT................................................................................................................ xxxvii CHAPTER 1 - INTRODUCTION...................................................................................... 1 CHAPTER 2 - RESPONSIVE SPACE .............................................................................. 3 CHAPTER 3 - ROBUST THERMAL CONTROL SUBSYSTEMS (TCS) FOR RESPONSIVE SPACE .................................................................................................. 9 CHAPTER 4 - MOTIVATION, OBJECTIVES, AND APPROACH.............................. 14 4.1 Motivation .............................................................................................................. 14 4.2 Objective ................................................................................................................ 15 4.3 Approach ................................................................................................................ 15 4.4 Unique Results of the Work................................................................................... 17 4.5 Organization of the Document ............................................................................... 19 CHAPTER 5 - DETERMINATION OF HOT- AND COLD-CASE DESIGN ORBITS FOR ROBUST THERMAL CONTROL SUBSYSTEM DESIGN............................. 20 5.1 Development of Weighting Matrices ..................................................................... 22 5.1.1 Temporal Variation of β ............................................................................... 22 5.1.1.1 Julian Day, JD ....................................................................................... 23 5.1.1.2 Sun Coordinates...................................................................................... 24 5.1.1.3 Orbital Elements..................................................................................... 25 5.1.2 Statistical Beta Angle versus Inclination Distribution ................................... 29 5.1.3 Historical Inclination Distribution ................................................................. 35
iv Page 5.1.4 Weighting Matrices ........................................................................................ 36 5.1.4.1 Statistical Weighting Matrices ............................................................... 36 5.1.4.2 Viable Weighting Matrices..................................................................... 37 5.2 Satellite Orbital Averaged Environmental Heat Loads.......................................... 39 ′′ ....................................................... 39 5.2.1 Orbital Averaged Direct Solar Flux, qsol ′′ ............................................................... 41 5.2.2 Orbital Averaged Albedo Flux, qalb ′′ ............ 47 5.2.3 Orbital Averaged Outgoing Longwave Radiation (OLR) Flux, qOLR 5.3 Hot- and Cold-Case Design Orbits Developed for Robust Thermal Control Subsystems ......................................................................................................... 53 5.4 Conclusion ............................................................................................................. 58 CHAPTER 6 - OPTIMAL DISTRIBUTION OF ELECTRONIC COMPONENTS TO BALANCE ENVIRONMENTAL FLUXES ............................................................... 60 6.1 Review of Satellite Layout Optimization Approaches........................................... 61 6.2 Numerical Experiment Investigations.................................................................... 63 6.2.1 Numerical Model Details ............................................................................... 63 6.2.2 Orbital Environmental Loads ......................................................................... 65 6.2.3 Hot- and Cold-Case Design Orbit Power Imbalance Case Studies................ 67 6.3 Methodology for Distributing Power Among Satellite Panels .............................. 73 6.4 Bin-Packing Problems............................................................................................ 75 6.5 Genetic Algorithm.................................................................................................. 78 6.5.1 Encoding......................................................................................................... 78 6.5.2 Fitness Evaluation .......................................................................................... 79 6.5.3 Evolution ........................................................................................................ 80 6.5.3.1 Elitist Strategies...................................................................................... 81 6.5.3.2 Reproduction .......................................................................................... 81 6.5.3.3 Best- and Worst-Fit Placement............................................................... 83 6.5.3.4 Mutation ................................................................................................. 83 6.6 Genetic Algorithm Tuning ..................................................................................... 83 6.6.1 Analysis of Convergence Criteria and Population Size ................................. 84
v Page 6.6.2 Evolution Parameter Tuning .......................................................................... 88 6.7 Algorithm Demonstration ...................................................................................... 91 6.8 Effect of Increasing Component Power ................................................................. 93 6.9 Conclusions ............................................................................................................ 95 CHAPTER 7 - OPTIMAL PLACEMENT OF ELECTRONIC COMPONENTS TO MINIMIZE HEAT FLUX NON-UNIFORMITIES..................................................... 97 7.1 Review of Component Placement Optimization Methods..................................... 98 7.2 Numerical Experiment Investigations.................................................................. 100 7.2.1 Numerical Model Details ............................................................................. 100 7.2.2 Component Placement versus Thermal Conductivity .................................. 100 7.3 Methodology ........................................................................................................ 102 7.4 Packing Problems................................................................................................. 104 7.5 Genetic Algorithm................................................................................................ 106 7.5.1 Encoding....................................................................................................... 106 7.5.2 Fitness Evaluation ........................................................................................ 107 7.5.3 Evolution ...................................................................................................... 110 7.5.3.1 Elitist Strategies.................................................................................... 110 7.5.3.2 Reproduction ........................................................................................ 111 7.5.3.3 Local Gradient Searches....................................................................... 112 7.5.3.4 Mutation ............................................................................................... 118 7.6 Genetic Algorithm Tuning ................................................................................... 118 7.6.1 Convergence Criteria.................................................................................... 119 7.6.2 Population Size............................................................................................. 120 7.6.3 Evolution Parameter Tuning ........................................................................ 120 7.7 Algorithm Demonstration .................................................................................... 123 7.8 Conclusions .......................................................................................................... 125 CHAPTER 8 - DEVELOPMENT AND EVALUATION OF REDUCED ORDER SATELLITE THERMAL MODELS......................................................................... 127 8.1 Methodology ........................................................................................................ 128
vi Page 8.2 Numerical Model Details ..................................................................................... 129 8.2.1 Description of Factors .................................................................................. 132 8.2.1.1 Orbit...................................................................................................... 132 8.2.1.2 Total Component Power....................................................................... 132 8.2.1.3 Component Dimension......................................................................... 132 8.2.1.4 Component Interface Heat Transfer Coefficient .................................. 132 8.2.1.5 Facesheet Material Transverse Thermal Conductivity......................... 133 8.2.1.6 Heat Pipes............................................................................................. 133 8.2.1.7 Panel-to-Panel Conductivity................................................................. 133 8.2.1.8 Surface Solar Absorptivity and Longwave Emissivity ........................ 133 8.2.1.9 Global Component Distribution and Local Component Placement ..... 135 8.3 Full-Factorial Screening Analysis........................................................................ 137 8.3.1 Main Effects versus Tmax Response ........................................................... 137 8.3.2 Interaction Effects versus Tmaxd Response ................................................ 141 8.3.3 Full-Factorial Screening Discussion ............................................................ 143 8.4 Reduced-Order Model Development ................................................................... 145 8.4.1 Reduced-Order Model.................................................................................. 146 8.4.2 Development of Test Cases.......................................................................... 149 8.4.2.1 Results .................................................................................................. 149 8.4.2.1.1 Factor Sweep Analysis ................................................................. 156 8.4.2.1.2 Evaluation of Reduced-Order Model for Other Distributions...... 160 8.4.2.1.3 Example of Reduced-Order Model Use ....................................... 163 8.5 Conclusions .......................................................................................................... 165 CHAPTER 9 - CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ....... 167 9.1 Determination of Hot- and Cold-Case Design Orbits for Robust Thermal Control Subsystem Design ............................................................................................ 167 9.2 Optimal Distribution of Electronic Components to Balance Environmental Fluxes .......................................................................................................................... 168
vii Page 9.3 Optimal Placement of Electronic Components to Minimize Heat Flux NonUniformities...................................................................................................... 169 9.4 Development and Evaluation of Reduced-Order Satellite Thermal Models ....... 170 9.5 Assumptions and Limitations of Work ................................................................ 171 9.5.1 Determination of Hot- and Cold-Case Design Orbits for Robust Thermal Control Subsystem Design...................................................................... 171 9.5.1.1 Assumptions ......................................................................................... 171 9.5.1.2 Limitations............................................................................................ 172 9.5.2 Optimal Distribution of Electronic Components to Balance Environmental Fluxes...................................................................................................... 173 9.5.2.1 Assumptions ......................................................................................... 173 9.5.2.2 Limitations............................................................................................ 174 9.5.3 Optimal Placement of Electronic Components to Minimize Heat Flux NonUniformities ............................................................................................ 174 9.5.3.1 Assumptions ......................................................................................... 174 9.5.3.2 Limitations............................................................................................ 175 9.5.4 Development and Evaluation of Reduced-Order Satellite Thermal Models 175 9.5.4.1 Assumptions ......................................................................................... 175 9.5.4.2 Limitations............................................................................................ 175 9.6 Suggestions for Future Work ............................................................................... 176 9.6.1 Research in Design Orbits for Robust Thermal Control Subsystem Design 176 9.6.2 Component Placement Optimization Research............................................ 177 9.6.3 Research in Reduced-Order Satellite Thermal Models................................ 178 LIST OF REFERENCES................................................................................................ 179 APPENDICES Appendix A – Review of Robust Satellite Architectures........................................... 193 A.1 TacSat ............................................................................................................. 193 A.2 Plug-and-Play Sat (PnPSat)............................................................................ 194 A.3 HexPak ........................................................................................................... 195
viii Page A.4 SMARTBus .................................................................................................... 196 A.5 MightySat ....................................................................................................... 197 A.6 CubeSat .......................................................................................................... 198 Appendix B – Review of Robust Thermal Control Subsystem (TCS) Approaches .. 199 B.1 Thermal Management for Modular Satellites (TherMMS) ............................ 199 B.2 Satellite Modular and Reconfigurable Thermal System (SMARTS) ............. 200 B.3 Forced Air Convection Thermal Switch (FACTS)......................................... 201 B.4 HexPak............................................................................................................ 201 B.5 SMARTBus .................................................................................................... 202 B.6 Integrated Thermal Energy Management System (ITEMS)........................... 202 Appendix C - Astrodynamics..................................................................................... 204 C.1 Equations of Motion ....................................................................................... 204 C.2 Sun Coordinates.............................................................................................. 205 C.3 Classical Orbital Elements.............................................................................. 207 C.4 Thermal Orbital Elements .............................................................................. 211 C.5 Perturbations................................................................................................... 214 C.6 Orbital Classifications .................................................................................... 215 Appendix D – Viable Weighting Matrices at Various Thresholds ............................ 219 Appendix E – Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits at Various Thresholds........................................................................................... 222 Appendix F – Numerical Model Investigation .......................................................... 228 F.1 Description of Software Tool.......................................................................... 228 F.1.1 Transient Solver ...................................................................................... 229 F.1.2 Steady-State Solver ................................................................................. 230 F.2 Description of Numerical Testing Model ....................................................... 231 F.3 Number of Orbital Simulations....................................................................... 231 F.4 Nodal Resolution Analysis.............................................................................. 240 F.5 Monte-Carlo Rays per Node Analysis ............................................................ 241 F.6 Analysis of Number of Orbital Positions ........................................................ 245
ix Page Appendix G – Characteristic Component Design Cases............................................ 249 G.1 Reference Missions ........................................................................................ 249 G.2 Design Case Distribution Development ......................................................... 256 G.3 Subsystems ..................................................................................................... 261 G.3.1 Attitude Determination and Control Subsystem (ADCS) ...................... 261 G.3.2 Orbit Determination and Control Subsystem (ODCS) ........................... 264 G.3.3 Telemetry, Tracking, and Command Subsystem (TTCS) ...................... 264 G.3.4 Command and Data Handling Subsystem (CDHS)................................ 265 G.3.5 Electrical Power Subsystem (EPS)......................................................... 265 G.3.6 Structure and Mechanisms Subsystem (SMS)........................................ 265 G.3.7 Payload ................................................................................................... 266 G.3.8 Thermal Control Subsystem (TCS) ........................................................ 266 Appendix H – Hot- and Cold-Case Temporal Flux Characteristics .......................... 267 Appendix I – Resulting Component Placement Values............................................. 270 Appendix J – Full Factorial Main and Interaction Effect Plots ................................. 291 Appendix K – Temperature Versus Time Results for Increasing Levels of Satellite Conductance ..................................................................................................... 304 Appendix L – Latin Hypercube Sampling Points ...................................................... 306 Appendix M – Summary of Simulation Test Cases................................................... 310 Appendix N – Summary of Reduced-Order Model Coefficients............................... 313 VITA ............................................................................................................................... 338
x
LIST OF TABLES
Table
Page
Table 1.1: Design and Development Schedule Requirements for Varying Spacecraft Complexities (Saleh and Dubos, 2007).......................................................................... 1 Table 1.2: Comparison of Traditional, Small, and Robust Satellite Design Approaches... 2 Table 2.1: Responsive Space Attributes a(Arritt et al., 2008), b(Saleh and Dubos, 2007), c
(U.S. GAO, 2008), d(Wegner and Kiziah, 2006). ......................................................... 5
Table 2.2: Responsive Space Mission Performance Requirements (ORSBS-001, 2007). . 5 Table 2.3: Responsive Space Satellite Architectures a(U.S. GAO, 2008), b(Fronterhouse, Lyke, and Achramowicz, 2007), c(Hicks, Enoch, and Capots, 2005), d(Hicks, Hashemi, and Capots, 2006), e(McDermott and Jordan, 2005), f(Freeman, Rudder, and Thomas, 2000), g(MightySat II DataSheet, 2005), h(Ince, 2005), i(CubeSat Design Specification, 2008). ...................................................................................................... 7 Table 5.1: Review of Hot- and Cold-Case Design Orbits Utilized for Robust TCS Development. ............................................................................................................... 21 Table 5.2: Surface Properties for the Four Surface Categories. ....................................... 39 Table 5.3: Albedo and OLR Values for Mission-Critical Hot and Cold Extreme Environments at Low, Medium or High Inclinations and Averaging Times from 16 seconds to 24 hours (Gilmore, 2002). .......................................................................... 45 Table 5.4: Average Correction Term, c , versus β . ......................................................... 46 Table 5.5: Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits for Each of Four Surface Categories at a 0.00 Threshold. .............................................................. 54 Table 5.6: Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits for Each of Four Surface Categories............................................................................................... 55
xi Table
Page
Table 5.7: Hot-Case Design Orbit Percent Difference Analysis (Utilizing Orbital Averaged Values and Viable Weighting Matrices). .................................................... 56 Table 5.8: Cold-Case Design Orbits Percent Difference (Utilizing Orbital Averaged Values and Viable Weighting Matrices). ..................................................................... 57 Table 6.1: Hot and Cold Case Design Orbit Environmental Parameters.......................... 66 Table 6.2: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values............................................................................................ 66 Table 6.3: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.10 / ε = 0.80). ...................................................................................................................................... 67 Table 6.4: Case C Computational Time and Maximum Temperature Difference Results for Optimized Distributions of 18, 36, and 54 Components at 600 W of Total Power. ...................................................................................................................................... 91 Table 6.5: Case C Results for Optimized, Even, and Worst-Case Component Distributions at 600 W of Total Power. ....................................................................... 92 Table 8.1: Summary of Fixed Numerical Model Parameters. ........................................ 130 Table 8.2: Summary of Factors and Corresponding Levels. .......................................... 130 Table 8.3: Hot- and Cold-Case Orbit Environmental Fluxes. ........................................ 134 Table 8.4: Full Factorial Summary of Lenth t-Ratio and Corresponding p-Values for Main Effects in Descending Order According to Absolute Effect Values. ............... 139 Table 8.5: Full Factorial Summary of Lenth t-Ratio and Corresponding p-Values for the First 15 Interaction Effects in Descending Order According to Absolute Contrast Values......................................................................................................................... 141 Table 8.6: Summary of Eight Reduced-Order Models versus Four Categorical Factors. .................................................................................................................................... 147 Table 8.7: Residual Normality Results for Three Response Variables. ......................... 156 Table 8.8: Summary of Nominal Factors Levels. ........................................................... 156 Table 8.9: Summary of Example Factor Levels. ............................................................ 164
xii Appendix Table
Page
Table C.1: Summary of Key Characteristics of RS Appropriate Orbits (Wertz, 2005; Wertz, 2007)............................................................................................................... 218 Table E.1: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.01 Threshold. ............................................................ 222 Table E.2: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.02 Threshold. ............................................................ 223 Table E.3: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.03 Threshold. ............................................................ 223 Table E.4: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.04 Threshold. ............................................................ 224 Table E.5: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.05 Threshold. ............................................................ 224 Table E.6: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.10 Threshold. ............................................................ 225 Table E.7: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.20 Threshold. ............................................................ 225 Table E.8: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.25 Threshold. ............................................................ 226 Table E.9: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.26 Threshold. ............................................................ 226 Table E.10: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.30 Threshold. ............................................................ 227 Table E.11: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.50 Threshold. ............................................................ 227 Table F.1: Three Panel Construction Types Used for Orbital Simulation Analysis....... 232 Table F.2: Numerical Model Parameters for Orbital Simulation Analysis. ................... 232 Table F.3: Orbital Simulation Analysis Hot-Case Summary.......................................... 239 Table F.4: Orbital Simulation Analysis Cold-Case Summary........................................ 239 Table F.5: Numerical Model Parameters for Nodal Resolution Analysis. ..................... 240
xiii Appendix Table
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Table F.6: Numerical Model Parameters for Monte-Carlo Rays per Node Analysis..... 243 Table F.7: Rays per Node Analysis Hot-Case Summary................................................ 244 Table F.8: Rays per Node Analysis Cold-Case Summary.............................................. 245 Table F.9: Numerical Model Parameters for Orbital Position Analysis......................... 247 Table F.10: Orbital Position Analysis Hot-Case Summary. ........................................... 247 Table F.11: Orbital Position Analysis Cold-Case Summary. ......................................... 247 Table G.1: FACTS LCB Distribution (Williams, 2005)................................................. 250 Table G.2: FACTS HCB Distribution (Williams, 2005). ............................................... 251 Table G.3: Representative Military Satellite Distribution. ............................................. 252 Table G.4: TherMMS LCB Component Distribution (Young, 2008). ........................... 253 Table G.5: TherMMS MCB Component Distribution (Young, 2008). .......................... 254 Table G.6: TherMMS HCB Component Distribution (Young, 2008)............................ 255 Table G.7: Summary of Reference Missions.................................................................. 256 Table G.8: Summary of s Values for Each Design Case .............................................. 258 Table G.9: m Values at Various Levels of s and N . ................................................... 260 Table G.10: Normalization Values for 18 Components for Each of Four Design Cases. .................................................................................................................................... 260 Table I.1: Summary of Hot- and Cold-Case Orbit (Case C-36) – Nominal Global Distribution / Nominal Local Placement. .................................................................. 271 Table I.2: Summary of Hot- and Cold-Case Orbit (Case C-36) – Nominal Global Distribution / Optimized Local Placement................................................................. 272 Table I.3: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.100)..................................................... 273 Table I.4: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.900)..................................................... 274 Table I.5: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.100)..................................................... 275 Table I.6: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.900)..................................................... 276
xiv Appendix Table
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Table I.7: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.100)..................................................... 277 Table I.8: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.900)..................................................... 278 Table I.9: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.100)..................................................... 279 Table I.10: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.900)..................................................... 280 Table I.11: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.100). ................................................. 281 Table I.12: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.900). ................................................. 282 Table I.13: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.100). ................................................. 283 Table I.14: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.900). ................................................. 284 Table I.15: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.100). ................................................. 285 Table I.16: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.900). ................................................. 286 Table I.17: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.100). ................................................. 287 Table I.18: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.900). ................................................. 288 Table I.19: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.123 / ε = 0.100). ........................................................................................................................ 289
xv Appendix Table
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Table I.20: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.561 / ε = 0.100). ........................................................................................................................ 289 Table I.21: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.123 / ε = 0.900). ........................................................................................................................ 290 Table I.22: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.561 / ε = 0.900). ........................................................................................................................ 290 Table L.1: Latin Hypercube Test Points for Space Filling Approach (1-32 of 128). ..... 306 Table L.2: Latin Hypercube Test Points for Space Filling Approach (33-64 of 128). ... 307 Table L.3: Latin Hypercube Test Points for Space Filling Approach (65-96 of 128). ... 308 Table L.4: Latin Hypercube Test Points for Space Filling Approach (97-128 of 128). . 309 Table M.1: Summary of Test Cases used to Evaluate Reduced-Order Models (1-33 of 100). ........................................................................................................................... 310 Table M.2: Summary of Test Cases used to Evaluate Reduced-Order Models (34-66 of 100). ........................................................................................................................... 311 Table M.3: Summary of Test Cases used to Evaluate Reduced-Order Models (67-100 of 100). ........................................................................................................................... 312 Table N.1: Summary of Reduced-Order (LH000 Model) Tmax Coefficients................ 314 Table N.2: Summary of Reduced-Order (LH000 Model) Tmin Coefficients. ............... 315 Table N.3: Summary of Reduced-Order (LH000 Model) Tmaxd Coefficients.............. 316 Table N.4: Summary of Reduced-Order (LH001 Model) Tmax Coefficients................ 317 Table N.5: Summary of Reduced-Order (LH001 Model) Tmin Coefficients. ............... 318 Table N.6: Summary of Reduced-Order (LH001 Model) Tmaxd Coefficients.............. 319 Table N.7: Summary of Reduced-Order (LH010 Model) Tmax Coefficients................ 320 Table N.8: Summary of Reduced-Order (LH010 Model) Tmin Coefficients. ............... 321 Table N.9: Summary of Reduced-Order (LH010 Model) Tmaxd Coefficients.............. 322 Table N.10: Summary of Reduced-Order (LH011 Model) Tmax Coefficients.............. 323
xvi Appendix Table
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Table N.11: Summary of Reduced-Order (LH011 Model) Tmin Coefficients. ............. 324 Table N.12: Summary of Reduced-Order (LH011 Model) Tmaxd Coefficients............ 325 Table N.13: Summary of Reduced-Order (LH100 Model) Tmax Coefficients.............. 326 Table N.14: Summary of Reduced-Order (LH100 Model) Tmin Coefficients. ............. 327 Table N.15: Summary of Reduced-Order (LH100 Model) Tmaxd Coefficients............ 328 Table N.16: Summary of Reduced-Order (LH101 Model) Tmax Coefficients.............. 329 Table N.17: Summary of Reduced-Order (LH101 Model) Tmin Coefficients. ............. 330 Table N.18: Summary of Reduced-Order (LH101 Model) Tmaxd Coefficients............ 331 Table N.19: Summary of Reduced-Order (LH110 Model) Tmax Coefficients.............. 332 Table N.20: Summary of Reduced-Order (LH110 Model) Tmin Coefficients. ............. 333 Table N.21: Summary of Reduced-Order (LH110 Model) Tmaxd Coefficients............ 334 Table N.22: Summary of Reduced-Order (LH111 Model) Tmax Coefficients.............. 335 Table N.23: Summary of Reduced-Order (LH111 Model) Tmin Coefficients. ............. 336 Table N.24: Summary of Reduced-Order (LH111 Model) Tmaxd Coefficients............ 337
xvii
LIST OF FIGURES
Figure
Page
Figure 3.1: Traditional Thermal Control Subsystem Design Procedure........................... 10 Figure 3.2: Robust Thermal Control Subsystem Design Procedure. ................................ 11 Figure 5.1: Overview of Analysis Methodology. ............................................................. 22 Figure 5.2: Illustration of β Relationship Over One Year for a Circular Orbit at an i of 0° and a Right Ascension of the Ascending Node of 0°. ............................................. 23 Figure 5.3: Illustration of Sun Coordinate Variables........................................................ 24 Figure 5.4: Illustration of the Six Classical Orbital Elements. ......................................... 26 Figure 5.5: β versus Time for Select Inclinations at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°). ......................................................................... 27 Figure 5.6: β versus Time for Select Inclinations at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°). .............................................................................. 28 Figure 5.7: β versus Time for Select Inclinations at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°). ............................................................................ 28 Figure 5.8: Normalized Frequency of Occurrence for β for Select i at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°). ....................................... 30 Figure 5.9: Normalized Frequency of Occurrence for β for Select i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°). ................................................ 30 Figure 5.10: Normalized Frequency of Occurrence for β for Select i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°). ....................................... 31 Figure 5.11: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°). ............ 31
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Figure 5.12: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°). .............. 32 Figure 5.13: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°). ............ 32 Figure 5.14: Normalized Frequency of Occurrence for β for Select i at Altitudes of 350 to 1000 km (Beginning on March 21st 2007 and RAAN of 0° to 360°). ..................... 33 Figure 5.15: Contour Plot of Normalized Frequency of Occurrence for β for i at Altitudes of 350 to 1000 km (Beginning on March 21st 2007 and RAAN of 0° to 360°)............................................................................................................................. 34 Figure 5.16: Historical Inclination Distribution based on Unclassified LEO Satellite Launches from 1957 to 2006........................................................................................ 35 Figure 5.17: Contour Plot of Statistical Weighting Matrix Based on Hadamard Product of
β versus i Distribution and Historical Inclination Distribution Information............. 37 Figure 5.18: Contour Plot of Viable Weighting Matrix for Combinations of β and i at a Threshold of 0.00. ........................................................................................................ 38 Figure 5.19: Contour Plot of Viable Weighting Matrix for Combinations of β and i at a Threshold of 0.03. ........................................................................................................ 38 Figure 5.20: Solar Cycle as Represented by Yearly Mean Sunspot Number (1900 - 2000) (Anderson and Smith, 1994). ....................................................................................... 40 Figure 5.21: Albedo versus OLR for Medium Inclination Orbits (30 to 60°) using 128 second Data Averaging (Anderson and Smith, 1994).................................................. 44 Figure 5.22: Orbital Averaged Albedo Flux Correction Factor, K, versus β .................. 47 Figure 5.23: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Absorber). .................................................................................................................... 49 Figure 5.24: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Absorber). ............................................................................................................ 49 Figure 5.25: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Reflector) Surfaces....................................................................................................... 50
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Figure 5.26: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Reflector). ............................................................................................................ 50 Figure 5.27: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Absorber)........................................................................................................... 51 Figure 5.28: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Absorber)........................................................................................................... 51 Figure 5.29: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Reflector)........................................................................................................... 52 Figure 5.30: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Reflector)........................................................................................................... 52 Figure 6.1: Illustration of Exterior and Interior of Six-sided Thermal Desktop® Satellite Model. .......................................................................................................................... 64 Figure 6.2: Mathematical Models for Each of Four Design Cases (Refer to Appendix G for a detailed explanation of the design cases)............................................................. 65 Figure 6.3: Maximum and Minimum Subsystem Temperature and Maximum Temperature Difference for Varying Levels of Power Imbalance (Hot-Case Orbit). . 68 Figure 6.4: Temporal Variation of Temperature for Six Subsystems at Imbalance of -1.0 for the Hot-Case Orbit.................................................................................................. 68 Figure 6.5: Temporal Variation of Temperature for Six Subsystems at Imbalance of 0.0 for the Hot-Case Orbit.................................................................................................. 69 Figure 6.6: Temporal Variation of Temperature for Six Subsystems at Imbalance of +1.0 for the Hot-Case Orbit.................................................................................................. 69 Figure 6.7: Maximum and Minimum Subsystem Temperature and Maximum Temperature Difference for Varying Levels of Power Imbalance (Cold-Case Orbit). 70 Figure 6.8: Temporal Variation of Temperature for Six Subsystems at Imbalance of -1.0 for the Cold-Case Orbit................................................................................................ 71 Figure 6.9: Temporal Variation of Temperature for Six Subsystems at Imbalance of 0.0 for the Cold-Case Orbit................................................................................................ 71
xx Figure
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Figure 6.10: Temporal Variation of Temperature for Six Subsystems at Imbalance of +1.0 for the Cold-Case Orbit................................................................................................ 72 Figure 6.11: Illustration of Bin-Packing Methodology for Optimized Component Distributions................................................................................................................. 78 Figure 6.12: Chromosome Representation. ...................................................................... 78 Figure 6.13: Maximum Temperature Difference versus Fitness for Increasing Number of Components. ................................................................................................................ 80 Figure 6.14: Maximum and Minimum Temperature versus Fitness for Increasing Number of Components. ............................................................................................................ 80 Figure 6.15: Illustration of Evolution Techniques Used to Obtain a Fully Populated New Generation of Solutions................................................................................................ 81 Figure 6.16: Illustration of Reproduction Methods. ......................................................... 82 Figure 6.17: Fitness and Computational Time versus Ending Criteria for Component Distribution A with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case A-18, Case A-36, Case A-54). ........................................................................... 85 Figure 6.18: Fitness and Computational Time versus Ending Criteria for Component Distribution C with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case C-18, Case C-36, Case C-54)............................................................................. 85 Figure 6.19: Fitness and Computational Time versus Ending Criteria for Component Distribution D with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case D-18, Case D-36, Case D-54). ........................................................................... 86 Figure 6.20: Fitness and Computational Time versus Population Multiplier for Component Distribution A with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case A-18, Case A-36, Case A-54). ..................................................... 86 Figure 6.21: Fitness and Computational Time versus Population Multiplier for Component Distribution C with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case C-18, Case C-36, Case C-54). ...................................................... 87
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Figure 6.22: Fitness and Computational Time versus Population Multiplier for Component Distribution D with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case D-18, Case D-36, Case D-54). ..................................................... 87 Figure 6.23: Normalized Fitness versus Computation Time for Increasing Values of Elitism Parameter Pb .................................................................................................... 88 Figure 6.24: Normalized Fitness versus Computation Time for Increasing Values of Reproduction Parameter Pr . ........................................................................................ 89 Figure 6.25: Normalized Fitness versus Computation Time for Increasing Values of BestFit Parameter Pbf . ........................................................................................................ 89 Figure 6.26: Normalized Fitness versus Computation Time for Increasing Values of Worst-Fit Parameter Pwf . ............................................................................................. 90 Figure 6.27: Normalized Fitness versus Computation Time for Increasing Values of Mutation Parameter. Pm ............................................................................................... 90 Figure 6.28: Case C-36 Temporal Results for Optimized Component Distributions at 600 W of Total Power in the Hot-Case Orbit. .................................................................... 92 Figure 6.29: Case C-36 Temporal Results for Even Component Distributions at 600 W of Total Power in the Hot-Case Orbit............................................................................... 93 Figure 6.30: Case C-36 Temporal Results for Worst-Case Component Distributions at 600 W of Total Power in the Hot-Case Orbit. ............................................................. 93 Figure 6.31: Maximum and Minimum Temperature versus Total Power for Optimized, Even, and Worst-case Component Distributions for Case C-36 in the Hot-Case Thermal Environment. ................................................................................................. 94 Figure 6.32: Maximum Temperature Difference versus Total Power for Optimized, Even, and Worst-case Component Distributions for Case C-36 in the Hot-Case Thermal Environment................................................................................................................. 95 Figure 7.1: Illustration of Component Placement for Offset Values of a) 0.00 m; b) 0.10 m; c) 0.20 m; d) 0.30 m; and e) 0.40 m...................................................................... 101
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Figure 7.2: Maximum Component Temperature at Steady-State for Offset Values of 0.00 m to 0.35 m and Thermal Conductivities of 100 to 10,000 W/m-K. ......................... 101 Figure 7.3: Magnified Maximum Component Temperature at Steady-State for Offset Values of 0.00 m to 0.35 m and Thermal Conductivities of 100 to 2,700 W/m-K.... 102 Figure 7.4: Illustration of Relationship Between Rectangular Domain and Resulting Effective Circular Area. ............................................................................................. 103 Figure 7.5: Arrangement of Components with Circular Effective Areas in a Domain. . 103 Figure 7.6: Chromosome Representation. ...................................................................... 107 Figure 7.7: Maximum Temperature Difference versus Normalized Fitness for Increasing Number of Components. ............................................................................................ 109 Figure 7.8: Maximum Temperature versus Normalized Fitness for Increasing Number of Components. .............................................................................................................. 109 Figure 7.9: Illustration of Evolution Techniques Used to Obtain a Fully Populated New Generation of Solutions.............................................................................................. 110 Figure 7.10: Illustration of Reproduction Methods. ....................................................... 112 Figure 7.11: Illustration of Local Gradient Searches for Five Components Initially in a Corner and in the Center. ........................................................................................... 117 Figure 7.12: Illustration of Normalized Fitness versus Generation Number for Three Levels of Pg (0.0, 0.01, and 0.10).............................................................................. 118 Figure 7.13: Normalized Fitness versus Number of Repeating Solutions Before Convergence for Increasing Number of Components................................................ 119 Figure 7.14: Time versus Number of Repeating Solutions Before Convergence for Increasing Number of Components. .......................................................................... 120 Figure 7.15: Normalized Fitness versus Computation Time for Increasing Values of Elitism Parameter Pb .................................................................................................. 121 Figure 7.16: Normalized Fitness versus Computation Time for Increasing Values of Reproduction Parameter Pr . ...................................................................................... 121
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Figure 7.17: Normalized Fitness versus Computation Time for Increasing Values of Local Gradient Parameter Pg ..................................................................................... 122 Figure 7.18: Normalized Fitness versus Computation Time for Increasing Values of Mutation Parameter Pm .............................................................................................. 122 Figure 7.19: Optimized Component Placement for 18 Uniform Components in a Square Domain....................................................................................................................... 123 Figure 7.20: Optimized Temperature Distribution Results for 18 Uniform Components Obtained from a Thermal Desktop® Finite Difference Model.................................. 124 Figure 7.21: Optimized Component Placement for 11 Non-Uniform Components in a Square Domain........................................................................................................... 124 Figure 7.22: Optimized Temperature Distribution Results for 11 Non-Uniform Components Obtained from a Thermal Desktop® Finite Difference Model. ........... 125 Figure 8.1: Illustration of Relationship between Input Conditions, TCS Approach, and Thermal Performance................................................................................................. 128 Figure 8.2: Illustration of Exterior and Interior of Six-sided Thermal Desktop® Satellite Model Detailing Multiple Components on Each Face............................................... 129 Figure 8.3: Cold-Case Orbit Energy Balance Results. ................................................... 134 Figure 8.4: Hot-Case Energy Balance Results................................................................ 135 Figure 8.5: Illustration of Four Global Component Distribution and Local Component Placement Cases for Select Surfaces in the Cold-Case Orbit with α = 0.123 and ε = 0.900........................................................................................................................... 136 Figure 8.6: Effects Pareto Chart for Tmaxd Responses (All Main Effects). .................. 139 Figure 8.7: Illustration of First Four Main Effects: a) EXT_ABS, b) F_T_CND, c) ORBIT, and d) GLBL_DIS and Including a Linear Regression Line and Horizontal Jitter for Clarity. ......................................................................................................... 140 Figure 8.8: Effects Pareto Chart for Tmaxd Responses (First 15 Interaction Effects). .. 142 Figure 8.9: Illustration of First Four Interaction Effects: a) EXT_ABS*ORBIT, b) GLBL_DIS*TOT_PWR, c) EXT_ABS*EXT_EMS, and d) ORBIT*TOT_PWR and Including a Linear Regression Line and Horizontal Jitter for Clarity. ...................... 143
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Figure 8.10: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmax Result versus Computer Simulation (CS) Tmax Results. .......................................... 149 Figure 8.11: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmin Result versus Computer Simulation (CS) Tmin Results............................................ 150 Figure 8.12: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmaxd Result versus Computer Simulation (CS) Tmaxd Results. ........................................ 150 Figure 8.13: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmax Results versus CS Tmax.................................... 151 Figure 8.14: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmin Results versus CS Tmin. .................................... 151 Figure 8.15: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmaxd Results versus CS Tmaxd................................ 152 Figure 8.16: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmax Results. ................................................. 152 Figure 8.17: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmin Results................................................... 153 Figure 8.18: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmaxd Results. ............................................... 153 Figure 8.19: Normal Probability Plot of Tmax Residual Results with 95% Confidence Intervals and Statistical Indices.................................................................................. 154 Figure 8.20: Normal Probability Plot of Tmin Residual Results with 95% Confidence Intervals and Statistical Indices.................................................................................. 154 Figure 8.21: Normal Probability Plot of Tmaxd Residual Results with 95% Confidence Intervals and Statistical Indices.................................................................................. 155 Figure 8.22: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmax versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.......................................................................................................................... 157
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Figure 8.23: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmin versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.......................................................................................................................... 157 Figure 8.24: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmaxd versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.......................................................................................................................... 158 Figure 8.25: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmax versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits............................................................................................. 158 Figure 8.26: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmin versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits............................................................................................. 159 Figure 8.27: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmaxd versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits............................................................................................. 159 Figure 8.28: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmax Result versus Computer Simulation (CS) Tmax Case B-36 Results. ........................ 160 Figure 8.29: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmin Result versus Computer Simulation (CS) Tmin Case B-36 Results. ......................... 161 Figure 8.30: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmaxd Result versus Computer Simulation (CS) Tmaxd Case B-36 Results. ...................... 161 Figure 8.31: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmax Case B-36 Results versus CS Tmax. ................. 162 Figure 8.32: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmin Case B-36 Results versus CS Tmin. .................. 162 Figure 8.33: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmaxd Case B-36 Results versus CS Tmaxd. ............. 163 Figure 8.34: Tmax and Tmin Temperature Results versus external emissivity (EXT_EMS) for both Cold- and Hot-Case Conditions.............................................. 165
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Figure A.1: TacSat-1 (Raymond et al., 2004)................................................................. 193 Figure A.2: TacSat-2 (Peck, 2006). ................................................................................ 194 Figure A.3: Plug-and-Play Satellite General Schematic (Spaceworks, Inc., 2008)........ 195 Figure A.4: Illustration of HexPak Deployment (Hicks, Enoch and Capots, 2005)....... 195 Figure A.5: Illustration of SMARTBus Concept (McDermott and Jordan, 2005). ........ 196 Figure A.6: Illustration of MightySat II.1 (MightySat II DataSheet, 2005). .................. 197 Figure B.1: TherMMS TCS Candidate Solutions (Young, 2008). ................................. 199 Figure B.2: SMARTS TCS Concept (Bugby, Zimbeck, and Kroliczek, 2008).............. 200 Figure B.3: ITEMS TCS Concept (Birur and O’Donnell, 2001).................................... 203 Figure C.1: Illustration of Sun Coordinate Variables. .................................................... 206 Figure C.2: Diagram of Six Classical Orbital Elements. ................................................ 207 Figure C.3: Semi-major Axis and Eccentricity Diagram................................................ 207 Figure C.4: Ecliptic and Equatorial Planes along with Vernal Equinox and Orbit Normals. .................................................................................................................................... 208 Figure C.5: Diagram of Orbital Inclination. ................................................................... 208 Figure C.6: Diagram of the Right Ascension of the Ascending Node (RAAN)............. 209 Figure C.7: Diagram of Argument of Perigee. ............................................................... 209 Figure C.8: Diagram of True Anomaly........................................................................... 210 Figure C.9: Diagram of Orbital Elements for Circular Orbits. ....................................... 210 Figure C.10: Diagram of β = 0° and i = 113.4°. .......................................................... 211 Figure C.11: Diagram of β = 0° and i = 0°. ................................................................. 211 Figure C.12: Diagram of β = 90° and i = 113.4°. ........................................................ 212 Figure C.13: Illustration of β Relationship Over One Year for a Circular Orbit at an Inclination of 0° and a RAAN of 0°........................................................................... 212 Figure C.14: Absolute Value of the Maximum β versus i (0 to 180°). ....................... 213 Figure C.15: Earth’s Oblateness. .................................................................................... 214 Figure C.16: Sun Synchronous Orbit with β = 0°......................................................... 216
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Figure C.17: β versus Time for Select Inclinations at an Altitude of 650 km (Beginning on March 21st 2007 and RAAN = 0°). ....................................................................... 217 Figure D.1: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.00, b) 0.01, c) 0.02, and d) 0.03. .................................................. 219 Figure D.2: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.04, b) 0.05, c) 0.10, and d) 0.20. .................................................. 220 Figure D.3: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.25, b) 0.26, c) 0.30, and d) 0.50. .................................................. 221 Figure F.1: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on Nominal Panel Construction (54.91 kg)............................................ 233 Figure F.2: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Conductivity Panel Construction (50.41 kg). .......................... 234 Figure F.3: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Capacitance Panel Construction (464.36 kg)........................... 235 Figure F.4: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on Nominal Panel Construction (54.91 kg)............................................ 236 Figure F.5: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Conductivity Panel Construction (50.41 kg). .......................... 237 Figure F.6: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Capacitance Panel Construction (464.36 kg)........................... 238 Figure F.7: Subsystem Temperature and Computational Time Results versus Nodal Resolution for the Hot-Case Orbit. ............................................................................ 241 Figure F.8: Hot- and Cold-Case Monte-Carlo Rays per Node Analysis Results for Five Levels of Rays per Node (2000, 3000, 4000, 5000, 10000, 20000, and 100000). .... 244 Figure F.9: Hot- and Cold-Case Orbital Position Analysis Results for Five Levels of Orbital Positions (3, 6, 9, 12, and 36). ....................................................................... 246 Figure G.1: Component Power versus Component Number for Each Reference Mission. .................................................................................................................................... 256
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Figure G.2: Normalized Component Power versus Normalized Component Number for Each of Six Reference Missions. ............................................................................... 257 Figure G.3: Mathematical Models for Each of Four Design Cases................................ 258 Figure G.4: Normalization Values for 18 Components for Each of Four Design Cases.261 Figure H.1: Hot- and Cold-Case Orbit Illustrations........................................................ 267 Figure H.2: Hot- and Cold-Case Total Heat Flux Over One Orbital Period for Each of Six Surfaces. ..................................................................................................................... 268 Figure H.3: Hot- and Cold-Case Direct Solar Heat Flux Over One Orbital Period for Each of Six Surfaces. .......................................................................................................... 268 Figure H.4: Hot- and Cold-Case Albedo Heat Flux Over One Orbital Period for Each of Six Surfaces................................................................................................................ 269 Figure H.5: Hot- and Cold-Case OLR Heat Flux Over One Orbital Period for Each of Six Surfaces. ..................................................................................................................... 269 Figure J.1: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus Each of Eleven Main Effects. ....................................................... 292 Figure J.2: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus ORBIT for Each of Ten Interaction Effects.................................. 293 Figure J.3: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus TOT_PWR for Each of Ten Interaction Effects. .......................... 294 Figure J.4: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus C_DIM for Each of Ten Interaction Effects. ................................ 295 Figure J.5: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus C_I_CND for Each of Ten Interaction Effects. ............................ 296 Figure J.6: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus F_T_CND for Each of Ten Interaction Effects. ........................... 297 Figure J.7: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus HT_PIPE for Each of Ten Interaction Effects. ............................. 298 Figure J.8: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus P2P_CND for Each of Ten Interaction Effects............................. 299
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Figure J.9: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus EXT_ABS for Each of Ten Interaction Effects. ........................... 300 Figure J.10: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus EXT_EMS for Each of Ten Interaction Effects. ..... 301 Figure J.11: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus GLBL_DIS for Each of Ten Interaction Effects. .... 302 Figure J.12: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus LCL_PLC for Each of Ten Interaction Effects. ...... 303 Figure K.1: Cold-Case Component Temperatures (18 of 36 components) versus Second Orbit Orbital Time for Increasing Magnitudes of Thermal Bus Conductivity Values (C_I_CND = F_T_CND = P2P_CND = x)................................................................ 304 Figure K.2: Hot-Case Component Temperatures (18 of 36 components) versus Second Orbit Orbital Time for Increasing Magnitudes of Thermal Bus Conductivity Values (C_I_CND (W/m2-K) = F_T_CND (W/m-K) = P2P_CND (W/K) = x). .................. 305
xxx
NOMENCLATURE
Acronyms AD
= Anderson-Darling statistic
ADCS
= Attitude determination and control subsystem
AFRL
= Air Force Research Laboratory
AFRL/RV
= Air Force Research Laboratory Space Vehicles Directorate
AI&T
= Assembly, integration and test
ANN
= Artificial neural network
APG
= Annealed pyrolytic graphite
AU
= Astronomical unit
BFT
= Blue force tracking
CAD
= Computer aided design
CDHS
= Command and data handling subsystem
COTS
= Commercial off-the-shelf
CS
= Computer simulation
DARPA
= Defense Advanced Research Projects Agency
DOD
= Department of Defense
DOE
= Design of experiments
EO
= Electrical-optical
EPS
= Electrical power subsystem
ERBE
= Earth radiation budget experiment
FACTS
= Forced air convection thermal switch
FEBSS
= Flexible, extensible bus for small satellites
GA
= Genetic algorithm
HEO
= Highly elliptical orbit
xxxi I&T
= Integration and testing
IR
= Infrared
ISET
= Integrated systems engineering team
ISR
= Intelligence, surveillance, and reconnaissance
ITEMS
= Integrated thermal energy management system
JPL
= Jet Propulsion Laboratory
LEO
= Low Earth orbit
MCM
= Multi-chip module
MDO
= Multi-disciplinary optimization
MOD
= Mean equinox of date
NASA
= National Aeronautics and Space Administration
NRL
= Naval Research Laboratory
ODCS
= Orbit determination and control subsystem
OLR
= Outgoing longwave radiation
ORS
= Operationally Responsive Space
PCB
= Printed circuit board
PERM
= Pruned-Enriched-Rosenbluth Method
PnP
= Plug-and-play
PnS
= Plug-and-sense
PSE
= Pseudo-standard random error
QD
= Quadrisection
RAAN
= Right ascension of the ascending node
RO
= Reduced-order
RS
= Responsive Space
RSATS
= Responsive Space advanced technology study
SAR
= Synthetic aperture radar
SCOUT
= Small, smart, spacecraft for observation and utility tasks
SMARTS
= Satellite modular and reconfigurable thermal system
SMS
= Structure and mechanisms subsystem
SPA
= Space Plug-and-play avionics
xxxii TCS
= Thermal control subsystem
TherMMS
= Thermal management for modular satellites
TOA
= Top of atmosphere
TSP
= Traveling salesman problem
TTCS
= Telemetry, tracking, and command subsystem
USAF
= United States Air Force
UT
= Universal time
Symbols
A
= Area
a
= Semimajor axis
c
= Step size
c
= Contrast value
c (θ )
= Albedo solar zenith angle correction coefficient
c
= Average correction term
cos (θ )
= Cosine solar zenith angle
cos (θ )
= Average cosine solar zenith angle
C_DIM
= Component side dimension
C_I_CND
= Component interface heat transfer coefficient
d
= Days from J2000.0
D
= Day
E ′′
= Earth emitted irradiation flux
end max
= Maximum generations for convergence
EXT_ABS
= Surface solar absorptivity
EXT_EMS
= Surface longwave emissivity
f
= Eclipse fraction
f
= Number of faces
fit
= Fitness function
xxxiii fitmax
= Maximum fitness function
fitnorm
= Normalized fitness function
fitrel ,s
= Relative fitness function
Fs − E
= View factor from spherical satellite to spherical Earth
Fg
= Force due to gravity
F_T_CND
= Facesheet material transverse thermal conductivity
g
= Mean anomaly
g
= Generation number
G
= Universal gravitational constant
GLBL_DIS
= Global component distribution factor
h
= Circular orbit altitude
HT_PIPE
= Heat pipe factor
i
= Inclination of orbit
JD
= Julian Day
k
= Number of factor effects
K
= Orbital averaged albedo flux correction factor
L
= Mean longitude of the Sun
LCL_PLC
= Local component placement factor
m
= Mass of satellite
m
= Linear multiplier for number of components
m
= Number of contrast values
M
= Month
M⊕
= Mass of Earth
N
= Total number of components
n
= Mean motion or average angular velocity of a satellite
n
= Number of components on a face
n
= Number of training data locations
ORBIT
= Orbit type factor
xxxiv ORBIT
ORBIT
+
−
= Orbit type factor mean response at high level = Orbit type factor mean response at low level
P
= Orbital period
P
= Orbital-averaged power
Pb
= Elitist percentage
Pbf
= Best-fit heuristic percentage
Pm
= Mutation percentage
Pr
= Reproduction percentage
Pwf
= Worst-fit heuristic percentage
popmax
= Maximum population number
P2P_CND
= Panel-to-panel thermal conductance
q′′
= Orbital averaged extreme external environmental heat load
′′ qalb
= Orbital averaged albedo flux
′′ qenv
= Averaged heat load from environment
q′′gain
= Average heat flux consisting of environmental and component loads
′′ qOLR
= Orbital averaged outgoing longwave radiation flux
′′ qsol
= Orbital averaged direct solar flux
r, r
= Orbital radius
r
= Acceleration
RE
= Earth’s radius (6378 km)
R⊕−
= Distance from the Earth to Sun
R ( X, θˆ )
= Correlation matrix
S ′′
= Solar irradiation
So′′
= Solar constant (1367 W/m2)
t
= Time
tLenth , j
= Lenth statistic
xxxv Tiso_max
= Maximum second-orbit temperatures reach by an equivalent isothermal structure
Tiso_min
= Minimum second-orbit temperatures reach by an equivalent isothermal structure
Tmax
= Maximum second-orbit temperature reached by any component
Tmin
= Minimum second-orbit temperature reached by any component
Tmaxd
= Maximum temperature difference reached between any two component at any second-orbit time
Tmaxd_p
= Maximum temperature difference reached between any two components on the same panel at any second-orbit time
TOT_PWR
= Total component power factor
TOT _ PWR + = High-level of total component power factor TOT _ PWR − = Low-level of total component power factor
V , Vcir
= Satellite velocity
w, h
= Width and height of domain
xi
= Training data at ith location
x*
= Non-sampled point
X
= Training data matrix
yi
= Training data response at ith location
yˆ(x* )
= Prediction model response at non-sample point
Y
= Training data response matrix
Y
= Year
z (xi )
= Gaussian Process
Greek Symbols
α
= Surface solar absorptivity
β
= Beta angle
β*
= Cutoff beta angle used to determine eclipse fraction
γ
= Reference direction in mean equinox of date system
xxxvi
δ
= Declination of the Sun
ε
= Obliquity of ecliptic
ε
= Surface longwave emissivity
ε
= Specific mechanical energy
θ
= Solar zenith angle
λ
= Ecliptic longitude
μ
= Earth’s gravitational constant (398,600.5 km3/s2)
μ
= Mean value used in Gaussian Process regression model
μˆ
= Estimate of mean value used in Gaussian Process regression model
ν
= True anomaly of orbit
ρ alb
= Albedo fraction
ρ alb
= Average albedo fraction
ρ alb (0°)
= Solar zenith angle dependent albedo fraction
ρ alb (θ )
= Solar zenith angle dependent albedo fraction
σ
= Stefan-Boltzmann constant
ϕ
= Angular orbital position (zero at the most-nearly sub-solar point)
Ω
= Right ascension of the ascending node
Ωo
= Right ascension of the ascending node at initial time
Ω sun
= Right ascension of the sun
Ω J2
= Orbital variation of RAAN (deg/day)
Superscripts/Subscripts hot , cold
= Hot- cold-case
i
= Face index
j
= Component index
s
= Chromosome solution index
xxxvii
ABSTRACT
Hengeveld, Derek William, Ph.D., Purdue University, December 2010. Development of a System Design Methodology for Robust Thermal Control Subsystems to Support Responsive Space. Major Professors: Dr. James E. Braun and Dr. Eckhard A. Groll, School of Mechanical Engineering. This dissertation presents design and implementation tools supporting the development of Responsive Space thermal control subsystems. An extended literature review revealed topics requiring immediate attention and others providing promising contributions. Consequently, several diverse works were carried out. Single hot- and cold-case design orbits that work well in the design of Responsive Space thermal control subsystems over a wide range of satellite surface properties and expected operating environments were identified. While hot- and cold-case design orbits for traditional missions are established from a well-defined spacecraft attitude and orientation, robust thermal control subsystem bounding orbit conditions must be based on a broad range of potential orbit definitions. From the analysis, a hot-case design orbit was found ( β = 72°, i = 52°, altitude = 350 km) which provided orbital averaged heat fluxes within 0.5% of the values determined using design orbits specific to individual surface types. A general cold-case design orbit was found ( β = 0°, i = 28°, altitude = 1000 km) which provided orbital averaged heat fluxes within 9.0% of the values determined using design orbits specific to individual surface types. If more accuracy is required, two additional cold-case design orbits were identified. For spacecraft with predominantly solar reflector surfaces, a cold-case design orbit ( β = 0°, i = 74°, altitude = 1000 km) provides minimum orbital averaged heat fluxes. For all other surface types, a cold-case design orbit ( β = 0°, i = 28°, altitude = 350 km) provided orbital averaged heat fluxes
xxxviii within 3.4% of the values determined using design orbits specific to individual surface types. A computational tool was developed which improves satellite thermal performance through intelligent component placement. Based on orbital-averaged electronic component power and environmental heat fluxes, the tool provides the optimal satellite face and location within that face for all components such that thermal gradients across the satellite are minimized. Results are determined using a two-step process. First, a global distribution optimization strategy was applied to determine the specific satellite face with which a component should be mated. After all component-face combinations are determined, a local placement algorithm was applied to determine specific component locations within each face. Both steps were based on genetic-algorithms that utilize a combination of elitist strategies, reproduction, local gradient searches, mutation, and bestand worst-fit heuristics. Tuning studies were carried out on the genetic-algorithms to determine appropriate convergence criteria, population size, and values for evolution parameters. The global distribution optimization strategy determines component-face pairs for multi-sided satellites, which achieve a more balanced distribution of heat flux. This depends upon panel orientation and the external thermal environmental fluxes placed upon them. Results showed that a strictly rule-based approach would provide the least computational expense and provide reasonable results for the cases considered here. However, this approach might not be suitable for cases not considered. Consequently, an approach that takes full advantage of the capabilities of the algorithm is recommended. However, this approach increases computational expense. Computational requirements, on average, are expected to be approximately 12 seconds or less for up to 54 components using a 2.5 gigahertz dual-core computer. The maximum computational requirements are expected to be less than approximately 22 seconds for up to 54 components using the same machine. Optimized, even, and worst-case distributions for a nominal distribution of 36 components in the hot-case orbit were found for total power of 100 W up to 1200 W. On average, optimized component distributions reduced maximum temperatures, increased minimum temperatures, and reduced maximum temperature
xxxix differences by 5.4 K, 7.1 K, and 12.6 K, respectively, over evenly distributed components. The largest and smallest maximum temperature difference reductions were 17.8 K (at 400 W) and 9.5 K (at 100 W), respectively. The second step provides rapidly optimized component placement on a given panel approaching a uniform heat flux distribution. Optimized results were obtained for 18 uniform and 11 non-uniform components within 20 s and 7 s, respectively, using a 2.5 gigahertz dual-core processor. Advantages of this method include no need for thermophysical properties and boundary conditions. Optimized results are obtained using only component averaged power and domain size. Consequently, this approach is ideally suited to situations where limited information is readily available. In addition, limiting the required inputs provides for relatively fast solutions. However, care should be taken to ensure that a uniform distribution of fluxes is required for optimized placement. This robust and fast approach can be utilized in a variety of applications including microelectronics and satellite development and is especially suited to those demanding low computational expense. Reduced-order models to predict satellite temperature responses for an 11-factor computer simulation model were developed. These surrogate models were shown to provide acceptable results over more computationally expensive computer simulations. Consequently, these models are ideally suited for rapid evaluation of a wide-range of satellite thermal control subsystem design approaches. Further, these models do not require users to have an extensive background in thermal modeling and/or access to expensive thermal simulation software. In total, eight reduced-order models were developed (one for each level of the categorical variables) and based computer simulations run at evaluation points determined using Latin Hypercube sampling. A Gaussian Process regression model approach was used to fit the subsequent data. The eight reduced-order models were tested using 100 randomly generated test cases. The reduced-order model results were compared to computer simulation results by subtracting the two results at all test points. Residuals, found for maximum temperature, minimum temperature, and maximum temperature difference over all components, had means of 0.1448 K, 0.06414 K, and 0.08643 K, respectively. Additionally, the maximum
xl temperature, minimum temperature, and maximum temperature difference responses had standard deviations of 1.547 K, 1.077 K, and 1.518 K, respectively.
1
CHAPTER 1 - INTRODUCTION
Space exploitation provides tremendous opportunities. Since 1957, satellites have been developed to take advantage of this new high ground by providing communication, scientific observation, weather monitoring, navigation, remote sensing, surveillance, and data-relay services (Gilmore, 2002). However, space presents extraordinary challenges. Consequently, satellites have become exceedingly complex and costly. Current satellites can take from 3 to 7 years to deploy (Table 1.1) and cost from millions to billions of dollars (Williams and Palo, 2006a). Table 1.1: Design and Development Schedule Requirements for Varying Spacecraft Complexities (Saleh and Dubos, 2007). Assemble, Test, and
Total Design and
Launch Operations Development Schedule [months]
[months]
Low Complexity Spacecraft
12
40 to 45
Medium Complexity Spacecraft
15
48 to 55
High Complexity Spacecraft
22
78 to 83
Increased complexity and cost can be traced to the growing importance and expectations of space. In addition, high launch costs push manufacturers to extend design life to reduce life-cycle costs (Noel, Excorpizo, and Jones, 2004). However, extended design life is accomplished through added redundancy that in turn increases complexity and cost. Until recently, affordable space exploitation has remained elusive. While current costs to place a kilogram of capability on orbit remains expensive, modern advances have significantly increased the capability resident in every kilogram thus opening the door for small satellites (i.e. < 500 kg) (Cebrowski and Raymond, 2005).
2 Matured in the late 1980s, small satellites have the capability to increase launch rates, increase space-based solutions to common terrestrial problems, and aid in developing a more robust aerospace industry (Noel, Excorpizo, and Jones, 2004). These advantages are realized through cost savings resulting from decreased mass (both bus and launch vehicle), reduced complexity, limited redundancy, utilization of existing and offthe-shelf technology, and shortened design schedules. In addition, reduced on-orbit lifetimes have a side effect of making technology insertion much easier than for larger programs (Bearden, 2001). Finally, small satellites spread risk. Whereas loss of a large complex system presents a potentially disastrous situation, loss of a small satellite does not since they are easier to replenish. More flexible and even cheaper satellites can be realized through robust design approaches. Robust satellites are designed to meet a broad range of mission requirements; consequently, they drastically reduce non-recurring engineering costs and greatly diminish design, development, and Assembly, Integration, and Test (AI&T) schedules (Table 1.2). Table 1.2: Comparison of Traditional, Small, and Robust Satellite Design Approaches. Design Approach
Mission
System
Risk
Flexibility Performance Tolerance
Development Cost Time
System Focus
Traditional/Military
Low
High
Low
High
High Performance
Traditional/Commercial
Low
High
Low
High
High
Profit
Traditional/Experimental
Low
High
Mid
Mid
High
Science
Small
Mid
Low
Mid
Mid
Mid
Cost
Robust
High
Mid
Mid
Low
Low1
Time
1
High initial cost for development units with low cost for subsequent (i.e. production) units. This table illustrates advantages of robust design above traditional and small satellite
approaches; the primary being reduced development time. Responsive Space (RS) is a modern example of small, robust satellites that challenge the boundaries of schedule limits while keeping costs low.
3
CHAPTER 2 - RESPONSIVE SPACE
The United States is more dependent on space assets than any other nation (Rumsfeld, 2001) especially for military applications such as surveillance, communication, navigation, meteorology, theatre support, and force application (Williams and Palo, 2006a). Although these assets currently provide an asymmetric advantage and has thrust the United States as the clear leader in the use of space, this is not guaranteed in the future (Cebrowski and Raymond, 2005). To reduce development time and cost while maintaining the United States’ military space authority, a Responsive Space (RS) vision has been initiated to revolutionize the methods in which satellites are designed, developed, and put into use. The objective of RS is to quickly deliver low cost, short-term tactical capabilities to address unmet warfighter and intelligence needs (U.S. GAO, 2008; Wegner, 2006). Achieving this will require a considerable departure from DOD’s traditional large space acquisition approach. Wegner et al. (2005) and Arritt et al. (2007) highlight specific objectives including satellites taskable by theatre commanders and data tasking using existing infrastructure. Three noteworthy pillars of this effort will have significant implications in the design of RS architectures. First, RS architectures must be designed with robustness to handle a wide range of conditions (i.e. components, technologies, environments) as opposed to traditional design, which is intended for a specific set of requirements. Second, RS will provide tactical space support in less than six days. In view of current approaches which take years (Table 1.1), this presents a major change. Finally, RS strives to keep mission life-cycle costs low. The Operationally Responsive Space (ORS) office has been directed to strive for satellite costs under $40 million and launch costs under $20 million (Arritt et al., 2008). This is a significant deviation from current average AFRL small satellite costs of $87 million and launch costs of $21 to $28 million for the
4 Minotaur launch vehicle and $65 million for the Evolved Expendable Launch Vehicle (U.S. GAO, 2008). RS will be a significant challenge, but the potential impact is vast. RS will increase the security of the United States and its allies through enhanced capabilities. An Integrated Systems Engineering Team (ISET) comprised of industry, government and academia representatives was established to define the range of missions needed for RS activities. These include: 1) ISR missions such as Electrical-Optical (EO), synthetic aperture radar (SAR), hyper-spectral imagery, signals intelligence, and blue force tracking; 2) communication; 3) weather sensing; 4) navigation; and 5) space superiority (ORSBS-001, 2007 and Summers, 2005). These missions lend themselves to two natural orbit regimes: low-Earth orbit (LEO) for missions such as EO and SAR where proximity of the sensor to the target is preferred and highly-elliptical orbit (HEO) for missions such as blue force tracking (BFT) and communications where long dwell is more important than resolution (ORSBS-001, 2007). In addition, RS has the potential to provide disaster relief, increase monitoring of earthquake activity and provide a teaching tool for Universities (Wegner et al., 2005). Further, it would most certainly provide low-cost, reliable spacecraft and launch vehicles, improving access to space. Currently, less than 20 percent of the Department of Defense (DOD) payloads make it into space (Cebrowski and Raymond, 2005). In addition, DOD’s Space Test Program, which is designed to help the science and technology community find opportunities to test in space relatively cost-effectively, has only been able to launch an average of seven experiments annually in the past four years (U.S. GAO, 2006). RS could improve these statistics. The National Security Presidential Directive NSPD-40 signed in 2004 states that the current RS program would show initial capability before 2010 (NSPD-40, 2005). The primary organization charged with tackling these problems is the ORS program office and supported by, among others, the AFRL Space Vehicles Directorate (AFRL/RV) at Kirtland Air Force Base. AFRL/RV has made RS one of its core thrusts and is developing and demonstrating spacecraft to meet those challenges (Wegner and Kiziah, 2006). To date, several programmatic attributes have been generated which are a significant departure from traditional approaches (Table 2.1).
5 Table 2.1: Responsive Space Attributes a(Arritt et al., 2008), b(Saleh and Dubos, 2007), c (U.S. GAO, 2008), d(Wegner and Kiziah, 2006). Attribute
Traditional
RS
Approach
Approach c
Spacecraft Cost
Average of $87 million
Less than $40 milliona
Launch Costs
$21 to $65 millionc
Less than $20 milliona
Architecture
Varies
Modular / Plug-and-playd
Development Schedule
40 to 83 monthsb
Within 6 days of call upd
On-orbit checkout
Days to months
Less than 4 hoursd
Mission Lifetime
Years
Approximately one yeard
Mission Type
Specific
Tailoredd
In addition to overall cost and development limits, a recent study was completed by the ISET classifying RS missions including mission type, spacecraft mass and power limits. Mission requirements were separated into three regimes: LEO, HEO, and space superiority missions (Table 2.2). Table 2.2: Responsive Space Mission Performance Requirements (ORSBS-001, 2007). Performance
LEO
HEO
Space
Requirement
Missions
Missions
Superiority Missions
Altitude
350 to 705 km 7800 x 525 km 400 to 800 km
Inclination
0 to 98.7°
63.4°
0 to 98.7°
Maximum Spacecraft Bus Mass
200 kg
200 kg
350 kg
Maximum Payload Mass
200 kg
200 kg
200 kg
Bus Orbital Average Power
220 W
210 W
220 W
Payload Orbital Average Power
250 W
250 W
250 W
Payload Peak Power
700 W
500 W
700 W
Pointing
Nadir
Nadir
Nadir
LEO missions will range from equatorial to Sun synchronous orbits. Also, altitude requirements were impacted by drag requirements. Drag at low altitudes must be overcome by propulsion lending to a minimum altitude of 350 km. To meet 25-year
6 atmospheric drag re-entry requirements, a maximum altitude of 705 km is required (ORSBS-001, 2007). The reference HEO mission is a 63.4° inclination elliptical orbit (i.e. 7800 x 525 km) to allow for long dwell times necessary for communications missions. Finally, a space superiority mission was developed to allow for instances where more capability will be required in a LEO environment. To achieve RS, all aspects must be reconsidered including: on-demand launch vehicles, responsive buses, and responsive payloads. Several efforts are underway in both the public and private sectors to develop new launch vehicles capable of placing satellites in LEO. Foust and Smith (2004) provide an exhaustive review of these efforts including those of SpaceX and their Falcon 1. This unit is capable of launching roughly 450 kg into LEO at a cost of $7.9 million (i.e. $16,667/kg) (SpaceX, 2008). To date, five Falcon 1 launches have been attempted with two successes. Responsive buses and payloads could be approached by rapidly maneuvering onorbit pre-deployed assets or rapidly reconfiguring on-orbit assets for new and tailored missions. Although these techniques have their own intrinsic merit, a more robust solution is focused on rapidly launching and operating satellites. To meet this challenge, the methodologies used to design, manufacture and test buses and payloads must radically change (Williams and Palo, 2006a; Lawlor, 2006). Several examples of RS enabling bus architectures were found in the literature (Table 2.3) and are summarized in Appendix A. Each of these satellites attempt to meet the needs of RS. Although none has achieved all RS objectives each has improved various aspects through technology developments. RS success will be based in part on three driving principles: modularity, standardization, and acceptance of risk (Jones, 2005). Modular satellite bus architectures are of great interest as an enabling approach for meeting the cost, time, and performance needs of RS (Young, 2008). This will require a shift from custom built to modular plugand-play (PnP) architectures (Lawlor, 2006). A modular PnP bus is assembled from modular components with standard interfaces and minimal interdependencies between modules. It uses open-standards and interfaces, self-describing components, and an autoconfiguring system (The Aerospace Corporation, 2006). Although in the early
7 developmental states extensive system integration and testing is required, system integration is simple and testing tasks are automated. This will lend itself to a flexible satellite bus upon which component types (i.e. powers) and configurations (i.e. location) are variable. Table 2.3: Responsive Space Satellite Architectures a(U.S. GAO, 2008), b(Fronterhouse, Lyke, and Achramowicz, 2007), c(Hicks, Enoch, and Capots, 2005), d(Hicks, Hashemi, and Capots, 2006), e(McDermott and Jordan, 2005), f(Freeman, Rudder, and Thomas, 2000), g(MightySat II DataSheet, 2005), h(Ince, 2005), i(CubeSat Design Specification, 2008). Satellite
Developer
Launch
Development
Year
Time
Cost
TacSat 1a
NRL
Not launched
1 year
$23 million
TacSat 2a
AFRL
2006
29 months
$39 million
a
AFRL
2008
Not available
$62.7 million
a
NRL
2009
Not available
$114 million
a
Several
Not available
Not available
Not available
b
AFRL
Not available
Not available
Not available
Lockheed Martin
Not launched
Not available
Not available
AeroAstro
Not available
Not available
Not available
USAF
2000
15 months (I&T)
$36.4 million
Varies
Varies
Up to two years
Not available
TacSat 3 TacSat 4 TacSat 5 PnP Sat
HexPakc,d SMARTBus MightySat
e
f,g
CubeSath,i
Standardization is crucial in building modular satellite bus architectures more quickly and at a lower cost (U.S. GAO, 2008; Young, 2008). Design_Net Engineering, in the development of the Floating Potential Probe satellite, also found that standards and PnP are a key component in rapid development and integration (Murphy, 2005). A programmatic effort is being led by the Naval Research Laboratory and the John Hopkins University Applied Physics Laboratory to develop RS bus standards (U.S. GAO, 2008). From this, the ISET developed a set of bus standards by which RS spacecraft can be manufactured and tested within a given performance envelope (ORSBS-002, 2007). Flagg, White, and Ewart (2007) provide a review of current RS standards. One example of RS standardization is Space PnP Avionics (SPA). SPA systems are a more ‘internet-
8 like’ method of communication between sensors or actuators and are reconfigurable, providing an easily expandable network that is robust to failures (Lyke et al., 2005). In addition, the SpaceWire standard (ECSS-E-50-12A) has been specified as part of such a payload-bus interface for high rate data (Jaffe, Clifford and Summers, 2008). The final driving principle for the success of RS is an underlying acceptance of risk. In effect, this emphasizes RS satellites that are good enough. As a result, RS will utilize ‘state-of-the-industry’ as opposed to ‘state-of-the-art’ technologies in order to keep costs low (Raymond et al., 2005). Although systems need only be good enough to align with the philosophy of RS, they must be robust enough to meet a considerably larger design space over traditional optimized systems. A significant evolution in satellite bus design and development is essential. Traditional thermal control architectures are not well suited to modular spacecraft and, as a result, a robust TCS capable of operating under a variety of scenarios is imperative (Young, 2008). Development of a robust TCS able to meet the challenges of RS will prove to be one of the most significant obstacles. Although it can rely on relaxed design methodologies to align with the philosophy of RS, it must be robust enough to meet a considerably larger design space over traditional optimized systems.
9
CHAPTER 3 - ROBUST THERMAL CONTROL SUBSYSTEMS (TCS) FOR RESPONSIVE SPACE
Traditional thermal control subsystem design methodologies provide highly optimized and capable systems that are strongly influenced by a specific set of mission requirements. These include spacecraft configuration (i.e. component and payload selection along with their placement and orientation) and orbit definition, which defines the external thermal environment. The sum of these requirements in turn dictates TCS design (Figure 3.1). Traditional, risk averse TCS are designed to accommodate a single combination of these design parameters and are not intended for ‘off-design’ conditions. Traditional TCS designs are intended for one and only one mission. Although this highly engineered and optimized point-design readily achieves its intent, this methodology will not work for RS. Specifically, robust TCS must be developed. Robust TCS design must be drastically different from traditional approaches for three reasons. First, it will be intended for not just one, but a broad range of missions. The LEO mission ranges from Table 2.2 are a prime example. Robust RS TCS must handle missions from 350 km to 705 km and 0 to 98.7° inclinations. Therefore, developing a robust TCS for RS is imperative (Young et al., 2008; Birur and O’Donnell, 2001). In addition to orbit and therefore thermal environment variations, it must be able to handle variations in the number, power distribution, and placement of components, and allow for rapid integration of new technologies. Second, since RS missions are based on quick response, not enough time is available to appropriately design, model, and test in a traditional manner. Long term programmatic goals have placed this at six days (Table 2.1). Consequently, robust RS TCS must be designed and developed long before mission call-up (Figure 3.2). The accelerated design process provides significant problems to the RS thermal engineer due to the relatively
10 large and not clearly defined design space created by combinations of payloads, bus components and orbits.
Figure 3.1: Traditional Thermal Control Subsystem Design Procedure. Finally, robust RS TCS should be designed to minimize programmatic costs. Cost constraints will prevent the use of some approaches such as developing and filling a warehouse of RS satellites to fill all mission needs, but will lead to flexible thermal control design concepts that enable faster and less expensive design cycles (Birur and O’Donnell, 2001). Several examples of robust RS TCS design and development efforts were found in the literature and are summarized in Appendix B. These include design approaches including TherMMS, SMARTS, FACTS and development efforts including
11 HexPak, SMARTBus, and ITEMS. All of these concepts attempt to meet the needs through modularization and standardization all while accepting risk.
Figure 3.2: Robust Thermal Control Subsystem Design Procedure. A robust RS TCS approach is a drastic departure from traditional point design approaches. Young (2008) evaluated how best to accomplish this. His analysis revealed that an isothermal bus architecture with thermal balance modulation on the heat rejection side will enable full modularity. This work is a significant step forward in developing a robust RS TCS but stops short of defining performance requirements and the resulting design envelope. Standardized requirements are needed to fill this void.
12 There are no officially designated thermal requirements for an RS thermal control system and very little thermal work had been done on the topic (Bugby, Zimbeck, and Kroliczek, 2008) prior to this thesis work. The ISET has initiated a standardization effort for RS but few thermal control standards have been developed. Those that exist include launch and testing requirements, spacecraft control by turning heaters on/off, and delivery of a geometric and thermal math model of the spacecraft bus (ORSBS-002, 2007). From the literature, it is apparent that developing standard RS TCS requirements is an area that needs significant development. For example, hot- and cold-case design orbits are selected based on experience and intuition leading to vastly different design values and therefore thermal environments between development programs. Consequently, there is a need to develop a uniform set of design requirements for RS TCS including well-defined hot- and cold-case design orbits. Within the robust thermal control subsystem design procedure (Figure 3.2), mission objectives/concepts dictate a required set of electronic components that must be placed on the satellite. Traditionally, components are placed throughout a satellite without concern for thermal implications. Any thermal problems that result are dealt with far into the design process and can be addressed through rearrangement of components and/or modification of the TCS. Due to the shortened timelines, rearrangement and modification will not be allowed for RS. Consequently, RS TCS designs must be robust enough to handle the wide range of component placements that are expected. As an alternative, optimization algorithms could be developed to thermally improve component placement. These tools currently do not exist. Consequently, there is a need to develop computationally inexpensive optimization tools that would reduce robust RS TCS requirements and/or provide additional margin to existing designs. In addition, defining the requirements for attaining isothermal bus architectures with thermal balance modulation on the heat rejection side is needed. An isothermal bus architecture is not realistic but it is practical to consider a design approaching that performance. A quasi-isothermal bus architecture could still provide an RS TCS but requirements and standardization must be developed. It is not clear what should characterize a quasi-isothermal bus and what these implications have on the performance
13 and design envelope of the satellite. For example, is a 10 K temperature difference across the satellite good enough or would a more or less strict standard provide ‘good enough’ results. Also, thermal balance modulation on the heat rejection side is necessary for the RS TCS. Again, the definition of a standard control authority along with the resulting design envelope needs to be addressed.
14
CHAPTER 4 - MOTIVATION, OBJECTIVES, AND APPROACH
Chapters 1 through 3 highlight the overall objectives of RS (Responsive Space) and the challenge of developing a robust TCS (thermal control subsystems). The following describes the current efforts undertaken to further this endeavor. 4.1 Motivation A robust RS TCS approach is a drastic departure from traditional point design approaches. Young (2008) evaluated how best to accomplish this. His analysis revealed that an isothermal bus architecture with thermal balance modulation on the heat rejection side will enable full modularity. This work is a significant step forward in developing a robust RS TCS but stops short of defining performance requirements and the resulting design envelope. There are no officially designated thermal requirements for an RS thermal control system and very little thermal work had been done on the topic (Bugby, Zimbeck, and Kroliczek, 2008). Consequently, there is a need for a uniform set of design requirements for RS TCS including well-defined hot- and cold-case design parameters. In addition, defining the requirements for attaining an isothermal bus architecture with thermal balance modulation on the heat rejection side is needed. Also, thermal balance modulation on the heat rejection side is necessary for the RS TCS. Again, the definition of a standard control authority along with the resulting design envelope needs to be addressed. Finally, RS TCS designs must be robust enough to handle the wide range of component placements that are expected. As an alternative, component locations could be thermally optimized. These optimization tools currently do not exist. Consequently, there is a need to develop computationally inexpensive optimization tools that would reduce robust RS TCS requirements and/or provide additional margin to existing designs.
15 4.2 Objective The objective of the current work was to establish standards, design requirements, and design tools for the design and development of robust thermal control subsystems (TCS) for Responsive Space (RS). The following section describes the approach to accomplish this objective. 4.3 Approach To achieve the objective of the current work, the following tasks were completed. TASK 1: LITERATURE REVIEW - A thorough literature review was done to investigate the current state of RS, RS thermal control subsystem work, and thermal control subsystem technologies among others. The results of this work are summarized in this document and provided the framework for all ensuing analyses. TASK 2: HOT- AND COLD-CASE DESIGN ORBIT DETERMINATION - Orbit definition plays a significant role since thermal boundary conditions are established from spacecraft attitude and orientation and provide the bounding hot and cold cases (McMordie, 2003; Karam, 1998). Of these, bounding orbit definitions provide an immediate challenge, due to the large number of potential orbit definitions. Previous works have utilized design orbit cases for evaluating RS TCS, but these were based on experience and intuition not analysis. As a result, resulting design values have been significantly different between programs. Therefore, single hot and cold case design orbits should be identified that work well in the design of TCS for RS missions over a wide range of satellite surface properties and expected operating orbits. Hot- and cold-case design orbits will ensure that the most extreme thermal environments expected to be encountered will be applied to potential RS TCS. In addition, these will provide a standard set of circumstances with which all other work can be designed. An approach using both statistical and historical data was used
16 to select hot and cold case design orbits appropriate for RS. The results of this work are summarized in CHAPTER 5. TASK 3: OPTIMAL DISTRIBUTION OF ELECTRONIC COMPONENTS TO BALANCE ENVIRONMENTAL FLUXES - The isothermal bus concept could be approached by simply optimizing electronic component placement. The RS effort provides an ideal proving ground to attempt this given the Plug-and-play (PnP) and therefore, flexible nature of component placement. To achieve this, an algorithm is required to optimally distribute components among several distinct satellite panels. Each panel is characterized not only by its shape, but also by environmental fluxes incident upon it. The algorithm should be robust enough to handle any number and power distribution of components, most satellite geometries, and any possible orbit. Genetic algorithms in conjunction with bin-packing method approaches were used as the backbone for this effort. The results of this work are summarized in CHAPTER 6. TASK 4: OPTIMAL PLACEMENT OF ELECTRONIC COMPONENTS TO MINIMIZE HEAT FLUX NON-UNIFORMITIES - After initial component distribution, the isothermal bus concept is further approached by simply optimizing component placement in order to evenly ‘spreadout’ fluxes. To achieve this, an optimization algorithm is required to optimally place components over a given panel in order to minimize heat flux non-uniformities. The algorithm should be robust enough to handle any number and power distribution of components along with most panel geometries. Genetic algorithms in conjunction with packing methods were used. The results of this work are summarized in CHAPTER 7. TASK 5: DEVELOPMENT OF REDUCED ORDER MODELS - The literature has shown that an isothermal bus (i.e. zero temperature gradients across the
17 TCS bus) coupled with thermal balance modulation on the heat rejection side can facilitate a robust RS TCS. However, true isothermal conditions are not realistic; only quasi-isothermal conditions can be achieved. Consequently, it is important to determine quasi-isothermal requirements that allow for successful TCS design without being too strict and forcing unreasonable TCS designs. In addition, thermal balance modulation requirements are directly related to TCS design. For example, the range of longwave emissivities required to ensure that a given satellite will meet thermal requirements is directly tied to a given surface solar absorptivity. Therefore, determination of the specific or range of thermal design variables (e.g. effective conductivity, surface optical properties, and optimized placement) necessary in achieving varying levels of quasiisothermal conditions and required control authority is necessary. Reduced order models were developed and tested for to accomplish this task. These models can be utilized as a primary investigation tool and allow for a wide-range of searches and analyses. 4.4 Unique Results of the Work The results of the work presented here yielded the following unique features: RESULT 1:
A standard definition of hot- and cold-case design orbits is not available for RS. To the author’s knowledge, no efforts of this type have ever been undertaken. Single hot- and cold-case design orbits that work well in the design of robust thermal control subsystems over a wide range of satellite surface properties and likely operating environments were determined. The hot- and cold-case design orbits have many potential applications. First, they provide consistent design criteria for the development and comparison of robust thermal control subsystem approaches. In the absence of clearly defined mission criteria, these design orbits also provide a reasonable surrogate. Finally, they are useful for the design of robust spacecraft over a wide
18 range of both satellite surface properties and low Earth orbit operating environments. RESULT 2:
A computational tool was developed which improves satellite thermal performance through intelligent component placement. Based on orbital-averaged electronic component power and environmental heat fluxes, the tool provides the optimal satellite face and location within that face for all components such that thermal gradients across the satellite are minimized. Results are determined using a two-step process. As a first step, a global distribution optimization strategy was developed to determine component-face pairs for multi-sided satellites that achieve a more balanced distribution of heat flux. This depends upon panel orientation and the external thermal environmental fluxes placed upon them. Computational requirements are minimal and optimized results provide a significant improvement over even distribution of power.
RESULT 3:
A local placement algorithm was developed to determine specific component locations within each face for all optimized componentface pairs (RESULT 2). This computational tool provides rapidly optimized component placements approaching a uniform heat flux distribution. Optimized results are obtained with minimal computational expense. Advantages of this method include no need for thermophysical properties and boundary conditions; optimized results are obtained using only component averaged power and domain size. Consequently, this approach is ideally suited to situations where limited information is readily available. In addition, limiting the required inputs provides for relatively fast solutions. This robust and fast approach can be utilized in a variety of applications including
19 microelectronics and satellite development and is especially suited to those demanding low computational expense. RESULT 4:
Reduced-order models to predict satellite temperature responses were developed. These surrogate models were shown to provide acceptable results over more computationally expensive computer simulations. These models are ideally suited for rapid evaluation of a broad-range of satellite thermal control subsystem design approaches. Further, these models do not require users to have an extensive background in thermal modeling and/or access to expensive thermal simulation software.
4.5 Organization of the Document CHAPTER 1 through CHAPTER 3 provides an overview of RS including the objective of this program and design and development of RS TCS. CHAPTER 4 addresses the motivation for this study and lists the specific objectives of the work presented here. CHAPTER 5 presents the entire work on design hot and cold case evaluation and includes key results. CHAPTER 6 details global while CHAPTER 7 investigates local component placement optimization efforts. CHAPTER 8 summarizes the reduced order models that were developed for evaluating items such as quasiisothermal design conditions and thermal modulation ranges. CHAPTER 9 is a summary of this document and includes conclusions and suggestions for future work. The remainder of this document includes references and appendices.
20
CHAPTER 5 - DETERMINATION OF HOT- AND COLD-CASE DESIGN ORBITS FOR ROBUST THERMAL CONTROL SUBSYSTEM DESIGN
Environmental heating plays a significant role in the design and development of spacecraft Thermal Control Subsystems (TCS). Consequently, orbital parameters - which influence direct solar, albedo, and Earth-emitted heat fluxes - must be carefully evaluated over their expected range. Traditionally, extreme heating environments are identified for a particular mission and hot- and cold-case design orbits are established. However, despite consistent data, there is not a commonly accepted methodology in design orbit development (Karam, 1998). Approaches include those where TCS design influences mission concepts (Diaz-Aguado et al., 2006) and others where a given mission dictates specific hot- and cold-case design orbits (Megahed and El-Dib, 2007). Recently, robust spacecraft have gained much attention. Although this approach has significant advantages, it also presents new challenges. While, hot- and cold-case design orbits for traditional missions are established from a well-defined spacecraft attitude and orientation, robust TCS bounding orbit conditions must be based on a broad range of potential orbit definitions. Consequently, determining limiting hot and cold cases in a manner appropriate for robust missions is problematic. Further, the range of methodologies and resulting design orbits that have been used is noteworthy. For example, in optimizing the Brazilian Multimission Platform, TCS hot- and cold- case design orbits were established for each of several mission classes (e.g. equatorial and sun-synchronous) (Muraoka et al., 2006). Several other robust TCS efforts include thermal management for modular satellites (TherMMS) by Young (2008); forced air convection thermal switches (FACTS) by Williams (2005); and satellite modular and reconfigurable thermal system (SMARTS) by Bugby, Zimbeck, and Kroliczek (2008). Although these approaches are intended for the same broad range of missions, they are based on significantly different hot- and cold-case design orbits (Table 5.1). Not only
21 does this lead to potentially inconsistent development efforts but without careful analysis, can result in over- or under-designed systems. Therefore, establishment of a uniform set of hot- and cold-case design orbits utilizing a consistent methodology is necessary in the development of robust TCS. Table 5.1: Review of Hot- and Cold-Case Design Orbits Utilized for Robust TCS Development. Orbit
Orbit
Name
Type
Altitude Inclination [km]
Beta
Solar
Angle
Flux
[degrees] [degrees] [W/m2]
Albedo
OLR
[---]
TherMMS (Young, 2008) Orbit A - Hot
Circular
400.0
60
40
1,414
0.28
260.5 K
Orbit A - Cold Circular
400.0
60
40
1,322
0.18
248.7 K
Orbit B - Hot
Circular
400.0
0
0
1,414
0.26
266.0 K
Orbit B - Cold Circular
400.0
0
0
1,322
0.14
251.8 K
Orbit C - Hot
Circular
800.0
Sun-synch
90
1,414
0.27
260.2 K
Orbit C - Cold Circular
800.0
Sun-synch
90
1,322
0.18
249.0 K
FACTS (Williams, 2005) Hot Case
Circular
200.0
Cold Case
Circular 1,000.0
NA
90
1,414
0.57
275.0 W/m2
NA
0
1,322
0.18
218.0 W/m2
SMARTS (Bugby, Zimbeck, and Kroliczek, 2008) Beta 90°
Circular
185.2
NA
90
1,354
0.35
225.0 W/m2
Beta 0°
Circular
185.2
NA
0
1,354
0.35
225.0 W/m2
NA: This information was not available in the cited work. The primary goal of the study presented in this chapter was to identify single hotand cold-case design orbits that work well in the design of TCS over a wide range of satellite surface properties and expected operating environments. To achieve this objective, a general approach was developed to identify worst case orbits employing a combination of statistical and historical data such that orbits with little potential of being utilized for robust missions are disregarded (Figure 5.1).
22
Figure 5.1: Overview of Analysis Methodology. First, temporal beta angle ( β ) relationships were developed based on traditional methods. From this, β versus inclination angle ( i ) relationships were developed which provided a statistical distribution. These results were combined with historical launch information to generate statistical and subsequently viable weighting matrices. The viable weighting matrices were applied to orbital-averaged thermal environmental models to eliminate statistically insignificant orbits. Using this approach, hot- and cold-case design orbits were found at distinct β and i combinations. For more detailed discussion of applicable variables, refer to Appendix C. 5.1 Development of Weighting Matrices Weighting matrices were developed based on statistical β development and historical i information. These were subsequently applied to orbital-averaged thermal environmental models to eliminate statistically insignificant orbits. This analysis started with a development of the temporal variation of β . 5.1.1 Temporal Variation of β
β is the minimum angle between the solar vector and the orbit plane and can vary from -90° to 90° where the sign describes the direction of orbit as viewed from the Sun.
23 Satellites with positive β appear to be going counterclockwise while negative β indicate a clockwise direction. β is a function of both Sun coordinates and orbital elements as follows (Gilmore, 2002)
β = sin −1 ( cos (δ ) ⋅ sin ( i ) ⋅ sin ( Ω − Ω sun ) + sin (δ ) ⋅ cos ( i ) ) .
(5.1)
Several of the parameters that are used to calculate β change with time. To better understand β , its temporal variation was investigated (Figure 5.2). The analysis begins by providing a method of describing time in orbit from which Sun coordinates and orbital elements are determined.
Figure 5.2: Illustration of β Relationship Over One Year for a Circular Orbit at an i of 0° and a Right Ascension of the Ascending Node of 0°. 5.1.1.1 Julian Day, JD Calendar time is not convenient for computations, especially over long time intervals. The universally adopted time measure for astronomical problems is the JD which is a continuous count of the number of days since Greenwich noon on January 1, 4713 BC (Wertz and Larson, 2005). The following algorithm yields an integer value of JD at noon universal time (UT) given a certain Y , M , and D (Wertz and Larson, 2005; Vallado, 2007)
24 JD = D − 32,075 + 1461 ⋅ (Y + 4800 + ( M − 14) / 12) / 4 +367 ⋅ ( M − 2 − ( M − 14) / 12 ⋅ 12) / 12 −3 ⋅ ((Y + 4900 + ( M − 14) / 12) / 100) / 4
.
(5.2)
The current epoch, or standard moment in time upon which orbital elements are referenced, is J2000.0 that is defined as January 1, 2000. Given the current JD , d was found as follows (Wertz and Larson, 2005) d = JD − 2451545.0 .
(5.3)
This value was used to directly determine the β for any given JD . 5.1.1.2 Sun Coordinates Unless otherwise noted, Sun coordinates were based on a low-precision method which provides the equation of time to a precision of 0m.1 between 1950 and 2050 due to expansion truncation and coordinates of the Sun to 0.01° (Vallado, 2007; Astronomical Almanac, 2007). The coordinate system, mean equinox of date (MOD), is based on the intersection of the ecliptic and equatorial planes (i.e. γ direction) on a given date (Vallado, 2007). The MOD reference frame accounts for motion of the coordinate system as a result of gravitational forces (Astronomical Almanac, 2007). All pertinent Sun coordinate variables are illustrated in Figure 5.3.
Figure 5.3: Illustration of Sun Coordinate Variables.
25 Ω sun is the angular distance measured eastward along Earth’s equator from the equinox to a plane perpendicular to the Earth’s equator and passing through the Sun (i.e. hour circle) (Vallado, 2007; Astronomical Almanac, 2007; Wiesel, 1997) as shown here
Ω sun = λ − (
ε 180 ε ) ⋅ tan 2 ( ) ⋅ sin(2 ⋅ λ ) + ( ) ⋅ (tan 2 ( )) 2 ⋅ sin(4 ⋅ λ ) . π 2 2 ⋅π 2
180
(5.4)
The δ is the angular distance north or south of Earth’s equator measured along the hour circle (Astronomical Almanac, 2007)
δ = sin −1 (sin(ε ) ⋅ sin(λ )) .
(5.5)
The λ is defined as the angular distance measured eastward along the ecliptic (i.e. the mean plane of the Earth’s orbit around the Sun) from the equinox to a plane passing through the poles of the ecliptic and the celestial object
λ = L + 1.915° ⋅ sin( g ) + 0.020° ⋅ sin(2 ⋅ g ) .
(5.6)
The ε is the angle between the planes of the equator and the ecliptic and varies with
JD due to gravitational perturbations (Astronomical Almanac, 2007)
ε = 23°.439 − 0°.0000004 ⋅ d .
(5.7)
Mean motion is a measure of the average angular velocity of an object. Therefore, L , approximates the ecliptic longitude. The g describes the angle from the perihelion
(i.e. the point where Earth is closet to the Sun) of an Earth moving with constant angular speed equal to the mean motion. This approximates the ν . These values were determined from the following and subsequently truncated to put them in the range of 0° to 360° (Astronomical Almanac, 2007) as shown here L = 280.461° + 0.9856474° ⋅ d
(5.8)
g = 357.529° + 0.9856003° ⋅ d .
(5.9)
5.1.1.3 Orbital Elements Orbital i and RAAN are two of the six classical orbital elements which describe an orbit about Earth. The i is the angle between the orbital and equatorial planes shown in
26 Figure 5.4. It can also be thought of as the angle between the angular momentum vector of the orbit and the vector parallel to the Earth’s spin axis. The i will remain constant unless subjected to an external force. These include man-made plane changes and naturally occurring luni-solar forces. Man-made plane changes were not included in this analysis and natural changes have been found to be very small (~1/100° per year) such that the i was assumed constant (Gopinath et al., 2004).
Figure 5.4: Illustration of the Six Classical Orbital Elements. The right ascension of the ascending node (RAAN) is the angle from the vernal equinox to the ascending node (Figure 5.4). The ascending node specifies the point at which the orbit crosses the equatorial plane traveling from south to north. For select i (e.g. 0°), the RAAN can be assumed constant (Figure 5.2). For all other i , changes in RAAN due to orbital perturbations are not negligible and must be included. The equations for Sun coordinates presented in the previous section were developed under the following assumptions: 1) gravity is the only force, 2) a spherically symmetric Earth, 3) the Earth’s mass is much greater than the satellite’s, and 4) the Earth and the satellite are the only two bodies in the system. Gravity is not the only force acting upon a satellite. Additional perturbing forces are present: solar radiation pressure, atmospheric drag, gravitational forces of the Sun and Moon, and forces as a result of a non-spherical Earth. Of these, perturbations due to a non-spherical Earth dominate the other sources; therefore, they were the only type of perturbations used in this study (Wertz and Larson, 2005).
27 The Earth has a bulge at the equator (oblateness) of approximately 22 km as a result of its rotation (Sellers, 2005). Because of this, periodic variations in the orbital elements are present that are dominated by variations in the RAAN and the argument of perigee. Precession due to Earth’s oblateness for circular orbits was approximated by Equation (5.10) and is a function a and i (Wertz and Larson, 2005) as follows:
Ω J 2 ≅ −2.06474 × 1014 ⋅ a
−
7 2
⋅ cos ( i ) .
(5.10)
Utilizing the orbit precession and given a RAAN at the beginning of the simulation, a corresponding RAAN at any subsequent time was calculated as follows: Ω = Ωo + Ω J 2 ⋅ t .
(5.11)
Figure 5.5 through Figure 5.7 illustrate select β variations over a period of one year for inclinations of 0°, 30°, 60°, 90° and 110° at an initial time of March 21, 2007. Figure 5.5 shows results for an altitude of 1000 km and RAAN of 0°.
Figure 5.5: β versus Time for Select Inclinations at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°). Figure 5.6 illustrates the effect of altitude on β history by simulating the results at the
28 lower end of altitudes (i.e. 350 km). Figure 5.7 illustrates the effect of RAAN on β history by applying a similar simulation at a RAAN of 90°.
Figure 5.6: β versus Time for Select Inclinations at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°).
Figure 5.7: β versus Time for Select Inclinations at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°).
29 As shown in Figure 5.5, the β varies continuously throughout the year because of orbital perturbations. For example, at an orbital inclination of 0° or 90°, perturbations caused by Earth’s oblateness are inconsequential and therefore, the resulting variation of
β follows the declination of the Sun over a year. At other inclinations, orbital perturbations play an important part in the resulting change of β versus time. At an altitude of 350 km (Figure 5.6), the effect of Earth’s oblateness becomes more pronounced and as a result, β changes more rapidly with respect to time. As before, orbital inclinations of 0° or 90° show no effect of perturbations. Figure 5.7 illustrates that a shift in initial RAAN causes a shift of the resulting β history for some inclinations. For other inclinations, such as 90°, a significant departure from previous results is shown. 5.1.2 Statistical Beta Angle versus Inclination Distribution For a given inclination, there is a unique probability distribution for the β . Some β may have zero probability at specific inclinations (e.g. inclination of 0° and β of 90°). For each integer inclination angle shown in Figure 5.5 through Figure 5.7, temporal β data was recorded every 0.001 days over one year (i.e. 365,000 data points). Each β was rounded to the nearest integer and placed within one-degree bins. The resulting frequency of occurrence values were normalized by the peak value (Figure 5.8 through Figure 5.10). For a given inclination, a maximum value is apparent for a particular β . For example, Figure 5.10 shows at an inclination of 60°, a maximum occurs at a β of approximately 36°. For this particular inclination, this is the β that occurred most often at an altitude of 350 km and a RAAN of 90°. In addition, the normalized frequency of occurrence becomes zero for combinations of β and inclination that are not possible. These results were expanded to include all inclinations from 0 to 110°. Figure 5.11 through Figure 5.13 provide contour plots of these results.
30
Figure 5.8: Normalized Frequency of Occurrence for β for Select i at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°).
Figure 5.9: Normalized Frequency of Occurrence for β for Select i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°).
31
Figure 5.10: Normalized Frequency of Occurrence for β for Select i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°).
Figure 5.11: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 1000 km (Beginning on March 21st 2007 and RAAN = 0°).
32
Figure 5.12: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 0°).
Figure 5.13: Contour Plot of Normalized Frequency of Occurrence for β for Range of i at an Altitude of 350 km (Beginning on March 21st 2007 and RAAN = 90°). Figure 5.11 through Figure 5.13 indicate a relationship between β and inclination (i.e. distinctive linear red areas). This was further refined by applying the same
33 methodology to a wider range of RAAN and altitude values. In effect, this ‘smoothed’ the resulting distribution. Figure 5.14 provides similar results to those of Figure 5.8 through Figure 5.10 but expands the analysis space to include summed results for integer β from 0 to 90°, select inclinations from 0 to 110°, integer RAAN from 0 to 360° and altitudes from 350 to 1000 km at a step size of 50 km. Figure 5.15 shows the same information but as a contour plot for all inclinations from 0 to 110°. Figure 5.14 and Figure 5.15 indicate that for a given inclination, there is a well defined peak in the normalized frequency of occurrence. Furthermore, there is a range of
β where the frequency of occurrence goes to zero. These figures clearly indicate a relationship of maximum normalized frequency versus β and inclination that follows ≈ ± ( i + 23.4 ) .
Figure 5.14: Normalized Frequency of Occurrence for β for Select i at Altitudes of 350 to 1000 km (Beginning on March 21st 2007 and RAAN of 0° to 360°). The expected distribution of β as a function of inclination presented in Figure 5.15 can be used to statistically predict the potential of encountering a specific β for a given inclination. For example, from Figure 5.15 at an inclination of 90°, the β most often
34 encountered is approximately 67°. It must be noted that this is based on a ‘blended’ distribution with information obtained from a range of altitudes (350 km to 1000 km) and a range of RAANs (0 to 360°) over a period of one year. In addition to capturing information over a broad range of orbital variables, this approach also had a tendency to smooth the resulting distribution. This can readily be verified by examining Figure 5.8 through Figure 5.10 versus Figure 5.14 for select i .
Figure 5.15: Contour Plot of Normalized Frequency of Occurrence for β for i at Altitudes of 350 to 1000 km (Beginning on March 21st 2007 and RAAN of 0° to 360°). A β at a given inclination can be disregarded if the continuous time spent at that combination is significantly less than typical LEO orbital periods of approximately 90 minutes. This would potentially occur for inclinations with relatively high perturbation frequencies (i.e. the number of oscillations over the course of one year). Investigation showed that these worst-case scenarios still yielded times on the order of hours and therefore all feasible combinations of β and inclination must be included in the analysis and cannot be disregarded. For example, at an i = 30° from Figure 5.5, which displays relatively high perturbation frequencies, continuous time spent at a given β of no less than approximately seven hours.
35 5.1.3 Historical Inclination Distribution Historically, satellites in LEO have been launched into a finite number of inclination regions. For example, inclinations around 63.4° will follow a Molniya orbit (Wertz and Larson, 2005) while those between 96.8° and 99.5° are sun-synchronous. Communication satellites such as the 66 Iridium satellites utilize near polar 86.4° inclinations and the 48 Globalstar satellites use 52° inclination orbits. Conversely, some inclinations are seldom, if ever, utilized. To account for variations in application, a probability distribution of inclinations was sought based on historical launch information. Data from 1957 to 2006 (Satcat.txt, 2007) was obtained for 29,493 unclassified orbiting objects (i.e. satellites, debris, and rocket fairings). To eliminate debris and rocket fairing information, this data set was cross-referenced with satellite launch information (Launchlog.txt, 2007). The resulting data set included 949 satellites launched into LEO with perigee greater than 350 km and apogee less than 1000 km. This included circular and elliptic orbits with maximum eccentricity of 0.03. For each object, specific orbital inclination information was obtained and the frequency of occurrence was tabulated and normalized for integer values of inclination from 0 to 110°. The results are shown in Figure 5.16.
Figure 5.16: Historical Inclination Distribution based on Unclassified LEO Satellite Launches from 1957 to 2006.
36 This figure provides an expected distribution of i based on a historical perspective that can be used to statistically predict the potential of encountering a specific i . This historical i and the previous statistical β versus i distributions were combined into weighting matrices that were subsequently used in the search for hot- and cold-case design orbits. 5.1.4 Weighting Matrices Weighting matrices helped focus the search by eliminating statistically insignificant orbits. A statistical weighting matrix was developed first and was found to eliminate much of the available search space. As a result, this approach was modified to include all viable combinations of β and i at a given threshold level. The result was the development of viable weighting matrices. 5.1.4.1 Statistical Weighting Matrices The probability that a particular combination of β and i angle will occur can be estimated from the product of the frequency that the β can occur at that particular i angle (Figure 5.15) and the probability of occurrence of the i angle based on historical data (Figure 5.16). With this in mind, a discrete statistical weighting matrix was developed using data from Figure 5.15 and Figure 5.16 with one-degree bin sizes for β and i angle. For β from 0 to 90° and i from 0 to 110°, this resulted in the Hadamard product (term by term) of a 91 x 111 matrix and a 1 x 111 matrix. The resulting 91 x 111 matrix was normalized and plotted as a contour plot as shown in Figure 5.17. As shown in Figure 5.17, the majority of β and i combinations were at or near zero. Consequently, resulting design orbits were found to be limited to those β and i combinations with values close to or at 1.0 from Figure 5.17 (e.g. β = 50°, i = 74°). In effect, utilizing the statistical weighting matrix eliminated most of the possible design orbit candidates. As a result, a modified approach was taken in order to increase the search space without losing the valuable information residing in the statistical weighting matrix. The result was the development of viable weighting matrices.
37
Figure 5.17: Contour Plot of Statistical Weighting Matrix Based on Hadamard Product of β versus i Distribution and Historical Inclination Distribution Information. 5.1.4.2 Viable Weighting Matrices The statistical weighting matrix includes values from zero (i.e. combinations not possible) up to one. To ensure that all possible combinations of β and i were given consideration, a viable weighting matrix was generated. This weighting matrix was created by converting all statistical weighting matrix values greater than a given threshold to one. In effect, the viable weighting matrix maximizes the search space at a given threshold. In addition, increasing threshold values reduce the size of the search space. Figure 5.18 and Figure 5.19 show viable weighting matrices for thresholds of 0.00 and 0.03; areas in yellow indicate the search space at the given threshold value. Viable weighting matrices at thresholds of 0.01, 0.02, 0.04, 0.05, 0.10, 0.20, 0.25, 0.26, 0.30, and 0.50 were also generated for use in developing hot- and cold-case design orbits and are included in Appendix D.
38
Figure 5.18: Contour Plot of Viable Weighting Matrix for Combinations of β and i at a Threshold of 0.00.
Figure 5.19: Contour Plot of Viable Weighting Matrix for Combinations of β and i at a Threshold of 0.03.
39 5.2 Satellite Orbital Averaged Environmental Heat Loads The next step in determining the design hot- and cold-case orbital parameters was to quantify orbital averaged maximum hot and minimum cold case external environmental heat loads. Four distinct satellite surface types were considered that cover the range of expected solar absorptivity and longwave emissivity properties as summarized in Table 5.2. Table 5.2: Surface Properties for the Four Surface Categories. Surface Category
Description
α
ε
Flat Absorber Flat Reflector Solar Absorber Solar Reflector
Black Z306 Polyurethane Paint (3 mils thick) Polished Aluminum Black Chrome on Nickel Foil Optical Surface Reflector (Diffuse)
0.95 0.15 0.90 0.10
0.87 0.05 0.10 0.80
These surface types were applied to a spherical satellite model in circular orbit. The metrics for hot- and cold-case heat loads were orbital-averaged external environment heat flux values consisting of three primary components: direct solar, albedo, and outgoing longwave radiation (OLR) flux (i.e. Earth emitted radiation). The characteristic temperature of space is 3 K and as a result, deep space radiation can be neglected as a radiation source (e.g. assumed to be at 0 K) with negligible error (Justus et al., 2001). In addition, collisions with atmospheric gasses can provide an additional heating source (Gilmore, 2002). This free molecular heating occurs only at exceptionally low orbit altitudes (below 180 km) and during fairing separation. Consequently, this heating source was disregarded for the analysis. Finally, charged particle heating is the result of near-Earth trapped charged particles in the Van Allen belts. This weak source of heating is typically disregarded for typical thermal analyses (Gilmore, 2002). ′′ 5.2.1 Orbital Averaged Direct Solar Flux, qsol Direct solar flux is a product of irradiation emanating from the Sun that arrives uninterrupted on a satellite, S ′′ , and surface solar absorptivity, α . Accounting for a
40 shape factor of ¼ (i.e. π ⋅ r 2 / 4 ⋅ π ⋅ r 2 ) and eclipse fraction, f , the orbital averaged direct solar flux, per unit surface area, can be calculated as follows: 1 ′′ = α ⋅ S ′′ ⋅ ⋅ (1 − f ) . qsol 4
(5.12)
Eclipse occurs for only certain orbits. For example, an orbit with a β of 0° will have an eclipse, while the orbit with a β of 90° will not. The fraction of time a circular orbit is in eclipse (i.e. eclipse fraction) was calculated using (Gilmore, 2002) 1 ⎡ 2 ⎤ 2 + ⋅ ⋅ h 2 R h ( ) 1 E ⎥ −1 ⎢ f = ⋅ cos ⋅ ⎢ 180 ( R + h ) ⋅ cos ( β ) ⎥⎥ ⎢⎣ E ⎦ f =0
if
β < β*
if
β ≥ β*
,
(5.13)
where
⎡
RE ⎤ ⎥ ⎣ ( RE + h ) ⎦
β * = sin −1 ⎢
for 0° ≤ β * ≤ 90° .
(5.14)
The spectral shape of solar irradiation, S ′′ , is approximated by a blackbody at 5777 K and as a result is primarily shortwave radiation. For an Earth orbiting spacecraft, the magnitude of this value is not constant. First, the Sun emitted energy varies with an 11year solar cycle although this effect is only a fraction of a percent (Anderson, Justus, and Batts, 2001). This phenomenon follows the mean sunspot number shown in Figure 5.20.
Figure 5.20: Solar Cycle as Represented by Yearly Mean Sunspot Number (1900 - 2000) (Anderson and Smith, 1994).
41 Second, since the Earth’s orbit is elliptical, the mean distance between the Earth and Sun varies approximately ±1.7 percent throughout the year. As a result, direct solar flux varies ±3.4 percent (Anderson, Justus, and Batts, 2001). The solar constant, So′′ , is defined as the radiation that falls on a unit area of surface normal to the line from the Sun, per unit time, and outside of the atmosphere at one astronomical unit (1 AU) (Anderson, Justus, and Batts, 2001). Based upon recommendations of the World Radiation Center, this has a value of 1367 W/m2 (Anderson, Justus, and Batts, 2001). Values at other locations and times are determined by the following: S ′′ =
So′′ . 2 R⊕−
(5.15)
The Earth/Sun distance, R⊕− , is measured in AU. Accounting for an additional ±5 W/m2 due to variations in the solar cycle, solar irradiation values range from ′′ ) up to 1419 W/m2 ( Shot ′′ ) depending upon the Earth’s position relative 1317 W/m2 ( Scold
to the Sun. ′′ 5.2.2 Orbital Averaged Albedo Flux, qalb
Albedo flux is defined as the amount of direct solar flux upon a planet that has been reflected back into space and lands upon a satellite’s surface in combination with surface optical properties. Since albedo flux emanates from solar irradiance, its spectral shape is similar to the Sun’s spectrum and therefore, is primarily composed of shortwave radiation. As a result, surface solar absorptivity becomes important. Assuming Earth is a diffusely reflecting sphere, incident albedo flux depends on a satellite’s view of Earth (i.e. sunlit and shaded portions). In addition, an altitude correction must be included due to view factor effects. Albedo flux is also dependent upon the solar zenith angle, θ , which is the angle between the solar vector and the Earth-satellite vector. On an orbitalaveraged basis, this becomes a function of β . The orbital averaged albedo flux takes the form ′′ = α ⋅ S ′′ ⋅ Fs − E ⋅ ρ alb ⋅ K . qalb
(5.16)
42 Assuming the satellite is much smaller than Earth, the view factor from a spherical satellite to a spherical Earth is given by
Fs − E
1 2 ⎡ 2⎤ 1 ⎢ ( h + 2 ⋅ h ⋅ RE ) ⎥ . = ⋅ 1− ⎥ RE + h 2 ⎢ ⎣ ⎦
(5.17)
Albedo, ρ alb , is defined as the fraction of incident solar energy reflected or scattered by a planet back into space and as a result, its value ranges from zero to one (Anderson, Justus, and Batts, 2001). A value of zero indicates a planet surface that is highly absorptive to shortwave radiation while a value of one indicates a planet surface that is highly reflective. Albedo values vary significantly depending upon the composition and properties of the planet and atmospheric surfaces. In general, continental areas have higher albedo values than those of oceanic regions while snow and ice covered continental regions have higher albedo than those without. In the atmosphere, dense cloud cover provides for higher albedo values. As a result, albedo increases towards the poles (i.e. increasing latitude) due to accumulation of snow and ice along with additional cloud cover associated with weather activity (NASA SP-8067, 1971). The strong relationship to Earth’s surface and atmosphere properties yields an expected seasonal variation of albedo values (NASA SP-8067, 1971). Diurnal variations in albedo are insignificant (Justus et al., 2001). In addition, albedo varies proportionally with solar zenith angle. This is caused, in part, by the optical characteristics of the Earth/atmosphere system. Small solar zenith angles result in relatively low optical thicknesses of the atmosphere and therefore, scattering effects are less and the solar irradiance is more easily transmitted to the Earth’s surface. At high solar zenith angles, the optical thickness increases and thus, scattering effects significantly increase, which increases albedo. Further, in a partly cloudy environment at low solar zenith angles, direct solar flux is more readily transmitted to the darker more absorbing Earth surface, which reduces albedo. At high solar zenith angles, this transmittance is decreased and reflectance, or albedo, from the cloud tops is increased.
43 Experimental data have shown that the β has limited effect on localized albedo (Justus et al., 2001). The β does, however, have an effect on orbital averaged albedo. At large β , a satellite spends a majority of the orbit at a large zenith angle, which increases albedo. Although albedo increases with increasing solar zenith angle, the opposite is true for albedo flux since this is a function of incident radiation on the Earth. Incident radiation decreases with the cosine of the angle from the subsolar point and approaches zero at the terminator (Gilmore, 2002). Albedo varies at different locations on Earth and as a result is a function of several variables, including local Earth and atmospheric conditions (which are correlated to latitude) and solar zenith angle. These values were found based upon the Earth Radiation Budget Experiment (ERBE) and are summarized in several sources (Karam, 1998; Justus et al., 2001). Average albedo, ρ alb , is obtained by averaging the solar zenith angle dependent albedo, ρ alb (θ ) , over non-eclipse portions of orbital period, P . When in eclipse, the value of ρ alb (θ ) becomes zero. Accounting for the effects of solar zenith angle (Justus et al., 2001), P
ρ alb =
∫ ρ alb (θ ) ⋅ S ′′ ⋅ cos (θ ) ⋅ dt 0
P
∫ S ′′ ⋅ cos (θ ) ⋅ dt 0
P
∫ ( ρ ( 0°) + c (θ ) ) ⋅ S ′′ ⋅ cos (θ ) ⋅ dt alb
=
0
P
∫ S ′′ ⋅ cos (θ ) ⋅ dt
,
(5.18)
0
which can be rewritten as,
ρ alb = ρ alb ( 0° ) + c .
(5.19)
Albedo data at a solar zenith angle of 0°, ρ alb ( 0° ) , were found from the ERBE, which includes data as a function of three distinct i bands (i.e. 0-30°, 30-60°, and 60110°) for several averaging periods (i.e. 16 seconds, 128 seconds, 896 seconds, 30 minutes, 90 minutes, 6 hours, and 24 hours). The 16 second time averaging data provides the most extreme values. For the analysis, 90 minute time averaged values were chosen, which were the closest to the period of low Earth orbit (LEO) orbits. A sample of
44 the data obtained for medium i orbits at 128 second data averaging is shown in Figure 5.21.
Figure 5.21: Albedo versus OLR for Medium Inclination Orbits (30 to 60°) using 128 second Data Averaging (Anderson and Smith, 1994). To reduce the size and corresponding complexity of available data, three extreme combinations were identified: albedo, combined, and OLR (Anderson and Smith, 1994). Albedo extreme values (ALB) consist of a pair of albedo and OLR values for which 0.04% of the values exceed the given albedo value. Combined extreme values (CMB) occur when 0.04% of the values are both above and to the right of the value for the hotcase extreme and 0.04% of the values are both below and to the left of the value for the cold-case extreme. OLR extreme values occur when 0.04% of the values exceed the given OLR value. These three are illustrated in Figure 5.21. For each of these extremes, mission critical values based on 3.3σ values in which the values are exceeded 0.04% of the time were utilized (Table 5.3). The average correction term, c , was determined by P
c=
∫ c (θ ) ⋅ cos (θ ) ⋅ dt 0
P
∫ cos (θ ) ⋅ dt 0
.
(5.20)
45 Table 5.3: Albedo and OLR Values for Mission-Critical Hot and Cold Extreme Environments at Low, Medium or High Inclinations and Averaging Times from 16 seconds to 24 hours (Gilmore, 2002). Mission-Critical Cold Case Mission-Critical Hot Case 0-30° 30-60° 60-110° 0-30° 30-60° 60-110° Alb OLR Alb OLR Alb OLR Alb OLR Alb OLR Alb OLR [---] [W/m2] [---] [W/m2] [---] [W/m2] [---] [W/m2] [---] [W/m2] [---] [W/m2] ALB ALB ALB ALB ALB ALB ALB CMB CMB CMB CMB CMB CMB CMB OLR OLR OLR OLR OLR OLR OLR
16 sec 128 sec 896 sec 30 min 90 min 6 hr 24 hr 16 sec 128 sec 896 sec 30 min 90 min 6 hr 24 hr 16 sec 128 sec 896 sec 30 min 90 min 6 hr 24 hr
0.06 0.06 0.07 0.08 0.11 0.14 0.16 0.13 0.13 0.14 0.14 0.14 0.16 0.16 0.40 0.38 0.33 0.30 0.25 0.19 0.18
273 273 265 261 258 245 240 225 226 227 228 228 232 235 150 154 173 188 206 224 230
0.06 0.06 0.08 0.12 0.16 0.18 0.19 0.15 0.15 0.17 0.18 0.19 0.19 0.20 0.40 0.38 0.34 0.27 0.30 0.31 0.25
273 273 262 246 239 238 233 213 213 217 217 218 221 223 151 155 163 176 200 207 210
0.06 0.06 0.09 0.13 0.16 0.18 0.18 0.16 0.16 0.17 0.18 0.19 0.20 0.20 0.40 0.38 0.33 0.31 0.26 0.27 0.24
273 273 264 246 231 231 231 212 212 218 218 218 224 224 108 111 148 175 193 202 205
0.43 0.42 0.37 0.33 0.28 0.23 0.22 0.30 0.29 0.28 0.26 0.24 0.21 0.20 0.22 0.22 0.22 0.17 0.20 0.19 0.19
182 181 219 219 237 248 251 298 295 291 284 275 264 260 331 326 318 297 285 269 262
0.48 0.47 0.36 0.34 0.31 0.31 0.28 0.31 0.30 0.28 0.28 0.26 0.24 0.24 0.21 0.22 0.22 0.21 0.22 0.21 0.21
180 180 192 205 204 212 224 267 265 258 261 257 248 247 332 331 297 282 274 249 245
0.50 0.49 0.35 0.33 0.28 0.27 0.24 0.32 0.31 0.28 0.27 0.26 0.24 0.23 0.22 0.22 0.20 0.20 0.22 0.22 0.20
180 184 202 204 214 218 224 263 262 259 260 244 233 232 332 331 294 284 250 221 217
Equation (5.20) is simplified by relating the values of θ , β , and ϕ (which defines the angular position around the orbit measured at zero at the most-nearly sub-solar point) using (Justus et al., 2001). cos(θ ) = cos( β ) ⋅ cos(ϕ ) .
(5.21)
For a circular orbit, ϕ varies linearly with time and thus, integration was made with respect to angular position. Noting that the value of c (θ ) becomes invalid in eclipse and substituting Equation (5.21) into Equation (5.20) yields:
46 π 2
c (θ ) ⋅ cos(ϕ ) ⋅ dϕ ∫ c (θ ) ⋅ cos( β ) ⋅ cos(ϕ ) ⋅ dt ∫π P
c=
=
0
P
−
2
∫ cos( β ) ⋅ cos(ϕ ) ⋅ dt
π 2
∫π cos(ϕ ) ⋅ dϕ
0
−
π
=
1 ⋅ 2
2
∫π c (θ ) ⋅ cos(ϕ ) ⋅ dϕ . (5.22)
−
2
2
The average correction term was solved numerically and results are given as a function of β in Table 5.4 (Justus et al., 2001). Table 5.4: Average Correction Term, c , versus β .
β
c
0 10 20 30 40 50 60 70 80 90
0.04 0.04 0.05 0.06 0.07 0.09 0.12 0.16 0.22 0.31
Assuming Earth is a diffusely reflecting sphere, incident albedo flux depends on a satellite’s view of Earth (sunlit versus shaded portions) in addition to solar zenith angle,
θ . For a more rigorous discussion, refer to Cunningham (1961). On an orbital averaged basis, these compounding effects were approximated as a function of β . A correction factor was developed using Monte-Carlo ray tracing software by solving Equation (5.16) for K over a range of β (0 to 90°) and altitudes (350 to 1000 km) for unit absorptivity, direct solar flux, and orbital-averaged albedo. The results are plotted in Figure 5.22.
47
Figure 5.22: Orbital Averaged Albedo Flux Correction Factor, K, versus β . A resulting cubic regression model fitted to this data revealed the correction factor K = 0.3132 + 0.7678E − 3 ⋅ β − 0.8034 E − 4 ⋅ β 2 + 0.4130 E − 6 ⋅ β 3
(5.23)
with a coefficient of determination, R2, of 99.8%. As shown, this model fits the data well with modest variance at large β . Due to its dependence on θ , orbital averaged albedo flux diminishes as β increases; therefore, these deviations will not significantly impact overall environmental fluxes. ′′ 5.2.3 Orbital Averaged Outgoing Longwave Radiation (OLR) Flux, qOLR Earth maintains an average effective temperature of -18°C and as a result, constantly emits longwave (i.e. infrared) radiation (Gilmore, 2002). In addition, the atmosphere is emitting radiation and absorbing Earth emitted energy. Because of this, the spectral distribution of Earth’s emitted radiation is relatively complex. However, it is accepted practice to model this source as a graybody spectrum according to a temperature range of 250 to 300K yielding wavelengths longer than 3 μm (Anderson, Justus, and Batts, 2001; Justus et al., 2001). As a result, the orbital averaged OLR flux is a result of diffuse irradiation emanating from Earth, E ′′ , that arrives on a satellite in combination with surface longwave emissivity, ε shown in the following:
48 ′′ = ε ⋅ Fs − E ⋅ E ′′ . qOLR
(5.24)
Although OLR flux is less than direct solar flux, it poses a unique problem since the source temperatures are very close to that of the spacecraft. As a result, it cannot be easily reflected away by selecting surface materials with appropriate optical properties since these same types of surfaces would not allow emittance of spacecraft heat. Although not as severe as albedo flux, OLR flux does vary over the globe and is primarily influenced by cloud cover and the Earth’s surface temperature (Anderson, Justus, and Batts, 2001). As expected the highest values of E ′′ occur in tropical and desert regions and decrease with latitude. Increasing cloud cover decreases E ′′ since cloud tops are typically cold and clouds impedes Earth emitted radiation (Anderson, Justus, and Batts, 2001). E ′′ decreases with increasing latitude but does not vary significantly with β (Justus et al., 2001). A decrease in E ′′ near the equator is attributed to increased cloud cover. As with albedo, E ′′ , was determined from the ERBE. The orbital averaged heat flux model for both the hot and cold cases is a combination of the three sources, ′′ = qsol ′′ ,hot + qalb ′′ ,hot + qOLR ′′ ,hot qhot
(5.25)
′′ = qsol ′′ ,cold + qalb ′′ ,cold + qOLR ′′ ,cold . qcold
(5.26)
Substituting the three sources provides
1 ′′ = α ⋅ Shot ′′ ⋅ ⎛⎜ ⋅ (1 − f ) + Fs − E ⋅ ρ alb,hot ⋅ K ⎞⎟ + ε ⋅ Fs − E ⋅ Ehot ′′ qhot ⎝4 ⎠
(5.27)
1 ′′ = α ⋅ Scold ′′ ⋅ ⎛⎜ ⋅ (1 − f ) + Fs − E ⋅ ρ alb,cold ⋅ K ⎞⎟ + ε ⋅ Fs − E ⋅ Ecold ′′ qcold ⎝4 ⎠
(5.28)
for the hot and cold cases, respectively (Justus et al., 2001). As these equations illustrate, the orbital averaged hot- and cold-case extreme external environmental heat loads are a function of altitude, β , i , ρ alb ( 0° ) , E ′′ , and surface properties ( α and ε ). Hot- and cold-case heat fluxes were identified for each surface type and combination of β and i by exhaustively searching over all other
49 applicable variables and finding the extreme (both maximum and minimum) values of Equation (5.27) and (5.28). Results from this analysis are shown in Figure 5.23 through Figure 5.30 for the four surface types.
Figure 5.23: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Absorber).
Figure 5.24: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Absorber).
50
Figure 5.25: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Reflector) Surfaces.
Figure 5.26: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Flat Reflector).
51
Figure 5.27: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Absorber).
Figure 5.28: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Absorber).
52
Figure 5.29: Hot Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Reflector).
Figure 5.30: Cold Case Orbital Averaged Extreme Heat Flux (W/m2) versus β and i (Solar Reflector). Hot-case orbital averaged extreme heat flux increases with increasing β until a local maximum is reached and then decreases. This is a direct result of orbital eclipse fractions. As the β increases, eclipse fractions decrease, which increases the orbital averaged direct solar and albedo fluxes. A maximum heat flux is reached at the point where eclipse
53 fractions reach zero. As the β continues to increase, the orbital average cosine solar zenith angle, cos (θ ) , approaches zero and therefore, albedo fluxes diminish. Because of the nature of albedo and OLR values that came out of the ERBE study, three distinct bands of values appear in the hot-case orbital averaged extreme heat flux figures. For a given β , the heat flux generally decreases with increasing i . This is the result of eclipse fractions that cause albedo fluxes to be seen over a portion of an orbit while OLR fluxes are seen over an entire orbit. As a result, OLR fluxes typically dominate albedo fluxes in the orbital averaged model. For large β and thus, small eclipse fractions, this trend is still prevalent since the orbit is near the terminator where albedo fluxes drop off due to the cos(θ ) factor. Cold-case orbital averaged extreme heat flux figures show similar trends to that of the hot case results. The only difference is the relative magnitude of fluxes that were found. The orbital averaged hot- and cold-case extreme external environmental heat load results provide insight into design orbits although they do not take into account the probabilities of occurrence for different combinations of β and i . The next section addresses this issue. 5.3 Hot- and Cold-Case Design Orbits Developed for Robust Thermal Control Subsystems The viable weighting matrix was applied to the orbital averaged heat load results. The use of the viable weighting approach provides a method of examining a subset of hot- and cold-case heat flux information without altering their values. The statistical weighting approach tends to smooth the heat flux information. Critical hot- and cold-case orbital parameters were identified by first applying the viable weighting matrix at a 0.00 threshold to the orbital averaged heat flux models for each of the four surface types. In essence, this method stripped away values that were either not possible (e.g. combinations of β and i ) or not statistically significant (e.g. i that historically have not been utilized). The resulting hot- and cold-case results are shown in Table 5.5. Results for all other threshold values are included in Appendix E. For each hot- and cold-case critical orbit and surface type, a single β was found in combination with a set of i that gave the same orbital averaged heat flux. However, the
54 goal was to define a single pair of β and i for each surface type or possibly for all surface types. As a next step, the threshold was increased for each surface type and hotand cold-case designs until the set of i was narrowed to one value. Table 5.6 provides resulting critical hot- and cold-case orbits for each of the four surface categories. As a result of applying increasing threshold values, the set of i have been reduced to one for the same critical hot- and cold-case heat fluxes from Table 5.5. As shown in Table 5.6, the threshold level needed was very low (0.03) in all cases except for the cold-case solar reflector. These results were verified within 3.2% of values obtained from Monte-Carlo ray tracing software. The results show that critical cold-case orbits occur at β of 0° in all cases. This is not surprising since the largest eclipse fractions occur here. Critical hot-case orbits occur at β of 72° since this is the angle at which eclipse fraction initially becomes zero at an altitude of 350 km. Table 5.5: Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits for Each of Four Surface Categories at a 0.00 Threshold. Orbit Type
Heat Earth IR Altitude β i Flux and Albedo1 [---] [---] [W/m2] [degrees] [degrees] [---] [km] Flat Absorber Hot Case 0.95 0.87 437.4 72 49-53,55-60 3 350 Cold Case 0.95 0.87 275.7 0 2-4,15,21,23-25,28-30 2 1,000 Flat Reflector Hot Case 0.15 0.05 60.9 72 49-53,55-60 2 350 Cold Case 0.15 0.05 37.3 0 2-4,15,21,23-25,28-30 1 350 Solar Absorber Hot Case 0.90 0.10 348.5 72 49-53,55-60 1 350 Cold Case 0.90 0.10 206.5 0 2-4,15,21,23-25,28-30 1 350 Solar Reflector Hot Case 0.10 0.80 112.2 72 49-53,55-60 3 350 62-76,79-83,85-87,893 1,000 Cold Case 0.10 0.80 63.5 0 91,93,94,97-100,105,108 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR)
α
ε
55 Table 5.6: Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits for Each of Four Surface Categories. Orbit Type
Heat Threshold Earth IR Altitude β i Flux Level and Albedo1 [---] [---] [W/m2] [---] [degrees] [degrees] [---] [km] Flat Absorber Hot Case 0.95 0.87 437.4 0.03 72 52 3 350 Cold Case 0.95 0.87 275.7 0.03 0 28 2 1,000 Flat Reflector Hot Case 0.15 0.05 60.9 0.03 72 52 2 350 Cold Case 0.15 0.05 37.3 0.03 0 28 1 350 Solar Absorber Hot Case 0.90 0.10 348.5 0.03 72 52 1 350 Cold Case 0.90 0.10 206.5 0.03 0 28 1 350 Solar Reflector Hot Case 0.10 0.80 112.2 0.03 72 52 3 350 Cold Case 0.10 0.80 63.5 0.26 0 74 3 1,000 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR)
α
ε
The altitudes at which critical cold-case orbits were found show interesting characteristics. Intuitively, it is expected that critical cold-case orbits would occur at high altitudes (i.e. 1000 km). This occurs for the flat absorber and solar reflector cold cases since these have relatively high OLR fluxes, which occur irrespective of eclipse fraction. Thus, high altitudes produce lower fluxes due to view factor considerations. For the other two surface types (flat reflector and solar absorber), critical cold-case orbits occur at altitudes of 350 km. This result is caused by the fact that each of these surfaces is driven by albedo fluxes (i.e. absorptivity significantly greater than emissivity). As a result, critical cold-case environments occur when eclipse fractions are the greatest. This occurs at the lowest altitudes. In order to consider the possibility of defining hot- and cold-case orbiting parameters that work well for any surface type, the design orbit parameters determined for each surface type in Table 5.2 were applied to the remaining three surface types. For example, the design hot-case orbit for a flat absorber occurred at β = 72°, i = 52°, Earth IR and albedo value = 3, and an altitude of 350 km. At this combination of orbit parameters, the
56 design hot-case heat flux was determined to be 437.4 W/m2. These same orbit parameters were applied to the flat reflector, solar absorber, and solar reflector surface types. The resulting hot case heat fluxes were determined and compared to the design hot case heat fluxes. Results for the hot-case design orbits are shown in Table 5.7. The percent differences in heat fluxes are in comparison to design values for each surface type and set of orbiting parameters. Average percent differences across the surface types are also given for each unique set of orbit parameters. Table 5.7: Hot-Case Design Orbit Percent Difference Analysis (Utilizing Orbital Averaged Values and Viable Weighting Matrices).
β i Earth IR and Albedo Values1 Altitude
72° 52°
72° 52°
72° 52°
72° 52°
3
2
1
3
350 km 350 km 350 km 350 km Heat % Heat % Heat % Heat % Flux Diff. Flux Diff. Flux Diff. Flux Diff. [W/m2] [%] [W/m2] [%] [W/m2] [%] [W/m2] [%]
Flat Absorber 437.4 0.0 434.3 0.7 (α = 0.95 | ε = 0.87) Flat Reflector 60.9 0.0 60.9 0.0 (α = 0.15 | ε = 0.05) Solar Absorber 346.7 0.5 348.0 0.1 (α = 0.90 | ε = 0.10) Solar Reflector 112.2 0.0 107.8 4.0 (α = 0.10 | ε = 0.80) Average --0.1 --1.2 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR)
421.0
3.7
437.4
0.0
60.4
0.9
60.9
0.0
348.5
0.0
346.7
0.5
93.6
16.6
112.2
0.0
---
5.3
---
0.1
As shown in Table 5.7, the average percent error in heat flux by using a single set of hot-case design orbit parameters ( β = 72°, i = 52°, Earth IR and Albedo Values = 3, Altitude = 350 km) is minimal (0.1%). In addition, this single hot-case design orbits provided orbital average heat fluxes within 0.5% of the values determined using design orbits specific to individual surface types. It appears that this combination of orbit parameters could be utilized as a single design point for all surface types including satellites having combined surface properties.
57 A similar analysis was completed for the cold-case design orbits. As shown in Table 5.8, the orbit parameters with the lowest average percent heat flux error corresponds to β = 0° and i = 28° with Earth IR and albedo value = 2 and an altitude of 1000 km. At this design point, the average percent error in heat flux by using a single set of cold-case design orbit parameters is -4.7%. Although still small, this value is notably influenced by the solar reflector results (i.e. percent difference of -9.0%). It could be argued that a separate design cold-case orbit ( β = 0°, i = 74°, Earth IR and Albedo Values = 3, Altitude = 1000 km) should be used for spacecraft with predominantly solar reflector surfaces. If this approach is used then a design cold-case orbit ( β = 0°, i = 28°, Earth IR and Albedo Values = 1, Altitude = 350 km) should be used for all other surface types with a maximum percent difference of -3.4%. Table 5.8: Cold-Case Design Orbits Percent Difference (Utilizing Orbital Averaged Values and Viable Weighting Matrices).
β i Earth IR and Albedo Values1 Altitude
0° 28°
0° 28°
0° 28°
0° 74°
2
1
1
3
1000 km 350 km 350 km 1000 km Heat % Heat % Heat % Heat % Flux Diff. Flux Diff. Flux Diff. Flux Diff. [W/m2] [%] [W/m2] [%] [W/m2] [%] [W/m2] [%]
Flat Absorber 275.7 0.0 285.2 -3.4 (α = 0.95 | ε = 0.87) Flat Reflector 38.6 -3.3 37.3 0.0 (α = 0.15 | ε = 0.05) Solar Absorber 220.1 -6.6 206.5 0.0 (α = 0.90 | ε = 0.10) Solar Reflector 69.2 -9.0 92.3 -45.5 (α = 0.10 | ε = 0.80) Average ---4.7 ---12.2 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR)
285.2
-3.4
279.8
-1.5
37.3
0.0
40.0
-7.1
206.5
0.0
230.3 -11.6
92.3
-45.5
63.5
0.0
---
-12.2
---
-5.0
Further analysis indicated that hot- and cold-case design orbits also work well for intermediate surface types. An analysis was conducted for a surface with α = 0.50 and ε = 0.50. The resulting critical hot-case orbit was identical to the design point found
58 previously ( β = 72°, i = 52°, Earth IR and Albedo Values = 3, altitude = 350 km). The critical cold-case orbit was identical to the design point identified for all surface types ( β = 0°, i = 28°, Earth IR and albedo value = 2, altitude = 1000 km). 5.4 Conclusion Single hot- and cold-case design orbits that work well in the design of robust thermal control subsystems over a wide range of satellite surface properties and likely operating environments were determined. A general approach was developed which employed a combination of statistical and historical data such that statistically insignificant orbits were disregarded. From the analysis, a hot-case design orbit was found ( β = 72°, i = 52°, altitude = 350 km) which provided orbital averaged heat fluxes within 0.5% of the values determined using design orbits specific to individual surface types. A general cold-case design orbit was found ( β = 0°, i = 28°, altitude = 1000 km) which provided orbital averaged heat fluxes within 9.0% of the values determined using design orbits specific to individual surface types. If more accuracy is required, two cold-case design orbits were identified. For spacecraft with predominantly solar reflector surfaces, a coldcase design orbit ( β = 0°, i = 74°, altitude = 1000 km) provides minimum orbital averaged heat fluxes. For all other surface types, a cold-case design orbit was found ( β = 0°, i = 28°, altitude = 350 km) which provided orbital averaged heat fluxes within 3.4% of the values determined using design orbits specific to individual surface types. These results were based on a spherical satellite model and therefore, care should be taken when significant departures from this assumption are encountered. This includes both geometric and attitude deviations such as those for a Sun-pointing solar panel. In addition, the design orbits are intended for LEO missions with altitudes in the range of 350 to 1000 km. Consequently, their use for highly elliptic orbits and those with relatively large altitudes is not recommended. The hot- and cold-case design orbits have many potential applications. First, they provide consistent design criteria for the development and comparison of robust thermal control subsystem approaches. In the absence of clearly defined mission criteria, these design orbits also provide a reasonable surrogate. Finally, they are useful for the design
59 of robust spacecraft over a wide range of both satellite surface properties and low Earth orbit operating environments.
60
CHAPTER 6 - OPTIMAL DISTRIBUTION OF ELECTRONIC COMPONENTS TO BALANCE ENVIRONMENTAL FLUXES
Robust satellites are required to handle a wide variety of missions. As a result, the underlying thermal control subsystem (TCS) must be robust enough to maintain appropriate thermal requirements for satellite components under the most severe (hot and cold) environments. Investigation has revealed that this could best be accomplished with a TCS utilizing an isothermal bus with some level of thermal energy balance control (Young et al., 2008; Shannon III et al., 2008). The isothermal bus concept is an idea focused on approaching a single thermal node satellite representation. In effect, heat is shared efficiently between cold and hot components and as a result, gradients across the satellite are minimized thus reducing temporal variations in temperature. This concept could be achieved in numerous ways. These include embedded heat pipes, integrated pumped fluid loops and integrated high conductivity face sheets such as annealed pyrolytic graphite (APG). One method that has not received much attention is simply optimized component placement. As a first approach, electronic components can be optimally distributed amongst the various exterior spacecraft panels to achieve a more balanced distribution of heat flux. This depends upon panel orientation and the external thermal environmental fluxes placed upon them. Further, components can be optimally located on each panel to more evenly distribute applied fluxes. In combination, these two measures will help reduce requirements for more expensive isothermal bus solutions. Consequently, a computational tool was developed which improves satellite thermal performance through intelligent component placement. Based on orbital-averaged electronic component power and environmental heat fluxes, the tool provides the optimal satellite face and location within that face for all components such that thermal gradients across the satellite are minimized. Results are determined using a two-step process.
61 CHAPTER 6 presents a global distribution optimization strategy to determine the specific satellite face with which a component should be mated. After all component-face pairs are determined, CHAPTER 7 presents a local placement algorithm used to determine specific component locations within each face. Although developed in response to a RS need for isothermal conditions, these methods could also be applied to traditionally designed spacecraft potentially reducing TCS performance requirements. 6.1 Review of Satellite Layout Optimization Approaches Layout optimization is one of the key techniques for improving global performance of satellites (Sun and Teng, 2003). Component placement in satellites is an important design process at the conceptual design stage, since it influences a range of fundamental characteristics of a satellite, and design change in the latter detailed design stage is difficult (Izui et al., 2007). Optimal layout design of a satellite module is a 3-D packing problem with performance constraints and is known to be an NP-hard problem in terms of computational complexity. NP-hard problems cannot be solved in polynomial time and are intrinsically difficult (Jackson and Carpenter, 2004). Polynomial time occurs when the execution time of a computation, t (n ) , is no more than a polynomial function of the problem size, n , and a constant k such that
t (n ) ≤ O( n k ) .
(6.1)
Generally put, polynomial time algorithms are reasonable to compute. An NP-hard problem also has the property that an algorithm for solving it may be translated into one for solving any other problem of similar complexity with only polynomially more work. The NP-hard class includes some of the most difficult types of optimization problems that can be conceived, and problems in NP-hard are said to be presumably intractable. In several studies, the inertial performance of a cylindrical satellite module was optimized through placement of n components. Sun and Teng (2002) utilized a two-stage procedure for component placement; the first step was a genetic-algorithm based solution method for global placement while the second step used ant colony optimization methods to further refine placements. Huo and Teng (2009) analyzed the same problem utilizing a co-evolutionary method with heuristic rules. The layout design of a satellite module was
62 decomposed into three stages: the distribution of the objects among the subspaces, the detailed layout design within the subspaces, and the whole layout scheme design. Chen, Shi, and Teng (2008) examined the same cylindrical layout problem utilizing a differential evolution algorithm with Gaussian-mutation methods. Zeng, Shi, and Teng (2006) solved the same problem utilizing game optimization methods. Izui et al. (2007) formulated a bi-level optimization procedure for inertial optimization of satellites. A genetic algorithm was used in the first step to obtain approximate location of components with a second step using successive quadratic programming to determine detailed component positions. A nine-component problem was studied with the goal of optimizing inertial properties. Langer, Pühlhofer, and Baier (2004) utilized a multi-objective evolutionary algorithm using clustering techniques to divide the population in groups. For each group, a response surface is estimated based on the population at hand and gradient-based optimization is utilized on the response surface. Parallelization was used to improve computational expense. These methods were applied for optimal placement of over 50 components with respect to center of gravity, collision, and functional constraints. Other approaches focused on optimization of payload and sensor locations. Ferebee Jr. and Allen (1991) investigated payload placement in order to optimize inertial properties to reduce propellant budget for gravity gradient-induced momentum desaturations. Jackson and Carpenter (2004) examined the utility of implementing a sun sensor placement optimization tool using a hybrid genetic algorithm. This algorithm combines genetic algorithms with simulated annealing methods. The task of determining sun sensor placement on a spacecraft is generally performed manually, based on prior experience and similarity to previous designs. This type of manual process tends to be highly iterative, consumes valuable time and resources, and generally produces suboptimal results. Jackson and Carpenter (2004) explored the solution of the multi-objective optimization problem that includes as constraints minimizing the number of sun sensors, providing continuous 4π steradian coverage, providing multiple sensor field-of-view overlap everywhere, and minimizing sensor blockage. The sun sensor placement problem is a difficult combinatorial optimization problem that cannot be solved optimally in
63 polynomial time due to the vastness of the solution space. The sun sensor placement problem may be considered as a variant of the classical traveling salesman problem (TSP). In the classical TSP, a salesman must make a complete tour of n cities, visiting each city only once, in the most efficient manner possible (i.e., traveling the least Euclidean distance). No general method of solution is known, and the problem is classified as NP-hard. The literature has shown that layout optimization has been used often to improve satellite performance. However, these approaches have neglected thermal performance. Only Jackson and Norgard (2002) were found to include thermal effects. They utilized multi-disciplinary optimization (MDO) methods to optimize the placement of avionics boxes (satellite components). They considered minimization of interconnections between components and balancing mass between opposite panels. In addition, they considered thermal effects by keeping uniform thermal loading (orbital average power) on each satellite panel. This thermal optimization approach is ideally suited when external environmental heating fluxes are uniform (i.e. the same on each external panel). However, this is not realistic and a more robust approach including the effects of nonuniform external heating environments is needed. 6.2 Numerical Experiment Investigations To investigate the advantages of optimized component distribution, simple numerical experiments were conducted. The focus was to evaluate the merits of redistribution of component power between panels of a common satellite architecture. The remainder of this section details the numerical model used in these experiments, the orbital environmental conditions, experimental approach, and subsequent results. 6.2.1 Numerical Model Details A numerical model was developed consistent with current robust satellite architectures (Arritt et al., 2008). A 1.0 m x 1.0 m x 1.0 m six-sided frame-and-panel construction satellite was used with a honeycomb panel construction having 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material (Figure 6.1). Material properties were based on Gilmore (2002). A uniform heat-flux
64 source was placed on the interior of each panel that represents electrical components with uniform heat flux spreading. Heat generation of all internal components totaled 600 W and was conducted to the panel only (i.e. not coupled through radiation) with a contact conductance of 110 W/m2-K (Gilmore, 2002). Panel to panel conductance was 12 W/K to simulate panel-to-panel longerons and bolted joints (Bugby, Zimbeck, and Kroliczek, 2008). All heat generation was ultimately dissipated to a deep space environment (0 K) through radiation from the exterior of the panel surfaces. Exterior panel surfaces were modeled as a solar reflector ( α = 0.1 and ε = 0.8) for the hot- and cold-case orbits.
Figure 6.1: Illustration of Exterior and Interior of Six-sided Thermal Desktop® Satellite Model. The case study was simulated using Thermal Desktop® as a CAD based interface to the SINDA/FLUINT finite-difference thermal analyzer. Thermal Desktop® and SINDA/FLUINT are widely accepted tools for spacecraft thermal design. All computational analysis was done using a 2.5-gigahertz dual-core processor. A complete description of modeling parameters are included in Appendix F. In order to test the algorithms and evaluate results, a comprehensive set of characteristic component design cases (Case A, B, C, and D) were developed (Figure 6.2). These were developed and based on Williams’ (2005) low-capability (LCB) and high-capability bus (HCB) FACTS approach; a representative military satellite; and Young’s (2008) low-capability (LCB), medium-capability (MCB), and high-capability (HCB) TherMMS concept. Case A was based on the FACTS HCB. Case B is an
65 averaged model based on all six reference missions. Case C is based on the representative military satellite. Case D, a limiting case, is a uniform distribution such that all components will have equal power. For each design case, both component power and number of components were normalized. Normalization provided a way of comparing reference missions on an equal basis and shows a generalized shape or distribution of power versus component number. A curve fit was then applied to each normalized case. The resulting normalized distributions can be scaled in both the x- and y-axes to accommodate various total power and component combinations. These design cases were used as needed. Refer to Appendix G for a detailed explanation of their development.
Figure 6.2: Mathematical Models for Each of Four Design Cases (Refer to Appendix G for a detailed explanation of the design cases). 6.2.2 Orbital Environmental Loads To appropriately test component distributions, an appropriate environmental load is required. Both hot- and cold-case orbital environments were selected based on previous design hot- and cold-case orbits. The environmental parameters are detailed in Table 6.1. These parameters were set within Thermal Desktop® to determine the associated incident direct solar, albedo, and OLR fluxes. Using this model and values from Table 6.1, the incident environmental flux sources were determined for each surface over
66 one orbit. Appendix H Figure H.2 through Figure H.5 show hot- and cold-case results for total, direct solar, albedo, and OLR incident fluxes. Table 6.1: Hot and Cold Case Design Orbit Environmental Parameters. Hot-Case Orbit Cold-Case Orbit Beta Angle
72°
Inclination
52°
0° 28° 2
Solar Constant
1,419 W/m
1,317 W/m2
Orbital-Averaged OLR
274 W/m2
228 W/m2
Orbital-Averaged Albedo
0.392
0.180
Altitude
350 km
1,000 km
These incident flux values were averaged over one orbit for each surface. The resulting orbital averaged values are shown in Table 6.2. These incident heat flux values were adjusted to account for panel surface properties. For example, a solar reflector surface ( α = 0.1 and ε = 0.8) yields results shown in Table 6.3. Table 6.2: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values. Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface +X Surface -Y Surface +Y Surface -Z Surface +Z Surface
239.81 238.93 1,454.68 95.51 139.22 434.64
139.22 139.22 1,349.55 0.00 139.22 139.22
16.80 16.65 23.01 11.19 0.00 49.85
83.79 83.05 82.12 84.32 0.00 245.57
388.75 329.45 387.94 329.45 58.62 0.00 59.06 0.00 418.15 418.15 300.81 74.31
14.54 14.54 14.46 14.43 0.00 56.01
44.77 43.94 44.16 44.63 0.00 170.50
AVERAGES
433.80
317.74
19.58
96.47
268.89 191.89
19.00
58.00
67 Table 6.3: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.10 / ε = 0.80). Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface +X Surface -Y Surface +Y Surface -Z Surface +Z Surface
82.64 82.03 202.95 68.58 13.92 215.36
13.92 13.92 134.95 0.00 13.92 13.92
1.68 1.67 2.30 1.12 0.00 4.99
67.03 70.21 66.44 69.55 65.69 36.77 67.46 37.15 0.00 41.81 196.45 149.43
32.95 32.95 0.00 0.00 41.81 7.43
1.45 1.45 1.45 1.44 0.00 5.60
35.81 35.15 35.33 35.70 0.00 136.40
AVERAGES 110.91
31.77
1.96
77.18
19.19
1.90
46.40
67.49
Notably, these values are within 1.1% and 2.5% of the hot- and cold-case design fluxes, respectively (Table 5.7 and Table 5.8). 6.2.3 Hot- and Cold-Case Design Orbit Power Imbalance Case Studies To understand the importance of satellite power distribution, numerical experiments were conducted for a fixed total internal power generation and varying distributions between the six panels. The imbalance in power distribution was expressed relative to an even power distribution (i.e. 100 W per panel) and the impact of the imbalances on temperature differences across the panels was considered. Power was evenly distributed across each panel and the experiments were conducted for both the hot- and cold-case design orbits. Under the hot-case thermal environment with solar reflector external surfaces ( α = 0.1 and ε = 0.8), the +Z panel receives the greatest incident radiation (i.e. direct solar, albedo, and OLR) while the opposite panel (i.e. -Z) receives the least. Therefore, power was shifted between these two panels while the remaining panels each received 100 W. Power imbalance linearly ranged from -1.0 (i.e. 0 W on the –Z panel and 200 W on the +Z panel) up to +1.0 (i.e. 200 W on the –Z panel and 0 W on the +Z panel). Twenty-one imbalance cases were run between these two extremes. For each case, maximum and minimum subsystem temperature along with maximum temperature difference were
68 recorded (Figure 6.3). Temporal variations of temperature for imbalances of -1.0, 0.0, and +1.0 are shown in Figure 6.4 through Figure 6.6.
Figure 6.3: Maximum and Minimum Subsystem Temperature and Maximum Temperature Difference for Varying Levels of Power Imbalance (Hot-Case Orbit).
Figure 6.4: Temporal Variation of Temperature for Six Subsystems at Imbalance of -1.0 for the Hot-Case Orbit.
69
Figure 6.5: Temporal Variation of Temperature for Six Subsystems at Imbalance of 0.0 for the Hot-Case Orbit.
Figure 6.6: Temporal Variation of Temperature for Six Subsystems at Imbalance of +1.0 for the Hot-Case Orbit. Under the cold-case thermal environment, the +Z panel receives the greatest incident radiation (i.e. direct solar, albedo, and OLR) while the opposite panel (i.e. -Z) receives the least. Power was shifted between these two panels while the remaining panels each
70 received 100 W. The imbalance of power linearly ranged from -1.0 (i.e. 0 W on the –Z panel and 200 W on the +Z panel) up to +1.0 (i.e. 200 W on the –Z panel and 0 W on the +Z panel). Twenty-one imbalance cases were run between these two extremes. For each case, maximum and minimum subsystem temperature along with maximum temperature difference were recorded (Figure 6.7). Temporal variations of temperature for imbalances of -1.0, 0.0, and +1.0 are shown in Figure 6.8 through Figure 6.10.
Figure 6.7: Maximum and Minimum Subsystem Temperature and Maximum Temperature Difference for Varying Levels of Power Imbalance (Cold-Case Orbit). Although this is a simple experiment trading power between only two of six panels, it does indicate the benefit of optimized component power distribution. Hot-case results showed the best performance (i.e. minimized maximum temperature difference) at an imbalance of +1.0. Consequently, redistribution of power between the –Z and +Z panels reduced temperature differences by 33.3 K, reduced maximum temperature by 13.9 K, and increased minimum temperature by 19.1 K. Cold-case results showed the best performance at an imbalance of 0.4. Compared to worst-case condition at an imbalance of -1.0, optimized component distribution reduced maximum temperature differences by 9.5 K, reduced maximum temperature by 4.5 K, and increased minimum temperature by 5.6 K.
71
Figure 6.8: Temporal Variation of Temperature for Six Subsystems at Imbalance of -1.0 for the Cold-Case Orbit.
Figure 6.9: Temporal Variation of Temperature for Six Subsystems at Imbalance of 0.0 for the Cold-Case Orbit.
72
Figure 6.10: Temporal Variation of Temperature for Six Subsystems at Imbalance of +1.0 for the Cold-Case Orbit. Operational requirements typically insist that components are maintained within a range of approximately 40 K (Wertz and Larson, 2005). Consequently, optimized results from this experiment provide reductions in maximum temperature differences of 83.3% and 23.8% of requirements for the hot- and cold-cases, respectively. Some specific components such as energy storage modules require even tighter control, therefore optimized power distribution would have an even greater impact in these cases. A practical alternative to optimized component distribution is an even distribution of power (i.e. 100 W per panel). This approach reduces maximum temperature differences by 25.1 K for the hot-case and 8.1 K for the cold-case experiment. Consequently, evenly distributed power results from this experiment provide reductions in temperature differences of 62.8% and 20.3% of requirements for the hot- and cold-cases, respectively. Thermal performance could also be remedied through properly tuning individual panel surface properties (i.e. reduce or increase environmental loads as needed), but there is a limit to its effectiveness. Surface emissivity can be increased to 1.0 before deployable radiators are required. Consequently, utilizing an optimization tool to appropriately distribute component power across a satellite is important.
73 6.3
Methodology for Distributing Power Among Satellite Panels Consider a satellite panel (i.e. face) i of area, Ai , that receives a uniformly-
′′ ,i , and dissipates energy to the distributed averaged heat load from the environment, qenv surroundings via longwave radiation (i.e. ε i ⋅ σ ⋅ Ti 4 ⋅ Ai ). A total of ni components are placed on each face each with orbital averaged power, Pi , j (1 ≤ j ≤ ni ). Therefore, each face has a total power generation from discrete component sources of
ni
∑P j =1
i, j
which is
assumed to be evenly distributed. The condition of uniform heat flux is addressed by the approach described in CHAPTER 7. Component heat gains occur on one side of the face while environmental heat loads and dissipation occur on the opposite side. Assuming negligible panel thickness, a surface energy balance yields, ni
∑P j =1
i, j
′′ ,i ⋅ Ai = ε i ⋅ σ ⋅ Ti 4 ⋅ Ai . + qenv
(6.2)
Now consider a structure consisting of f faces (1 ≤ i ≤ f ). The goal of the present work is to isothermalize this structure such that the temperature of each face is the same. Further, it is assumed that each face has the same optical properties; therefore, ni
∑P j =1
i, j
′′ ,i ⋅ Ai = ε ⋅ σ ⋅ T 4 ⋅ Ai . + qenv
(6.3)
At isothermal conditions, conductive and radiative coupling between faces can be ignored. Summing equation (6.3) for all faces yields f ⎛ ni ⎞ f ′′ ,i ⋅ Ai ) = ∑ ( ε ⋅ σ ⋅ T 4 ⋅ Ai ) . ⎜ ∑ Pi , j ⎟ + ∑ ( qenv ∑ i =1 ⎝ j =1 i =1 ⎠ i =1 f
(6.4)
Rearranging equation (6.4), provides ⎛ ni ⎞ f ′′ ,i ⋅ Ai ) P ⎜ ∑ i , j ⎟ + ∑ ( qenv ∑ i =1 ⎝ j =1 ⎠ i =1 = ε ⋅σ ⋅ T 4 . f ∑ Ai f
i =1
(6.5)
74 For the i th face equation (6.3) becomes, ni
∑P
i, j
j =1
′′ ,i = ε ⋅ σ ⋅ T 4 . + qenv
Ai
(6.6)
Setting equations (6.5) and (6.6) equal to one another yields,
∑P
i, j
j =1
Ai
⎛ ni ⎞ f ′′ ,i ⋅ Ai ) P ⎜ ∑ i , j ⎟ + ∑ ( qenv ∑ 1 i =1 ⎝ j =1 i = ⎠ . = f ∑ Ai f
ni
′′ ,i + qenv
(6.7)
i =1
The right-hand-side of equation (6.7) is an average heat flux for the entire satellite considering total environmental and component heat loads for all faces. Setting the righthand-side equal to q′′gain , this equation is simplified and becomes ni
∑P j =1
Ai
i, j
′′ ,i = q′′gain . + qenv
(6.8)
The left-hand-side of the same equation is the average heat flux for a given face. This balance is based on an assumption of isothermality; departure from this balance will generate temperature non-uniformities across the structure. Assuming environmental heat sources for any face and panel size are constant, all variables in equation (6.7) are fixed except for the distribution of components. f
The goal of the present work is optimizing the distribution of N = ∑ ni components i =1
over the f faces such that the heat flux resulting from environmental and component sources approaches the average flux value of the entire satellite. That is, we want to identify the distribution of components over the discrete panels such that, 2 ⎡ ⎛ ⎛ ni ⎞⎞ ⎤ f ⎢ ⎜ Pi , j ⎟⎟ ⎥ . min ⎢ ∑ ⎜ q′′gain − ⎜ ∑ = j 1 ⎜ ′′ ,i ⎟⎟ ⎟ ⎥ ⎜ A + qenv ⎢ i =1 ⎜ ⎟ ⎥ i ⎝ ⎠⎠ ⎦ ⎣ ⎝
(6.9)
75 This discrete problem quickly becomes unwieldy as the number of panels and components grow. The number of possible panel-component combinations is O( f N ) . For a six-sided satellite with 36 components, this becomes ~1.0E28 potential combinations. One class of methods that appears suitable for handling this problem is bin-packing methods. 6.4 Bin-Packing Problems Bin-packing problems involve placing objects of different values (e.g. volumes) into a finite number of bins. They are an extension of traditional packing problems that involve determining the optimal layout of a certain set of objects placed within a larger container (usually 2 or 3 dimensions). Traditionally, the goal of bin-packing is to place the objects in order to minimize the number of bins used. As with traditional packing, application of bin-packing is broad and diverse. It has been used in conjunction with genetic algorithms for optimal loading of multiple parts into the build cylinder of a rapid prototyping machine (Lewis et al., 2005). It has been used in operations research for line balancing (Falkenauer and Delchambre, 1992). Line balancing involves taking a given set of tasks of various lengths and constraints (i.e. some tasks require other tasks to be completed first) and determining how the tasks should be distributed over workstations (i.e. assembly line) so that cycle times are minimized. Another area where bin-packing is utilized is in production scheduling. Traditionally, these methods are used to optimize such tasks as storage of goods or allocation of products, but other approaches focus on the quality of a product. In one work, binpacking was utilized to allocate different part types in an ion plating batch process in order optimize the quality of coating (Chan et al., 2007). In general, these problems fall in the category of NP-hard (Békési, Galambos, and Kellerer, 2000). Due to the difficulty of solving packing problems, several solution methods exist. Many of these focus on one-dimensional bin-packing problems. The goal of the onedimensional bin-packing problem is to pack a given set of items with different sizes into a minimum number of equal-sized bins. Several methods have been developed to solve this type of problem including heuristic approaches.
76 Several heuristics exist which provide placement strategies that can provide satisfactory results. The main strategies include next-fit, first-fit, best-fit, and worst-fit strategies (Scholl, Klein, and Jürgens, 1997), which can be categorized by class or the order in which items are packed. On-line algorithms process information item by item. That is, the algorithms do not have any knowledge of the order in which the items will be processed. Conversely, off-line algorithms first build an order of the items before processing. Order can be simply non-increasing or non-decreasing order of value (i.e. weight or size) (Scholl, Klein, and Jürgens, 1997). In the next-fit strategy, the first item is packed into the first bin. Subsequent items are placed in this bin until the remaining capacity is insufficient for placing the next item. The next item is placed into the next bin and the process is repeated. The off-line and on-line next-fit algorithms have time complexities of O( n ) and O( n log n ) , respectively. The first-fit strategy places each subsequent item to a partially filled bin with the smallest index with sufficient capacity. If not available, it is placed into a new bin. The on-line and off-line first-fit heuristics can both take time O( n log n) . In the best-fit approach, each subsequent item is placed in the partially filled bin with the smallest sufficient residual capacity. When multiple bins have equal residual capacities, the item is placed in the bin with the smallest index. As with first-fit heuristics, the time complexity for the on-line and off-line best-fit versions is
O( n log n ) . The worst-fit algorithm is the opposite of the best-fit approach, where items are placed in the partially filled bin with the largest sufficient residual capacity. The time complexity of the worst-fit heuristics is the same as with best-fit methods. The minimum bin slack approach attempts to find a set of items, which can be placed into a bin most efficiently (Gupta and Ho, 1999). This has also been referred to as the subset sum heuristic (Caprara and Pferschy, 2004). Fleszar and Khalil (2002) developed several derivatives including a relaxed, perturbation, and sampling minimum bin slack method. Most published algorithms have computational requirements of O( n log n ) , but much work has been done to reduce these time complexities. These include a group fit heuristics approach (Johnson, 1974), on-line heuristics (Ramanan et al., 1989), and the H4
77 and H7 approaches of Martel (1985) and Békési, Galambos, and Kellerer (2000), respectively. Other alternative approaches include branch and bound procedures (Martello and Toth, 1990) and a hybrid procedure, consisting of several heuristics, reduction procedures, and branch and bound procedures (Scholl, Klein, and Jürgens, 1997). In addition, linear programming approaches have been utilized to solve cutting stock problems (Valério de Carvalho, 2002). Another approach is to utilize genetic algorithms in combination with heuristic approaches. Iima and Yakawa (2003) developed a genetic algorithm, which utilizes the first-fit and minimum bin slack method into crossover operations. Results showed that the accuracy of the methodology was better than the approaches developed by Fleszar and Khalil (2002) and Scholl et al. (1997). Other genetic algorithms developed for binpacking include those of Chan et al. (2007), Falkenauer and Delchambre (1992), and Lewis et al. (2005). Other approaches have been developed for packing problems with unequally-sized bins. The variable sized bin-packing problem is NP-hard and occurs when bins of different capacities are available for packing a set of items. Correia, Gouveia, and Saldanha-da-Gama (2008) provide a literature review of variable sized bin-packing problems. Although past work on this subject is relatively scarce, they found work which minimized the total capacity of the bins used (Friesen and Langston, 1986), greedy algorithm approaches (Kang and Park, 2003), branch and bound approaches (Monaci, 2002), and column generating techniques for both one-dimensional (Belov and Scheithauer, 2002; Alves and de Carvalho, 2007) and two-dimensional problems (Pisinger and Sigurd, 2005). The problem presented here is a variable sized bin-packing problem where bin size is proportional to face area (Figure 6.12) with two significant differences. First, ‘overflow’ (i.e. filling a bin past q′′gain ) is allowed. Thermal penalties are the same if a bin is over- or under-filled (i.e. too hot or too cold, respectively). Second, the resulting heat flux contribution of a component is dependent upon which bin it is placed. For example, an item placed on a larger face will yield a smaller heat flux than if it was placed on one that
78 was smaller. As a result, power for a given component is ‘poured’ into a given bin as shown in Figure 6.12.
Figure 6.11: Illustration of Bin-Packing Methodology for Optimized Component Distributions. The problem considered in this chapter involves determining the best approach to filling bins using a discrete number of components. The solution produces the order and location of components that are ‘poured’ into the bins. To solve this unique problem, a genetic algorithm approach with heuristics was developed. 6.5 Genetic Algorithm Genetic algorithms (GA) are often utilized as a tool for determining a global optimum solution. GAs are stochastic optimization methods inspired by natural evolution where individuals represent a feasible solution in the search space. GAs require three basic processes: encoding, fitness evaluation, and evolution. 6.5.1 Encoding Encoding is the process by which a particular potential solution is represented. In keeping with the vernacular of a genetic algorithm, a chromosome is utilized. For this particular problem, the chromosome (i.e. solution) represents the placement of all components. The chromosome for a particular solution with n components is portrayed in Figure 6.12.
Figure 6.12: Chromosome Representation.
79 The population of solutions for any generation, g , consists of popmax number of unique chromosomes. The goal is to determine which of these chromosomes provides the best solutions to the problem and how best to create new generations of solutions given an initial population. 6.5.2 Fitness Evaluation A fitness function assigns a figure of merit to each encoded solution (i.e. chromosome). For the problem at hand, the figure of merit is based on the distribution of components over the available faces. That is, a fitness function, fit , was created which calculates the sum-of-squares between the average heat flux resulting from the f environmental and N component heat sources over all satellite surfaces and the average flux on an individual face based on the orbital averaged environmental flux along with the components placed upon it. Thus, the fitness function becomes 2
⎛ ⎛ ni ⎞⎞ f Pi , j ⎜ ′′ ∑ ⎜ ⎟⎟ . fit = ∑ ⎜ qgain − j =1 ⎜ ′′ ,i ⎟⎟ ⎟ i =1 ⎜ ⎜ A + qenv ⎟ i ⎝ ⎠⎠ ⎝
(6.10)
Equation (6.10) shows that fitness is a function of the environmental heat sources on each panel along with panel dimensions and component powers. Studies were conducted to determine the relationship between fitness and thermal performance. At each of three discrete maximum component numbers ( N = 18,36,54 ), uniformly powered components (Case D) totaling 300 W were randomly distributed over the satellite previously described in the case study. For each unique distribution, fitness, maximum temperature difference between faces, and maximum/minimum face temperature were determined using the model for the hot-case orbit study. The results of these 120 tests are summarized in Figure 6.13 and Figure 6.14. These figures clearly show a proportional relationship between thermal performance and fitness values. For this particular example, the figures show that the proposed fitness value is a good measure of thermal performance in terms of uniform heat dissipation and achieving an isothermal structure.
80
Figure 6.13: Maximum Temperature Difference versus Fitness for Increasing Number of Components.
Figure 6.14: Maximum and Minimum Temperature versus Fitness for Increasing Number of Components. 6.5.3 Evolution Evolution is a crucial component in which new generations of solutions are developed and influences the efficiency by which improved solutions are discovered. For a given generation, g , the number of potential solutions (i.e. chromosomes) is equal to
81 popmax . Using these solutions, various evolutionary techniques are used to populate the
next generation, g + 1 . Methods borrowed from biological and sociological concepts were implemented that include elitist strategies, reproduction, and mutation. In addition, best-fit and worst-fit techniques were used to improve the capability of finding optimum solutions. Figure 6.15 illustrates the process of creating a new generation from the previous one utilizing these five techniques. The following subsections describe these techniques in more detail.
Figure 6.15: Illustration of Evolution Techniques Used to Obtain a Fully Populated New Generation of Solutions. 6.5.3.1 Elitist Strategies Elitist strategies involve copying some of the best chromosomes from one generation to the next. They have an advantage over traditional reproductive genetic algorithms in that the best solution is guaranteed to be monotonically improving. For any given generation, the chromosomes are sorted from best to worst according to their normalized fitness function. Then, the top Pb percent of a population is selected and carried over to the next generation. 6.5.3.2 Reproduction The next Pr percent of any given generation are created utilizing reproduction techniques. Reproduction occurs when two parent chromosomes are selected and are utilized to create an offspring. The methods for parent selection vary considerably and
82 can include both random and non-random algorithms. A biased roulette wheel (i.e. proportional selection) approach similar to the methods of Leung, Chan, and Troutt (2003) is utilized in the present study. The goal of this method is to determine a relative fitness, fitrel ,s , for each chromosome (1 < s < popmax ) in a population as illustrated by fitrel ,s =
1 − fits popmax
∑ 1 − fit i =1
.
(6.11)
i
In this approach, normalized fitness for each chromosome in a population is determined. The chromosomes are then sorted from best (lowest) to worst (highest) normalized fitness. Each of the normalized fitness values are subtracted from one to ensure that lower (i.e. more optimized) values receive a higher weight. Each resulting value is divided by the sum of all values to obtain relative fitness. Then the interval [0,1] is partitioned into n subintervals each equal to the relative fitness of a chromosome. Thus, chromosomes with the best fitness (i.e. lowest value) maintain the largest subintervals while those with the worst fitness maintain relatively small subintervals. Upon completion of this step, two random numbers [0,1] are chosen (one for each parent). Chromosome selection is based on the subinterval a random number falls into. Thus, this method favors those chromosomes with higher fitness for parent selection. Once the parents are selected, a random integer [1, 2 ⋅ n ] is generated to select a splice point for each parent. Thus, all genes to the left of this point for the first parent and those to the right for the second parent are combined to create a child (Figure 6.16).
Figure 6.16: Illustration of Reproduction Methods.
83 As previously discussed, the creation of each child by this method is biased towards parents of better fitness. The remaining population of each new generation is obtained through mutation techniques. 6.5.3.3 Best- and Worst-Fit Placement Best- and worst-fit placement are heuristics used to place a given order of components into available bins (i.e. face). In this algorithm, Pbf and Pwf percent of the next generation are created using best-fit and worst-fit methods, respectively. For each, a queue consisting of a randomly generated order is used to determine the next component available for placement. Best-fit placement puts the component in the bin such that the resulting flux for that bin is closest to the average for the structure. Worst-fit placement puts a component in the bin such that the resulting flux for that bin is smallest. Best- and worst-fit heuristics are also utilized to generate four solutions to the initial population. These include ascending-order best- and worst-fit placement along with descending-order best- and worst-fit solutions. 6.5.3.4 Mutation Mutation is the process by which random alterations to genetic materials are allowed to pass onto future generations. This technique is critical in ensuring that the algorithm does not become stuck in local minima. If all chromosomes of a population are similar, reproduction and local-gradient search techniques will only provide similar solutions at each iteration. Mutation allows for drastic changes in chromosomes. In this algorithm, Pm percent of the next generation is created using mutations which are created by randomly generating new chromosomes. It should be noted that Pb + Pr + Pbf + Pwf + Pm = 1 . 6.6 Genetic Algorithm Tuning The ability of the genetic algorithm to find an optimal solution and the required computational effort both depend on parameters of the algorithm. As a first step in investigating algorithm performance, both stopping criteria and population size were examined.
84 6.6.1 Analysis of Convergence Criteria and Population Size The algorithm populates successive generations until convergence. The algorithm compares the best solution for a given generation to one obtained in the previous generation. If the difference is less than 10-4, a counter is incremented by one and a new generation is created. This tolerance is small enough to capture the redistribution of a component that is 0.01% of total component power over a 6 m2 satellite (e.g. the case study structure). If not, the counter is reset to zero. Convergence criteria are reached when a given best solution is not improved beyond the tolerance over end max generations. An experiment was conducted to determine the effect of end max on both resulting fitness values and the time required to achieve these values. The algorithm was tested at three unique component power distributions (Case A, Case C, and Case D) and 11 different values of end max (1, 10, 25, 50, 100, 250, 500, 1000, 2500, 5000, and 10000). In addition, population size, popmax , was chosen to be a linear function of the number of components, m ⋅ N . Values of m included 1, 3, 5, 7, and 9. At each combination of parameters, 10 runs were conducted which would provide insight into any variability between runs. Consequently, 1,650 different algorithm runs were completed. Fitness values and computational time were plotted versus end max as shown in Figure 6.17 through Figure 6.19 and versus the linear multiplier, m , in Figure 6.20 through Figure 6.22. As these figures show, increasing end max had little impact on fitness. Only for Case C-18 did it have any noticeable effect and even in this case the improvement was ~0.2%. Increasing m did not have any noticeable impact on fitness. However, increasing these parameters significantly increased required computational time. As a first approach, setting end max and m equal to 1 would seem appropriate. For the cases that were analyzed, this would provide optimal results except for Case C-18 with minimized computational time. However, this limits the optimization routine to a simple rule-based approach and the full capability of the algorithm is not utilized. Although this appears appropriate for the cases considered here, it might not be for cases not considered.
85
Figure 6.17: Fitness and Computational Time versus Ending Criteria for Component Distribution A with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case A-18, Case A-36, Case A-54).
Figure 6.18: Fitness and Computational Time versus Ending Criteria for Component Distribution C with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case C-18, Case C-36, Case C-54).
86
Figure 6.19: Fitness and Computational Time versus Ending Criteria for Component Distribution D with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case D-18, Case D-36, Case D-54).
Figure 6.20: Fitness and Computational Time versus Population Multiplier for Component Distribution A with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case A-18, Case A-36, Case A-54).
87
Figure 6.21: Fitness and Computational Time versus Population Multiplier for Component Distribution C with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case C-18, Case C-36, Case C-54).
Figure 6.22: Fitness and Computational Time versus Population Multiplier for Component Distribution D with 18, 36, and 54 Total Components in the Hot-Case Design Orbit (Case D-18, Case D-36, Case D-54). Consequently, the results were analyzed with the aim of finding a combination of these parameters, which had a relatively low computational expense and still allowed the utilization of all capabilities of the algorithm. This was achieved by investigating the interaction between end max and m . For example, small end max combined with large m values could act similarly to a combination of large end max and small m . Case C-18 was
88 examined to determine the combination of parameters that most consistently found the lowest fitness values with minimum computational expense. Results showed that a combination of end max = 2500 and popmax = 1 provided good performance at reasonable computational expense. Over 10 runs, this combination consistently (10 of 10) found solutions better than the heuristic solutions. On average, computational expense was 1.9 seconds with maximum and minimum run times of 3.0 and 0.8 seconds, respectively. It should be noted that for the cases considered, a strictly rule-based approach provided reasonable results for the least computational expense. 6.6.2 Evolution Parameter Tuning A study was conducted to determine the effect of evolution parameters (i.e. Pb , Pr , Pbf , Pwf , Pm ) on performance and computation requirements. For Case C-18 at
300 W of total component power in the hot-case environment, each evolution parameter was varied while the remaining parameters were set equal to each other. At each combination of parameters, 10 runs were conducted which would provide insight into any variability between runs. At discrete values of the evolution parameters, the resulting converged fitness and computation time requirements were recorded and plotted versus one another (Figure 6.23 through Figure 6.27).
Figure 6.23: Normalized Fitness versus Computation Time for Increasing Values of Elitism Parameter Pb .
89
Figure 6.24: Normalized Fitness versus Computation Time for Increasing Values of Reproduction Parameter Pr .
Figure 6.25: Normalized Fitness versus Computation Time for Increasing Values of BestFit Parameter Pbf .
90
Figure 6.26: Normalized Fitness versus Computation Time for Increasing Values of Worst-Fit Parameter Pwf .
Figure 6.27: Normalized Fitness versus Computation Time for Increasing Values of Mutation Parameter. Pm The results provide little evidence that general rules of thumb can be established for evolution parameter values that provide the best results. It does appear that the highest values of the elitist, reproduction, and best-fit parameters do not lead to as good results as lower values. It also appears that an even distribution of these parameters (i.e. 0.2 each) provides reasonable fitness values with low computational time. It should be noted that fitness value differences are relatively small (~ 0.2%) and therefore, even appropriately
91 tuned parameters have little effect on solutions. Consequently, an even distribution of these parameters was deemed appropriate. 6.7 Algorithm Demonstration To demonstrate the tuned algorithm, the Case C distribution for 18, 36, and 54 components was applied to the previously described case study parameters in the hot-case thermal environment. Since the solution algorithm is stochastic in nature, the tuned algorithm was run 99 times in order to provide insight into any variability between runs. Results from the 99 runs are shown in Table 6.4. Computational requirements, on average, are expected to be approximately 12 seconds or less for up to 54 components using a 2.5 gigahertz dual-core computer. The maximum computational requirements are expected to be less than approximately 22 seconds for up to 54 components using the same machine. The thermal performance was also evaluated over the 99 runs. For the maximum, minimum, and median fitness results, the maximum temperature difference between components was evaluated utilizing Thermal Desktop® and summarized in Table 6.4. Thermal performance results varied less than 0.3 K between maximum, minimum, and median fitness values. Table 6.4: Case C Computational Time and Maximum Temperature Difference Results for Optimized Distributions of 18, 36, and 54 Components at 600 W of Total Power. Case
Case C-18 Case C-36 Case C-54
Computational Time Maximum Minimum Average [seconds] [seconds] [seconds] 4.2 14.1 22.3
0.9 4.2 11.8
1.6 7.0 12.3
Maximum Temperature Difference* Maximum Minimum Median [ΔK] [ΔK] [ΔK] 8.2 8.0 8.1
7.9 8.1 8.1
8.2 8.1 8.1
*Based on maximum, minimum, and median fitness numbers Optimized distributions were compared to even and worst-case distributions for the Case C component distribution for 18, 36, and 54 components in the hot-case environment for 600 W of total component power. An even distribution was generated by running the optimization algorithm assuming uniform environmental fluxes on all faces. A worst-case distribution was obtained by applying all component power to the +Z face.
92 The resulting thermal performance for these cases is also shown in Table 6.5. The optimized distribution provided a reduction in maximum temperature difference of 12.8 to 13.1 K over even and 84.3 to 84.9 K over worst-case component distributions. Note that due to slight variations in total power between cases, the worst-case distribution results are not constant. Table 6.5: Case C Results for Optimized, Even, and Worst-Case Component Distributions at 600 W of Total Power. Case
Case C-18 Case C-36 Case C-54
Maximum Temperature Difference Optimized Even Worst-case Distribution Distribution Distribution [ΔK] [ΔK] [ΔK] 8.2 21.3 92.5 8.1 21.0 93.0 8.1 20.9 92.8
Temporal variation of temperature for the 36-component case is shown in Figure 6.28 through Figure 6.30.
Figure 6.28: Case C-36 Temporal Results for Optimized Component Distributions at 600 W of Total Power in the Hot-Case Orbit.
93
Figure 6.29: Case C-36 Temporal Results for Even Component Distributions at 600 W of Total Power in the Hot-Case Orbit.
Figure 6.30: Case C-36 Temporal Results for Worst-Case Component Distributions at 600 W of Total Power in the Hot-Case Orbit. 6.8 Effect of Increasing Component Power A study was conducted to determine the effect of increasing total power on optimized results. Optimized, even, and worst-case distributions for Case C-36 in the hot-
94 case orbit were found for total power of 100 W up to 1200 W. The resulting maximum and minimum temperatures were generated utilizing Thermal Desktop® and shown in Figure 6.31. On average, optimized component distributions reduced maximum temperatures by 5.4 K over evenly distributed components. Maximum and minimum temperature reductions were 8.2 K (at 400 W) and 2.8 K (at 100 W), respectively. Optimizing component distribution increased minimum temperatures by 7.1 K, on average, over evenly distributed components. Maximum and minimum temperature increases were 9.3 K (at 400 W) and 5.7 K (at 1200 W), respectively.
Figure 6.31: Maximum and Minimum Temperature versus Total Power for Optimized, Even, and Worst-case Component Distributions for Case C-36 in the Hot-Case Thermal Environment. The resulting maximum temperature differences were generated utilizing Thermal Desktop® and shown in Figure 6.32. On average, optimized component distributions reduced maximum temperature differences by 12.6 K over evenly distributed components. The largest and smallest maximum temperature difference reductions were 17.8 K (at 400 W) and 9.5 K (at 100 W), respectively.
95
Figure 6.32: Maximum Temperature Difference versus Total Power for Optimized, Even, and Worst-case Component Distributions for Case C-36 in the Hot-Case Thermal Environment. 6.9 Conclusions A computational tool was developed which provides rapidly optimized component distributions over multi-sided structures each with unique environmental loading. The tool is based on a genetic algorithm utilizing a combination of elitist strategies, reproduction, best- and worst-fit heuristics, and mutation. A tuning study revealed appropriate convergence criteria, population size, and values for evolution parameters. Computational requirements are small; on average, approximately 12 seconds was required using a 2.5 gigahertz dual-core computer for 54 components over a six-sided structure. The maximum computational requirements are expected to be less than approximately 22 seconds for up to 54 components using the same machine. Optimized, even, and worst-case distributions for Case C-36 in the hot-case orbit were found for total power of 100 W up to 1200 W. On average, optimized component distributions reduced maximum temperatures by 5.4 K over evenly distributed components. Optimizing component distribution increased minimum temperatures by 7.1 K, on average, over evenly distributed components. Finally, optimized component distributions reduced maximum temperature differences by 12.6 K, on average, over
96 evenly distributed components. The largest and smallest maximum temperature difference reductions were 17.8 K (at 400 W) and 9.5 K (at 100 W), respectively.
97
CHAPTER 7 - OPTIMAL PLACEMENT OF ELECTRONIC COMPONENTS TO MINIMIZE HEAT FLUX NON-UNIFORMITIES
The methodology outlined in CHAPTER 6 was used to identify the specific satellite face that each component must be placed for optimized thermal performance. However, optimal locations within each face were not provided. The current chapter details a local placement algorithm used to determine specific component locations within each face. Although developed as an integral part of a satellite component placement optimization tool, this approach has other applications. Electronic devices have permeated almost every aspect of modern society. Examples are numerous and include those that are conspicuous (e.g. computers) to those that are not (e.g. satellites). Because of our ever-increasing dependence on the services they provide, we expect future electronic products to have improved reliability and increased performance. Developing reliable electronic components requires placing them within an acceptable thermal environment. This challenge is compounded by the fact that we require more capable and therefore more powerful devices in ever-decreasing package sizes. Consequently, both power and power density values have consistently increased; several studies have indicated that microprocessor power doubles approximately every three years (Mahajan et al., 2000; Mahajan et al., 2002; Krishnan et al., 2007; Moore, 2003) while satellite raw power doubles every five to six years (Hoeber and Kim, 2000). Realizing acceptable thermal conditions can be achieved through a combination of technology and design. For instance, Young et al. found that a robust satellite thermal control subsystem (TCS) could be achieved, in part, with an isothermal bus. The isothermal bus concept is a novel idea focused on approaching a single thermal node satellite representation. In effect, heat is shared efficiently between cold and hot components and as a result, gradients across the satellite are minimized, thus reducing temporal variations in temperature. This concept could be achieved in numerous ways
98 including embedded heat pipes, integrated pumped fluid loops, and integrated high conductivity face sheets such as annealed pyrolytic graphite (APG). One method that has not received much attention is simply optimized component placement; however, it has been studied for a wide-range of other applications. 7.1 Review of Component Placement Optimization Methods Quinn and Breuer (1979) provide one of the earliest attempts at optimized component placement for printed circuit boards (PCBs). In their study, they utilized a Hooke’s Law-like attractive and repulsive force between components. A solution routine was utilized to find component placement that approaches net zero forces on each component. These components are then expanded to fit a discrete set of component placement locations. Huang and Fu (2000) expanded the conventional force-directed placement technique by Quinn and Breuer and renamed it the thermal force-directed method. Net forces between chips are related to the magnitude of the chip’s power dissipation (i.e. larger power dissipation results in larger repulsive forces). Simultaneous linear equations are solved to determine equilibrium locations for thermal placement. The objective is to reduce heat density and evenly distribute power dissipation. They found their method produced better results than other methods including the quadrisection (QD) method of Lee and Chou (1996). Lee and Chou focused on tradeoffs between optimizing wireability and reliability. A hierarchical design was utilized. Components with higher heat dissipation rate were evenly distributed using a recursive quadrisection algorithm. This helped reduced maximum temperature and temperature variation on the substrate. A force-directed algorithm was then used to optimize wireability. Some studies have focused on discrete component locations. Although these optimization problems could be handled with combinatorial optimization methods, this becomes intractable (i.e. large computational effort) with increasing component numbers. Eliasi, Elperin, and Bar-Cohen (1990) utilized two stochastic heuristic optimization methods (simulated annealing and cluster optimization) in order to optimally place up to 100 components amongst a fixed number of discrete locations on a PCB. A few works have proposed using Genetic Algorithms (GAs). Jeevan et al. (2005) were the first to utilize GAs as an optimization tool for chip placement in multi-chip
99 modules (MCMs) and PCBs. They used a coarse finite element meshing scheme to calculate the temperature distribution on MCM and PCBs along with a GA to find the optimal position of power generating components/chips. Their discretized approach limited placement locations (non-continuous); but is based on physical properties (i.e. the GA fitness value is temperature). They found they were able to obtain better thermal distribution over QD approaches. In addition, their GA optimization runs required 2-4 minutes of CPU time (600MHz Pentium) to converge. Madadi and Balaji (2008) used GAs to find optimal placement of three discrete heat sources that could be placed continuously within a ventilated cavity and cooled by forced convection. The solution scheme utilized artificial neural networks using a Bayesian regularization algorithm to predict fitness while the GA was used to find optimal location. The capability of the GA was tested using a Rastrigin’s function. In a study by Suwa and Hadim (2007), component placement about a convectively cooled PCB is optimized simultaneously for thermal and electrical performance. Component junction temperatures are predicted using artificial neural networks (ANNs) combined with superposition methods. Two GAs are implemented in a cascade for efficient operation. The first is a coarse thermal optimization; the second is a fine thermal optimization. In this study, each component had a different size, thermal resistance, and amount of heat generation. In addition, the optimization scheme allowed for continuous placement across the domain. Queipo and Gil (2000) focused on optimal placement of conductively and convectively cooled electronic components subject to thermal (i.e. failure rate dictated by Arrhenius relation) and non-thermal (i.e. total wire length) optimization criteria. Temperatures and therefore failure rate was estimated using a nodal heat balance approach. Some approaches were based heavily on specific thermal conditions (i.e. temperature, thermophysical properties) and provide good results. Others take a relatively long time to generate results. The present approach provides a fast method for determining reasonable component placement that approaches a uniform distribution of heat flux without the use of thermophysical properties. This approach is especially useful in situations where limited or no thermophysical and environmental conditions are
100 readily available for the problem at hand and can be utilized in a variety of industries including microelectronics and satellite development. 7.2 Numerical Experiment Investigations To investigate the advantages of optimized component placement, simple numerical experiments were conducted. The focus was to evaluate the merits of rearrangement of components within a common satellite architecture panel. The remainder of this section details the numerical model used in these experiments, experimental approach, and subsequent results. 7.2.1 Numerical Model Details To investigate the advantages of optimized component placement, a case study was developed consistent with current robust satellite architectures (Arritt et al., 2008). A single 1.0 m x 1.0 m honeycomb panel was used with 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material. Material properties were based on Gilmore (2002). Multiple heat-generating components, each with a contact area of 10 cm x 10 cm, were attached to one side of the panel. Heat generation of all components totaled 100 W and was conducted to the panel only (i.e. not lost through radiation) with a contact conductance of 110 W/m2-K (Gilmore, 2002). All heat generation was ultimately dissipated to a deep space environment (0 K) through radiation from the bottom surface of the panel with an emissivity of 1.0. The case study was simulated using Thermal Desktop® as a CAD based interface to the SINDA/FLUINT finite-difference thermal analyzer. Thermal Desktop® and SINDA/FLUINT are widely accepted tools for spacecraft thermal design. All computational analysis was done using a 2.5 gigahertz dual-core processor. 7.2.2 Component Placement versus Thermal Conductivity To illustrate the advantages of optimized component placement, a simple parametric study was developed. Four evenly-powered components were symmetrically placed on a panel at varying offset values. Offset, defined as the distance each component symmetrically travels from the center, was increased from 0.00 m up to 0.40 m (Figure 7.1). For each placement, the thermal conductivity of the panel facesheets was
101 varied from 100 up to 10,000 W/m-K while keeping the remaining properties unchanged. Due to symmetry, all components maintained the same temperature, but the maximum component temperature varied because of different placements and panel thermal conductivities. The objective is then to minimize local hot spots (i.e. component temperatures) in order to approach isothermal conditions.
Figure 7.1: Illustration of Component Placement for Offset Values of a) 0.00 m; b) 0.10 m; c) 0.20 m; d) 0.30 m; and e) 0.40 m. Results of this parametric study show maximum component temperature at steady state for varying levels of offset and thermal conductivity (Figure 7.2). Figure 7.3 provides the same results over a smaller range of temperature and conductivity. These figures show that reduced maximum component temperatures can be achieved through improved component placement, increased thermal conductivities, or a combination of the two. The advantage of the approach presented here is the relatively low cost in implementation as opposed to other approaches that are technology based.
Figure 7.2: Maximum Component Temperature at Steady-State for Offset Values of 0.00 m to 0.35 m and Thermal Conductivities of 100 to 10,000 W/m-K.
102
Figure 7.3: Magnified Maximum Component Temperature at Steady-State for Offset Values of 0.00 m to 0.35 m and Thermal Conductivities of 100 to 2,700 W/m-K. 7.3
Methodology Consider a rectangular domain with dimensions w and h upon which n
components are placed. Each component has a distinct averaged power, Pj which is dissipated to the domain only (i.e. not radiated to the surroundings). Therefore, the average heat flux, q′′ , of all components over the domain is found from n
q′′ =
∑P j =1
j
w⋅h
.
(7.1)
In order for each component to achieve this average heat flux, it must evenly distribute heat over an area proportional to its power. This effective area, Aj , is calculated for each component by Aj =
Pj . q′′
(7.2)
The sum of these effective areas equals the area of the rectangular domain as described in the following:
103 n
Pj Pj ∑ j =1 = w⋅h . Aj = ∑ = ∑ q′′ j =1 j =1 q′′ n
n
(7.3)
The effective areas are assumed circular in shape since heat should be spread evenly in all directions. Therefore, this relationship is illustrated in Figure 7.4 for a threecomponent system.
Figure 7.4: Illustration of Relationship Between Rectangular Domain and Resulting Effective Circular Area. The effective radii of each effective area is found by
rj =
Aj
π
.
(7.4)
The goal of the present work is placing the effective areas (i.e. components) within the rectangular domain in a manner that evenly spreads heat flux. For an arbitrary placement within the domain, intersections result (i.e. overlapping green areas) between effective areas and between an effective area and the area outside of the domain (Figure 7.5).
Figure 7.5: Arrangement of Components with Circular Effective Areas in a Domain. An optimum arrangement would be one in which no intersection is encountered and therefore the average heat flux over the entire domain is uniform. Due to the circular
104 shape of the effective areas, this is not possible and thus the challenge becomes how to arrange these components such that the area of intersection is minimized. The inspiration for the solution to this problem was found in packing problems. 7.4 Packing Problems Packing problems involve determining the optimal layout of a certain set of objects placed within a larger container (usually 2 or 3 dimensions). The container and objects can take a variety of shapes including circles, squares, rectangles and other irregular shapes. The goal of these problems is typically to maximize packing efficiency (packing density). The application of packing problems is broad and diverse. In some industries (textile, glass, wood, paper, etc.), packing equal or unequal circles into a two dimension rectangular container without overlap is often encountered. The objective is to maximize material utilization (i.e. minimize scrap) and becomes an NP-complete problem (Huang et al., 2005). Other applications include: placing radio towers in a geographical region such that the coverage is maximal with minimal interference, determining the smallest box necessary to pack a given number of bottles and planting trees in a given region as to maximize foliage density but also allowing the trees to grow up to their maximum size (Szabó et al., 2007) In three dimensions, these types of problems have medical applications. Given a set of spheres and a three-dimension bounded region, fill the space with a minimal set of spheres and maximizing the occupied volume. This NP-hard problem coincides with the Gamma Knife treatment of tumor and vascular malformations within the head (Li and Ka-Lok, 2003). The literature shows several types of packing problems. One of the most prevalent is circle packing in either a circular or square domain. This NP-hard problem, typically involves situations upon which overlap is not allowed. Markót (2004) developed a new interval branch-and-bound algorithm that was used to solve the previously unsolved problem of packing 28 equal circles in a unit square without overlap. This was extended when Markót and Csendes (2005) utilized interval methods to determine the optimal packing of 28, 29, and 30 equal circles in a square
105 domain without overlap. Markót (2007) later completed the optimality proofs for these cases by determining all optimal solutions with mathematical rigor. For an exhaustive overview of circle in square packing without overlap, Szabó et al. (2007) provide an exhaustive overview. The placement of equal circles would result if all components had equal power values. More appropriate for the problem at hand (i.e. component placement) involves placing unequal circles in a domain. For unequal circle packing, several works are available. Wang et al. (2002) utilized an adaptive algorithm of the Tabu search for determining optimum packing of unequal circles in a larger containing circle without overlap. This method was improved by Huang and Chen (2006) by accelerating the search process. Addis, Locatelli, and Schoen (2008) approached this problem by mixing local and global optimization strategies with random search and local moves. Rule based methods are also prevalent. Huang et al. (2006) proposed two heuristics in order to solve the problem. In addition, Huang et al. (2005) proposed two greedy algorithms for solution of placing unequal circles in a rectangular container: a maximum-hole degree rule and self-look-ahead search strategy. One of the significant challenges of these types of problems is getting ‘trapped’ in local minima. Xu and Ren-Bin (2006) approached this problem by periodically reinitializing circle positions to avoid being trapped in these local minimums and utilized simulated annealing algorithms as an acceptance policy. Experimental results showed that their approach achieved good performance in terms of solution quality and computational time. Lü and Huang (2008) also used a simple heuristic (e.g. principle of maximum cave degree for corner-occupying actions) and coupled this to a Pruned– Enriched-Rosenbluth Method (PERM) algorithm. When looking ahead at all possible orders of packing the circles, PERM provides a strategy to prune the branches with low weights and enrich those with high weights. Other packing problems that have been investigated include: rectangles in a square (Caprara et al., 2006), irregular packing (Bennell and Dowsland, 1999 and 2001) spheres in a tetrahedron (Li and Ka-Lok, 2003) and bin-packing (Caprara, Lodi, and Monaci, 2005; Csirik et al., 2000).
106 Due to the difficulty of solving packing problems, several solution methods exist. One such method is interval analysis which is used extensively to solve the equal circle packing problem. Genetic algorithms have been utilized to determine the most efficient packing order for a set of unequal shapes. Combined with placement heuristics, such as ‘bottom left’ (BL), these approaches have attained good results. Hopper and Turton (1999) and Gonçalves (2007) used these methods for packing rectangular objects in a rectangular domain. Simulated annealing approaches also provide viable solution methodologies for these types of problems. Several works have combined the efforts of genetic algorithms with simulated annealing in order to prevent the premature convergence of the solution method. Leung, Chan and Troutt (2003) utilized these hybrid approaches for minimizing trim loss in cutting rectangular pieces from a rectangular domain. This approach was also taken by Soke and Bingul (2006) when they applied genetic algorithms, simulated annealing with an improved bottom left algorithm. A novel approach was developed by Mahanty et al. (2007). In this work they utilized both a genetic algorithm and a hybrid approach using a genetic algorithm and a local optimization algorithm based on a Coulomb potential model. The hybrid approach was found to give better results although computationally more expensive. 7.5 Genetic Algorithm As previously discussed, packing problems similar to the type encountered here are often difficult to solve due to the presence of local minima. As a result, genetic algorithms (GA) are often utilized as a tool for determining a global optimum solution. GAs are stochastic optimization methods inspired by natural evolution where individuals represent a feasible solution in the search space. GAs require three basic processes: encoding, fitness evaluation, and evolution. 7.5.1 Encoding Encoding is the process by which a particular potential solution is represented. In keeping with the vernacular of a genetic algorithm, a chromosome is utilized. For this particular problem any chromosome (i.e. solution) is a representation of the x j and y j
107 coordinates of each component. Since there are a total of n components, the chromosome for a particular solution is portrayed by the following:
Figure 7.6: Chromosome Representation. For the evaluations presented in this thesis, components were allowed to freely move to more optimized locations. However, in some instances some of the components will have fixed positions. This scenario is achievable using the algorithm presented here by simply fixing the x and y location for that particular component and optimizing the remaining components. The population of solutions for any generation, g , consists of
popmax number of unique chromosomes. The goal is to determine which of these chromosomes provides the best solutions to the problem and how best to create new generations of solutions given the initial population. 7.5.2 Fitness Evaluation A fitness function assigns a figure of merit to each encoded solution (i.e. chromosome). For the problem at hand, the figure of merit is directly related to the intersections amongst the effective areas of the components. That is, a fitness function,
fit , was created which calculates the intersection of the effective areas between all components. For two arbitrary effective circular areas, the fitness function is notated by the following:
fit[ Ai , Aj ] = Ai ∩ Aj .
(7.5)
Three distinct cases are isolated with this fitness function: 1) no intersection, 2) full intersection and 3) partial intersection. As expected, Case 1 occurs when two effective areas do not intersect. In this case the fitness function between these two components would be zero. Case 2 occurs when the smaller effective area lies fully inside the larger. In this case the fitness function would be equal to the smaller effective area. The first two cases were easily handled, but Case 3 provides some difficulty. Here, the two effective
108 areas have partial intersection. The intersection area was derived based on geometric principles. The resulting figure of merit, fit[ Ai , Aj ] , is shown as follows: 2 2 2 ⎛ d 2 + ri 2 − rj2 ⎞ 2 ⎛ ⎞ −1 d i − j − ri + rj cos Ai ∩ Aj = ri 2 ⋅ cos −1 ⎜ i − j r + ⋅ ⎟ ⎜ j ⎜ 2⋅d ⋅r ⎟ ⎜ 2 ⋅ d ⋅ r ⎟⎟ i− j i i− j j ⎝ ⎠ ⎝ ⎠. 1 ( − d i − j + ri + rj )( d i − j + ri − rj )(d i − j − ri + rj )(d i − j + ri + rj ) − 2
(7.6)
Extending this to n components required sweeping through and summing the fitness function for all possible pairs of effective areas. Therefore, the total fitness is found by cycling through all combinations of components; therefore, 2
1
fit[ A1 ,..., An ] = ∑ ∑ [ Ai ∩ Aj ] .
(7.7)
i = n j =i −1
The maximum fitness will occur when all effective circles lie coincident with one another in the domain corner. This is the limiting worst case and the intersection is given by n 3 n fitmax = ∑ (i − 1) ⋅ Ai + ⋅ ∑ Ai . 4 i =1 i =2
(7.8)
The fitness is then normalized by dividing by the maximum value
fitnorm =
fit[ A1 ,..., An ] . fitmax
(7.9)
After normalization, the maximum fitness becomes 1.0 while the minimum is a nonzero (due to the circular effective area assumption) number. Studies were conducted to determine the relationship between normalized fitness and thermal performance. At each of six discrete maximum component numbers ( n = 3,6,9,12,15,18 ), components were randomly placed over the honeycomb panel of the case study previously described. For each unique placement, normalized fitness, maximum temperature difference between components, and maximum component temperature were recorded. The results of these 240 tests are summarized in Figure 7.7 and Figure 7.8.
109
Figure 7.7: Maximum Temperature Difference versus Normalized Fitness for Increasing Number of Components.
Figure 7.8: Maximum Temperature versus Normalized Fitness for Increasing Number of Components. These figures clearly show a linear relationship between thermal performance and fitness values. This is especially apparent for larger numbers of components (i.e., larger values of n ). For this particular example, the figures show that the proposed normalized
110 fitness value is a good measure of thermal performance in terms of uniform heat dissipation and achieving an isothermal panel. 7.5.3 Evolution Evolution is a crucial component in which new generations of solutions are developed and influences the efficiency by which improved solutions are discovered. For a given generation, g , the number of potential solutions (i.e. chromosomes) is equal to
popmax . Using these solutions, various evolutionary techniques were implemented to populate the next generation, g + 1 . Methods borrowed from biological and sociological concepts include elitist strategies, reproduction, and mutation. In addition, steepestdescent gradient search techniques were used to improve the capability of finding optimum solutions. Figure 7.9 illustrates the process of creating a new generation from the previous one utilizing these four techniques. The following describes these techniques in more detail.
Figure 7.9: Illustration of Evolution Techniques Used to Obtain a Fully Populated New Generation of Solutions. 7.5.3.1 Elitist Strategies Elitist strategies involve copying some of the best chromosomes from one generation to the next. They have an advantage over traditional reproductive genetic algorithms in that the best solution is guaranteed to be monotonically improving. For any given generation, the chromosomes are sorted from best to worst according to their normalized
111 fitness function. Then, the top Pb percent of a population is selected and carried over to the next generation. 7.5.3.2 Reproduction The next Pr percent of any given generation are created utilizing reproduction techniques. Reproduction occurs when two parent chromosomes are selected and are utilized to create an offspring. The methods for parent selection vary considerably and can include both random and non-random algorithms. A biased roulette wheel (i.e. proportional selection) approach similar to the methods of Leung, Chan and Troutt (2003) is utilized. The goal of this method is to determine a relative fitness, fitrel ,s , for each chromosome (1 < s < popmax ) in a population as illustrated by
fitrel ,s =
1 − fitnorm ,s popmax
∑ 1 − fit i =1
.
(7.10)
norm ,i
In this approach, normalized fitness for each chromosome in a population is determined. The chromosomes are then sorted according to this value from best (lowest) to worst (highest). Each of the normalized fitness values are subtracted from one to ensure that lower (i.e. more optimized) values receive a higher weight. Each resulting value is divided by the sum of all values to obtain relative fitness. Then the interval [0,1] is partitioned into n subintervals each equal to the relative fitness of a chromosome. Thus, chromosomes with the best fitness (i.e. lowest value) will maintain the largest subintervals while those with the worst fitness will maintain relatively small subintervals. Upon completion of this step, two random numbers [0,1] are chosen (one for each parent). Chromosome selection is based on the subinterval a random number falls into. Thus, this method favors those chromosomes with higher fitness for parent selection. Once the parents are selected, a random integer [1, 2 ⋅ n ] is generated to select a splice point for each parent. Thus, all genes to the left of this point for the first parent and those to the right for the second parent are combined to create a child (Figure 7.10).
112
Figure 7.10: Illustration of Reproduction Methods. As previously discussed, the creation of each child by this method is biased towards parents of better fitness. The remaining population of each new generation is obtained through mutation techniques. 7.5.3.3 Local Gradient Searches It was found that improved solutions can be achieved using local search techniques. A specified percent of solutions for any given generation, Pg , were determined based on a gradient based local search routine. Recalling the definition of the problem, we have n components placed in a rectangular domain with dimensions w by h . Each component
is characterized by several variables including effective radii, rj . The placement of each component is defined by two-dimensional coordinates ( x j , y j ), such that the optimization problem is 2 ⋅ n dimensional. The free variables were defined as
x = x1 ,..., xn {x j : 0 ≤ x j ≤ w, j = 1, n}
(7.11)
y = y1 ,..., yn { y j : 0 ≤ y j ≤ h, j = 1, n} .
(7.12)
A fitness function assigns a figure of merit to each encoded solution (i.e. chromosome). For the problem at hand, the figure of merit was directly related to the intersections amongst the effective areas of the components. That is, a fitness function, fit , was created which calculated the intersection of the effective areas of all components by
113 2
1
fit[ A1 ,..., A n ] = ∑ ∑ [ Ai ∩ Aj ] .
(7.13)
i = n j =i −1
Therefore, the objective function for the local search routine is the fitness function that is a function of the x j , y j locations of the components. For notation simplicity, the objective or fitness function will be referred to as f ( x, y ) . Therefore, the gradient of
f ( x, y ) is a 2 ⋅ n dimensional vector ⎛ ∂f ⎞ ⎜ ∂x ( x, y ) ⎟ ⎜ 1 ⎟ ⎜ ∂f ⎟ ⎜ ∂y ( x, y ) ⎟ ⎜ 1 ⎟ ∇f = ⎜ ⎟. ⎜ ∂f ⎟ ⎜ ( x, y ) ⎟ ⎜ ∂xn ⎟ ⎜ ⎟ ⎜ ∂f ( x, y ) ⎟ ⎜ ∂y ⎟ ⎝ n ⎠
(7.14)
An arbitrary solution (i.e. chromosome) consists of the x j , y j locations for all components. For a given generation, g , this vector was denoted by s g . Peng, Thompson, and Li (2002) proposed a formulation of the steepest-descent method defined by
s g +1 = s g − η ⋅ c ⋅ ∇f (s g ) .
(7.15)
In this formulation,η , is a damping factor which is chosen depending on the objective function. For this problem, η was set to 1 although other values could be used to potentially improve performance. The variable c is a step size chosen to minimize the norm of the corresponding local gradient (i.e. method of steepest-descent). In traditional approaches to this algorithm, iterations are utilized to ‘walk’ towards to local minima. If c is sufficiently small, convergence can be guaranteed.
In the current method, a single step approach is utilized to improve algorithm performance. This is accomplished by finding the value of c which minimizes the norm
114 of the local gradient. Utilizing the previous notation, x 0j , y 0j defines the locations of the components at point 0. Expanding the Taylor Series about this point yields f ( x, y ) = f 0 ( x, y ) +
∂f ∂f ( x, y )( x1 − x10 ) + ... + ( x, y )( yn − yn0 ) ∂x1 0 ∂yn 0
⎤ ∂2 f 1 ⎡ ∂2 f 0 2 + ⎢ ( x, y )( x1 − x1 ) + ... + ( x, y )( yn − yn0 ) 2 ⎥ + O ( x 3 , y 3 ) ∂yn ∂yn 0 2! ⎣⎢ ∂x1∂x1 0 ⎦⎥
.
(7.16)
As proposed by Peng and Thompson (2002), this expression was simplified by
∂2 f neglecting the ‘mixed’ second order partial derivative terms, , {i ≠ j} , ∂xi ∂x j ∂2 f ∂2 f , {i ≠ j} and ,{i ≠ j} . The resulting expression is given as ∂xi ∂y j ∂yi ∂y j n ⎡ ⎤ ∂f ∂f f ( x, y ) ≈ f 0 ( x, y ) + ∑ ⎢ (x, y )( xi − xi0 ) + ( x, y )( y j − y 0j ) ⎥ ∂y j j =1 ⎢ ∂x j ⎥⎦ 0 0 ⎣
⎡ ∂2 f ⎤ 1 ∂2 f 0 2 ( x , y )( ) + ∑ ⎢ 2 ( x, y )( x j − x 0j ) 2 + − y y ⎥ j j 2! j =1 ⎢ ∂x j ∂y j 2 ⎥⎦ 0 0 ⎣
.
(7.17)
n
In a simplified form, this becomes
1 f ( x, y ) ≈ f 0 ( x, y ) + bT r + r T Mr , 2
(7.18)
⎛ ∂f ⎞ ( x, y ) ⎟ ⎜ ⎜ ∂x1 0 ⎟ ⎜ ⎟ ⎜ ∂f ( x, y ) ⎟ b = ∇f 0 ( x, y ) = ⎜ ∂y1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ∂f ⎟ ⎜ ( x, y ) ⎟ ⎜ ∂yn ⎟ 0 ⎝ ⎠
(7.19)
where
115 and ⎛ x1 − x10 ⎞ ⎜ 0 ⎟ ⎜ y1 − y1 ⎟ ⎟. r=⎜ ⎜ ⎟ 0 ⎜ xn − xn ⎟ ⎜ y − y0 ⎟ n⎠ ⎝ n
(7.20)
Typically, the Hessian, M , would be fully populated, but since the off diagonal terms were neglected, this becomes ⎛ ∂2 f 0 ⎜ 2 ( x, y ) ∂ x ⎜ 1 0 ⎜ ∂2 f ⎜ 0 ( x, y ) ∂y12 0 ⎜ ⎜ M= ⎜ ⎜ ⎜ ⎜ 0 0 ⎜ 0 0 ⎜ ⎜ ⎝
⎞ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ ⎟. ⎟ ∂2 f ⎟ ( x, y ) 0 ∂xn 2 0 ⎟ ⎟ ∂2 f 0 ( x, y ) ⎟⎟ 2 ∂yn 0 ⎠
(7.21)
Taking the gradient of this function yields, 1 ⎛ ⎞ ∇f ≈ ∇ ⎜ f 0 ( x, y ) + bT r + r T Mr ⎟ 2 ⎝ ⎠ ∇f ≈ ∇ ( f
) + ∇ ( b r ) + ∇ ⎛⎜⎝ 12 r Mr ⎞⎟⎠
(7.23)
1 ⎡∇ ( r T ) Mr + r T ∇ ( M ) r + r T M∇ ( r ) ⎤ ⎦ 2⎣
(7.24)
T
0
∇f ≈ ∇ ( r T ) b + r T ∇ ( b ) +
(7.22)
T
1 ∇f ≈ b + ⎡⎣ Mr + r T M ⎤⎦ 2
(7.25)
1 ∇f ≈ b + ⎡⎣ Mr + MT r ⎤⎦ 2
(7.26)
116 1 ∇f ≈ b + ⎡⎣ M + MT ⎤⎦ r . 2
(7.27)
MT = M .
(7.28)
But M is diagonal such that,
Therefore, ∇f ≈ b +
1 [ 2M ] r . 2
(7.29)
Finally,
∇f ≈ b + Mr .
(7.30)
According to Peng and Thompson (2002), the local minimum of a smoothly continuous function is characterized by zero first-order derivatives or a zero norm gradient. A step size, c , should be chosen such that the local gradient is minimized for each move. Each move can be rewritten as r = −cb such that
(
)
min ( ∇f T ∇f ) = min [ b − Mcb ] [ b − Mcb ] ,
(7.31)
[ b − Mcb]T
(7.32)
c
c
T
where T = ⎡ bT − c ( Mb ) ⎤ = bT − cbT MT . ⎣ ⎦
Substituting, this becomes
[ b − Mcb]T [b − Mcb] = ⎡⎣bT − cbT MT ⎤⎦ [b − Mcb] .
(7.33)
[ b − Mcb]T [ b − Mcb] = bT b − 2cbT Mb + c 2bT M 2b .
(7.34)
= b T b − b T M c b − c b T M T b + c b T M T Mcb
Therefore,
Taking the first derivative with respect to c and setting this equal to zero yields ∂ ( bT b − 2cbT Mb + c 2bT M 2b ) ∂c
= −2bT Mb + 2cbT M 2b = 0 .
(7.35)
117 Solving for c , this becomes bT Mb c= T 2 b Mb
(7.36)
and therefore, 2n
c=
∑b m
jj
∑b m
2 jj
j =1 2n j =1
2 j 2 j
.
(7.37)
Therefore, each solution can be improved utilizing this approach by determining the step size, c , and updating component locations utilizing Equation (7.15). This subroutine was tested by placing five uniform components within a domain and developing improved solutions using only the local gradient search method. The results of component placement - indicated by the circles - both in a corner and center of a domain are shown in Figure 7.11.
Figure 7.11: Illustration of Local Gradient Searches for Five Components Initially in a Corner and in the Center. This figure shows the outline of the effective areas of each component for incremental generations. The local gradient search subroutine appears to spread the heat flux over a domain and thus work towards optimal solutions. To illustrate the impact of local gradient searches on the path to solution, the placement of 18 uniformly powered components were optimized over a square domain and the resulting minimum normalized fitness value was tracked over 10,000 generations.
118 This was done for Pg of 0.0, 0.01, and 0.10 with the remaining evolution parameters set equal to one another. The results are plotted in Figure 7.12. This plot readily shows the impact of local gradient searches. For even a modest introduction of this approach (i.e. Pg = 0.01), the path to solution is greatly improved.
Figure 7.12: Illustration of Normalized Fitness versus Generation Number for Three Levels of Pg (0.0, 0.01, and 0.10). 7.5.3.4 Mutation Mutation is the process by which random alterations to genetic materials are allowed to pass onto future generations. This technique is critical in ensuring that the algorithm does not become stuck in local minima. If all chromosomes of a population are similar, reproduction and local-gradient search techniques will only provide much of the same. Mutation allows for drastic changes in chromosomes. In this algorithm, Pm percent of the next generation is created using mutation which are simply created by randomly generating new chromosomes. 7.6
Genetic Algorithm Tuning The ability of the genetic algorithm to find an optimal solution and the required
computational effort both depend on parameters of the algorithm. As a first step in
119 investigating algorithm performance, both stopping criteria and population size were examined. Next, evolution parameters were investigated to determine their optimal proportion such that Pb + Pr + Pg + Pm = 1 . All tuning criteria were based on a Case D component distribution. 7.6.1 Convergence Criteria The algorithm populates successive generations until convergence. The algorithm compares the best solution for a given generation to one obtained in the previous generation. If the difference is less than 10-4, a counter is incremented by one and a new generation is created. If not, the counter is reset to zero. Convergence criteria are reached when a given best solution is repeated for end max generations in succession. An experiment was conducted to determine the effect of end max on both resulting normalized fitness values and the time required to achieve these values (Figure 7.13 and Figure 7.14)
Figure 7.13: Normalized Fitness versus Number of Repeating Solutions Before Convergence for Increasing Number of Components.
120
Figure 7.14: Time versus Number of Repeating Solutions Before Convergence for Increasing Number of Components. As these figures show, increasing end max improves the final solution but increases the time required to obtain it. Through inspection, an end max value of 500 was selected to minimize solution time without sacrificing accuracy. 7.6.2 Population Size Population size, popmax , for each generation was chosen to be a linear function of the number of components, m ⋅ n . The value of m was varied from 3 up to 9. It was found that increasing this parameter had little if any impact on results for large n but slightly increased the computation time required to obtain these results. For small n , increasing this parameter had the opposite effect; it improved results but had no noticeable effect on computation time. Therefore, a population size of 50 was utilized. This value was approximately the minimum population size that was considered for a large number of components (i.e. m ⋅ n = 3 ⋅ 18 ) and greater than the maximum population size considered for a small number of components (i.e. m ⋅ n = 9 ⋅ 3 ). 7.6.3 Evolution Parameter Tuning A study was conducted to determine the effect of evolution parameters (i.e. Pb , Pr , Pg , Pm ) on performance and computation requirements. For a uniform 18-
component problem (i.e. Case D), each parameter was varied from 0.01 up to 0.90 while
121 the remaining parameters were made equal to each other. At these discrete data points, the resulting normalized fitness and computation time requirements were recorded and plotted versus one another (Figure 7.15 through Figure 7.18).
Figure 7.15: Normalized Fitness versus Computation Time for Increasing Values of Elitism Parameter Pb .
Figure 7.16: Normalized Fitness versus Computation Time for Increasing Values of Reproduction Parameter Pr .
122
Figure 7.17: Normalized Fitness versus Computation Time for Increasing Values of Local Gradient Parameter Pg .
Figure 7.18: Normalized Fitness versus Computation Time for Increasing Values of Mutation Parameter Pm . The results illustrate general ranges of each parameter that provide relatively low normalized fitness and time values. Specific parameter values were identified by preferentially selecting those that provide minimum normalized fitness. These occur at values of Pb = 0.475; Pr = 0.475; Pg = 0.250; and Pm = 0.100. Since the sum of these parameters must be equal to 1, these were normalized to values of Pb = 0.365; Pr = 0.365; Pg = 0.200; and Pm = 0.070. Evolution parameter tuning was completed on
123 an 18-component system. However, it is found that these tuned values provide satisfactory results for other component numbers. 7.7 Algorithm Demonstration Applying the tuned algorithm to the case study, 18 uniformly powered components were optimally placed. Total power was 100 W (i.e. each component is 5.5 W). Figure 7.19 shows the placement results from the algorithm after 18.5 s of computation time and Figure 7.20 gives two-dimensional temperature distributions utilizing Thermal Desktop®. Utilizing the optimized component placement, the resulting maximum and minimum temperatures were 212.4 K and 210.3 K, respectively. In addition, 11 non-uniformly powered components (i.e. 20.3, 17.3, 14.6, 12.2, 10.0, 8.1, 6.3, 4.7, 3.3, 2.1, and 1 W) were optimally placed. Figure 7.21 shows the placement results from the algorithm after 6.5 s of computation time and Figure 7.22 gives twodimensional temperature distributions utilizing Thermal Desktop®. Utilizing the optimized component placement, the resulting maximum and minimum temperatures were 231.8 K and 204.1 K, respectively.
Figure 7.19: Optimized Component Placement for 18 Uniform Components in a Square Domain.
124
Figure 7.20: Optimized Temperature Distribution Results for 18 Uniform Components Obtained from a Thermal Desktop® Finite Difference Model.
Figure 7.21: Optimized Component Placement for 11 Non-Uniform Components in a Square Domain.
125
Figure 7.22: Optimized Temperature Distribution Results for 11 Non-Uniform Components Obtained from a Thermal Desktop® Finite Difference Model. 7.8 Conclusions A computational tool was developed which provides rapidly optimized component placement approaching a uniform heat flux distribution. The tool is based on a genetic algorithm utilizing a combination of elitist strategies, reproduction, local gradient searches, and mutation. A tuning study revealed appropriate convergence criteria, population size, and values for evolution parameters. Optimized results were obtained for 18 uniform and 11 non-uniform components within 20 s and 7 s, respectively, using a 2.5 gigahertz dual-core processor. Advantages of this method include no need for thermophysical properties and boundary conditions. Optimized results are obtained using only component averaged power and domain size. Consequently, this approach is ideally suited to situations where limited information is readily available. In addition, limiting the required inputs provides for relatively fast solutions. However, care should be taken to ensure that a uniform distribution of fluxes is required for optimized placement. Due to low computational expense and limited required information, it could be used to initialize optimization problems not requiring uniform flux distributions (e.g. edge cooled panels). Additionally, this approach does not consider shape and size of components. Therefore, deviations from symmetric shapes
126 (e.g. rectangular components with high aspect ratios) could be problematic not only in terms of the thermal solution but also interference issues. These challenges aside, this robust and fast approach can be utilized in a variety of applications including microelectronics and satellite development and is especially suited to those demanding low computational expense.
127
CHAPTER 8 - DEVELOPMENT AND EVALUATION OF REDUCED ORDER SATELLITE THERMAL MODELS
Evaluating satellite thermal control subsystem (TCS) performance can be done through physical and/or computer experiments. Although physical experiments provide empirical evidence, they can be expensive. Significant costs can be incurred during fabrication (i.e. time and money) and once built, results are limited by the time it takes to complete all experiments. Additionally, physical experiments are limited by the flexibility of a test setup; consequently, parametric studies can be challenging. Computer experiments are an attractive option to overcome the challenges of physical experiments. Constructed correctly, computer experiments can easily accommodate parametric studies and are only limited by processing power. They are especially useful during design stages, although they too have inherent costs. Development of a nominal satellite thermal model can take days to weeks to develop with run times on the order of hours. Comparing and evaluating multiple TCS approaches, especially important in early design stages, amplifies these timelines. Considering the myriad TCS design approaches available, computational expense can become unwieldy. For example, consider a TCS design with five design parameters of interest each evaluated at 10 levels. Evaluating each combination of parameters at all levels would require 1.0E5 simulations. At 30 minutes per simulation, this would take over 5 years of computational time. Consequently, there is a need for the development of reduced order satellite models that can capture the effect of a high-resolution computer experiments without incurring significant computational expense. Reduced order models can then be used to evaluate different TCS approaches and provides a relatively quick means of evaluating design trade-offs. This chapter investigates reduced-order modeling approaches for satellite TCS evaluation. Along the way, this work examines satellite thermal performance for a wide-
128 range of TCS approaches of particular interest to AFRL. Consequently, an evaluation of these different TCS approaches is provided. 8.1 Methodology A satellite TCS approach is a combination of thermal technologies, thermal design variables, and inherent bus characteristics. Each specific TCS approach bridges the gap between input conditions (e.g. environmental fluxes, component characteristics) and resulting thermal performance (Figure 8.1). Satellite thermal performance can include metrics such as maximum/minimum component temperatures and maximum temperature difference between components. The TCS dictates the allowable range of input conditions that will meet a given set of thermal performance requirements.
Figure 8.1: Illustration of Relationship between Input Conditions, TCS Approach, and Thermal Performance. A design of experiments (DOE) evaluation approach was initially utilized to capture these interactions. The approach began by analyzing a wide range of input conditions, thermal design variables, and thermal technologies. The relative importance will vary and therefore, factor screening was used to investigate these effects. Given the relatively large number of factors to investigate, a factorial design approach was chosen as an evaluation tool. Factor screening was conducted using a 2k full-factorial approach. Full-factorial results were also used to develop reduced order satellite thermal models. Full-factorial based reduced-order models are ideally suited for extreme design values; however, their performance is limited at interior design points. Consequently, an
129 additional approach, based on space-filling designs, was utilized to develop reducedorder models. The following sections provide an overview of these approaches. First, an overview of the numerical model is presented followed by a discussion of the full-factorial screening analysis. Finally, an evaluation of reduced-order models, based on space-filling designs is presented. 8.2 Numerical Model Details A numerical model was developed consistent with current robust satellite architectures (Arritt et al., 2008). A 1.0 m x 1.0 m x 1.0 m six-sided frame-and-panel construction satellite was used with a honeycomb panel construction having 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material (Figure 8.2).
Figure 8.2: Illustration of Exterior and Interior of Six-sided Thermal Desktop® Satellite Model Detailing Multiple Components on Each Face. Material properties were based on Gilmore (2002). The satellite is nadir facing and has 36 components with a Case C power distribution (Case C-36). As a result of realistic space limitations of individual panels, the number of components per face was limited to six. However, the power distribution and component layout on each face were determined using the optimization routines. Each component was modeled as a uniform heat flux source on the interior of each panel. All heat generation was ultimately dissipated to a
130 deep space environment (0 K) through radiation from the exterior of the panel surfaces. Table 8.1 summarizes these fixed modeling parameters. Table 8.1: Summary of Fixed Numerical Model Parameters. Factor
Factor Description
Satellite Shape Satellite Size Satellite Structure Satellite Pointing Total Number of Components Component Power Distribution Panel Construction Facesheet Thickness Core Thickness Core Material
Cube 1.0 m x 1.0 m x 1.0 m Frame and Panel Nadir 36 Case C Honeycomb 0.00127 m 0.0254 m Al 5052 Honeycomb
Table 8.2 summarizes the factors that were evaluated throughout this work. In total, 11 factors were each evaluated at and/or between the two indicated levels. Table 8.2: Summary of Factors and Corresponding Levels. Label
Factor
A B C
Orbit Total Component Power Component Side Dimension Component Interface Heat Transfer Coefficient Facesheet Material Transverse Thermal Conductivity Heat Pipes Panel-to-Panel Thermal Conductance Surface Solar Absorptivity Surface Longwave Emissivity Global Component Distribution Local Component Placement
D E F G H I J K
Variable Name ORBIT TOT_PWR C_DIM
Low Value Cold-case 60 W 0.1 m
High Value Hot-case 600 W 0.2 m
C_I_CND
110 W/m2-K 700 W/m2-K
F_T_CND
170 W/m-K 1000 W/m-K
HT_PIPE
0
10 per panel
P2P_CND
12 W/K
36 W/K
EXT_ABS EXT_EMS GLBL_DIS LCL_PLC
0.123 0.100 Nominal Nominal
0.561 0.900 Optimized Optimized
131 Computer experiments were run using Thermal Desktop® as a CAD based interface to the SINDA/FLUINT finite-difference thermal analyzer. Thermal Desktop® and SINDA/FLUINT are widely accepted tools for spacecraft thermal design. Before computer experiments were run to evaluate the 11 factors, a complete analysis of the modeling parameters was conducted to ensure that simulation noise was minimized. Modeling parameters included the number of orbital simulations, nodal resolution, Monte-Carlo rays per node, and number of orbital positions. The results of these analyses can be found in Appendix F. Numerical model results (response values) were based on second-orbit temperature versus time values obtained for each of the 36 components. Responses include: maximum temperature, minimum temperature, maximum temperature difference, maximum panel temperature difference, and isothermal maximum/minimum temperatures. Descriptions of each are provided here. •
MAXIMUM TEMPERATURE (Tmax): Maximum second-orbit temperature reached by any component.
•
MINIMUM TEMPERATURE (Tmin): Minimum second-orbit temperature reached by any component.
•
MAXIMUM TEMPERATURE DIFFERENCE (Tmaxd): Maximum temperature difference reached between any two components at any second-orbit time.
•
MAXIMUM PANEL TEMPERATURE DIFFERENCE (Tmaxd_p): Maximum temperature difference reached between any two components on the same panel at any second-orbit time.
•
ISOTHERMAL MAXIMUM TEMPERATURE (Tiso_max): Maximum secondorbit temperatures reach by an equivalent isothermal structure. The isothermal structure was modeled as a single-node satellite and thus simulates a satellite with infinite bus conductance.
•
ISOTHERMAL MINIMUM TEMPERATURE (Tiso_min): Minimum secondorbit temperatures reach by an equivalent isothermal structure. The isothermal structure was modeled as a single-node satellite and thus, simulates a satellite with infinite bus conductance.
132 According to Young (2008), a robust RS TCS can be developed, in part, through an isothermal bus architecture. Consequently, reducing Tmaxd through technology improvements is critical and therefore, was the focus of the analyses presented here. 8.2.1 Description of Factors The following sections describe each of these 11 factors. Additionally, a description of the low- and high-values is included. The range over which the variable factors were evaluated were based on typical expected technology ranges. 8.2.1.1 Orbit Hot- and cold-case orbit parameters were based on well-defined orbital parameters and include specific altitudes, β , and heating rate factors (Hengeveld et al, 2009). The details of this work can be found in CHAPTER 5. 8.2.1.2 Total Component Power Total component power was modeled at two levels. At the high level, the total component power is 600 W, which is of similar magnitude to RS mission performance requirements for LEO missions (470 W) (ORSBS-001, 2007). Heat generation of all internal components was conducted to the panel only (i.e. not coupled through radiation). Low-level power was modeled at 10% of maximum (60 W) which is used to model ‘survival power’ conditions. Williams (2005) used a value of 50 W in his work. 8.2.1.3 Component Dimension Component footprint was varied between two levels: 10 cm x 10 cm and 20 cm x 20 cm. These values provide a typical range of component footprint sizes. 8.2.1.4 Component Interface Heat Transfer Coefficient The interface heat transfer coefficient between components and panels was varied from 110 W/m2-K to 700 W/m2-K. The lower value approximates bare interface contact between components and panel (Gilmore, 2002). The high level, providing an approximately 7x increase, was a target value by Hafer et al. (2008).
133 8.2.1.5 Facesheet Material Transverse Thermal Conductivity Facesheet transverse thermal conductivity was varied from 170 W/m-K to 1,000 W/m-K. The low value is typical of Aluminum 6061-T6 facesheet while the upper value is similar to the transverse thermal conductivity of k-Core. 8.2.1.6 Heat Pipes Embedded heat pipes were modeled at two levels: 0 and 10 per panel. Traditional heat pipe construction was used. The staff at Advanced Cooling Technologies was consulted to determine typical heat pipe modeling parameters. Based on discussion, 10 0.9525 cm (0.375 in) heat pipes per panel each with a vapor core of 0.5080 cm (0.200 in) diameter were selected. The evaporator and condenser heat transfer coefficients were set at 11,356 W/m2-K and 5,678 W/m2-K, respectively. Conductance between heat pipes and panel construction was included by applying a heat transfer coefficient (HP_IC) of 20,000 W/m2-K (2 W/cm2-K). Values above the design value provide minimal improvement as described by Bugby et al. (2010). 8.2.1.7 Panel-to-Panel Conductivity The low value of panel-to-panel conductance was selected to be 12 W/K to simulate panel-to-panel longerons and bolted joints (Bugby, Zimbeck, and Kroliczek, 2008). A panel-to-panel conductivity of 36 W/K was selected and simulates improved panel-topanel thermal performance. 8.2.1.8 Surface Solar Absorptivity and Longwave Emissivity Satellite exterior conditions were varied based on solar absorptivity and longwave emissivity. These parameters were applied to all panels. Individual exterior panel modifications were not pursued due to computational expense. Optical property bounds used for developing a reduced order satellite model were evaluated. Emissivity bounds were based on current AFRL research efforts (i.e. 0.100 < ε < 0.900). Based on these emissivity values, absorptivity bounds were evaluated. As a first approach, a first-order approximation based on hot- and cold-case design orbits was completed. The raw environmental flux values for hot- and cold-case orbits are shown in Table 8.3.
134 Table 8.3: Hot- and Cold-Case Orbit Environmental Fluxes. Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface 239.81 139.22 +X Surface 238.93 139.22 -Y Surface 1,454.68 1,349.55 +Y Surface 95.51 0.00 -Z Surface 139.22 139.22 +Z Surface 434.64 139.22 AVERAGES 433.80 317.74
16.80 16.65 23.01 11.19 0.00 49.85 19.58
83.79 83.05 82.12 84.32 0.00 245.57 96.47
388.75 387.94 58.62 59.06 418.15 300.81 268.89
329.45 329.45 0.00 0.00 418.15 74.31 191.89
14.54 44.77 14.54 43.94 14.46 44.16 14.43 44.63 0.00 0.00 56.01 170.50 19.00 58.00
Based on these orbital-averaged fluxes, both a hot-case and cold-case energy balance were created. The energy balance was based on a 1 m x 1 m cubic satellite (i.e. 6 m2 of dissipating area). However, the hot- and cold-cases include 600 W and 60 W, respectively, of internal energy dissipation to account for components. The resulting energy balance was completed for a range of absorptivities and emissivities. For each combination of optical properties, the resulting balance temperature was calculated and plotted in Figure 8.3 and Figure 8.4.
Figure 8.3: Cold-Case Orbit Energy Balance Results.
135
Figure 8.4: Hot-Case Energy Balance Results. These figures illustrate a range of absorptivity values (i.e. 0.123 < α < 0.561) that provide for an orbital-averaged balance temperature of 20°C over the range of emissivity values (i.e. 0.100 < ε < 0.900). However, these figures do not illustrate the range of temperatures that will occur over an orbit for these combinations of surface optical properties. For example, a combination of surface optical properties of α = 0.561 and ε = 0.900 yields an orbital-averaged balance temperature of approximately 20°C, but the temporal variations are still unknown. 8.2.1.9 Global Component Distribution and Local Component Placement Global and local component placement were modeled at two distinct levels. Consequently, four component placement cases arise. Figure 8.5 provides an illustration of these four cases for select surfaces in the cold-case orbit with surface properties of α = 0.123 and ε = 0.100. 1) Nominal Global Distribution / Nominal Local Placement – Globally, component power is evenly distributed between panels. Therefore, the same power is provided to each panel regardless of its interaction with space. Locally, components are symmetrically placed across each panel. For six components per panel, a 3x2 array of components were placed on each panel.
136
1) Nominal Global / Nominal Local
2) Nominal Global / Locally Optimized
3) Globally Optimized / Nominal Local 4) Globally Optimized / Locally Optimized Figure 8.5: Illustration of Four Global Component Distribution and Local Component Placement Cases for Select Surfaces in the Cold-Case Orbit with α = 0.123 and ε = 0.900. 2) Nominal Global Distribution / Optimized Local Placement – Globally, component power is evenly distributed between panels. Locally, optimization routines were run to optimally place components on each panel. 3) Optimized Global Distribution / Nominal Local Placement – Globally, optimization routines were run to optimally distribute components among six panels. Locally, components were evenly spread across each panel. For six components per panel, a 3x2 array of components were placed on each panel. 4) Optimized Global Distribution / Optimized Local Placement – Globally, optimization routines were run to optimally distribute components among six
137 panels. Locally, optimization routines were run to optimally place components on each panel. Resulting global and local optimal distribution and placement values were subsequently updated as shown in Table I.1 through Table I.18 in the Appendices. These distributions and locations were based on hot- and cold-case environmental fluxes as shown in Table I.19 through Table I.22. 8.3 Full-Factorial Screening Analysis Full-factorial screening designs investigate all possible combinations of factor levels. The work presented here examined 11 factors. Consequently, 2,048 cases were simulated (211). The following provides a screening analysis based on the results of these simulations. As mentioned previously, of critical importance is minimizing maximum temperature differences across satellites (i.e. approaching an isothermal bus). Consequently, results and discussion are based on Tmaxd. 8.3.1 Main Effects versus Tmax Response Variable factors were first evaluated by examining the impact each factor has on response. Main effects, the difference between factor means at high and low levels, provide a typical means of evaluating the impact of a given factor (Lenth, 1989). The +
main effect of ORBIT is a function of its mean at the high level, ORBIT , and low level, −
ORBIT , is shown by +
−
ORBIT = ORBIT − ORBIT .
(8.1)
Results, shown in Table 8.4, were developed using the statistical packages Minitab 15 and JMP 8. Minitab 15 results, including effects and their absolute value, were used to develop all figures in this work. JMP 8 provided the pure statistical analysis and includes contrasts, Lenth statistic, and resulting p-values. Contrasts are merely half of the effect values and provide the framework for developing Lenth t-Ratios and resulting p-values. Computer experiments typically do not have any experimental error due to replication. Consequently, traditional statistical approaches are not well suited to evaluating results. Lenth’s method provides a formal approach for the analysis of
138 unreplicated factorials (e.g. deterministic computational experiments). It relies on simple formulas in determining standard error of contrast estimates (Lenth, 1989). Within this procedure, the m contrast values, c1 , c2 ,...cm , corresponding to the m = 2k − 1 factor effects are considered normally distributed random variables with potentially different means but equal variances. Lenth (1989) defined a pseudo-standard random error, PSE, as shown here: PSE = 1.5 × median c j
(8.2)
c j < 2.5× so
so = 1.5 × median c j .
(8.3)
The PSE is then used to generate a ‘t-like’ statistic which is commonly referred to as the Lenth statistic as shown in tLenth , j =
cj PSE
.
(8.4)
For the current data, a PSE of 0.07757 was found. Lenth suggested using a t-statistic with m/3 degrees of freedom as an approximate reference distribution to this statistic.
However, it was found that this does not work well (Ye and Hamada, 2000). Ye and Hamada (2000) proposed generating a set of m estimated contrasts from a normal distribution and calculating a Lenth statistic (i.e. Equation (8.4)) for each. This process is repeated n times (e.g. JMP 8 uses a Monte Carlo simulation with n = 10,000 runs), which provides an approximation of the Lenth distribution. P-values can then be obtained by comparing the Lenth statistics from the contrasts in question to this distribution. P-values can then be used as a guide in selecting important factors. Table 8.4 shows that these 11 main effects have small p-values based on Lenth’s method analysis. Consequently, each factor has a significant impact on Tmaxd. For example, improving facesheet conductivity from 170 W/m-K to 1,000 W/m-K will reduce Tmaxd, on average, by approximately 33 K for the cases considered here. However, it should be noted that although total power was found to have a significant pvalue, the overall impact is quite small (~0.33 K).
139 Table 8.4: Full Factorial Summary of Lenth t-Ratio and Corresponding p-Values for Main Effects in Descending Order According to Absolute Effect Values. Factor
Effect
EXT_ABS F_T_CND ORBIT GLBL_DIS C_DIM HT_PIPE EXT_EMS C_I_CND P2P_CND LCL_PLC TOT_PWR
[K] 54.64 -32.86 22.57 -15.56 -9.72 -8.53 -7.89 -7.69 -6.24 -4.62 -0.33
Absolute Effect [K] 54.64 32.86 22.57 15.56 9.72 8.53 7.89 7.69 6.24 4.62 0.33
Contrast [K] 27.32 -16.43 11.28 -7.78 -4.86 -4.27 -3.95 -3.85 -3.12 -2.31 -0.17
Lenth Statistic [---] 340.45 -204.77 140.62 -96.98 -60.58 -53.15 -49.17 -47.94 -38.91 -28.76 -2.06
p-value [---] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0403
A pareto chart of these effects is shown in Figure 8.6 and illustrates the relative impact each factor has on satellite Tmaxd. Figure 8.6 shows that external absorptivity, facesheet transverse thermal conductivity, orbit, and global distribution of components have the most significant impact on Tmaxd based on overall averages of all simulations.
Figure 8.6: Effects Pareto Chart for Tmaxd Responses (All Main Effects).
140 Factor effects for the first four factors are plotted in Figure 8.7. The figure indicates that these main effects are based on bulk averages of the complete set of computer simulations. All main effect plots are included in Appendix J. It is therefore important to investigate interaction effects to provide improved understanding of these factors.
a) Main Effect EXT_ABS
b) Main Effect F_T_CND
c) Main Effect ORBIT d) Main Effect GLBL_DIS Figure 8.7: Illustration of First Four Main Effects: a) EXT_ABS, b) F_T_CND, c) ORBIT, and d) GLBL_DIS and Including a Linear Regression Line and Horizontal Jitter for Clarity.
141 8.3.2 Interaction Effects versus Tmaxd Response Although the main effects can be a good indicator of relative performance of factors, it is important to investigate their interaction. For example, the response of a given factor might be significantly impacted by the level of another factor. Consequently, the interaction effects were examined further. Interaction effects describe the average difference of a targeted factor at two levels of a secondary factor. For example, the interaction effect of TOT_PWR and ORBIT (i.e. ORBIT*TOT_PWR) is the difference between the main effect of ORBIT at the high level of TOT_PWR (i.e. TOT _ PWR + ) and low level of TOT_PWR (i.e. TOT _ PWR − ). This is shown in +
−
ORBIT * TOT _ PWR = [(ORBIT − ORBIT )TOT _ PWR +
−
−
+
−(ORBIT − ORBIT )TOT _ PWR ] / 2
.
(8.5)
Interaction effects results for the first 15 interaction effects are shown in Table 8.5. Table 8.5: Full Factorial Summary of Lenth t-Ratio and Corresponding p-Values for the First 15 Interaction Effects in Descending Order According to Absolute Contrast Values. Factor
Effect
EXT_ABS*ORBIT GLBL_DIS*TOT_PWR EXT_ABS*EXT_EMS ORBIT*TOT_PWR EXT_ABS*F_T_CND F_T_CND*ORBIT C_DIM*TOT_PWR EXT_ABS*TOT_PWR ORBIT*EXT_EMS C_I_CND*TOT_PWR ORBIT*GLBL_DIS F_T_CND*EXT_EMS F_T_CND*GLBL_DIS LCL_PLC*TOT_PWR C_DIM*C_I_CND
[K] 12.08 -11.74 -10.19 -9.25 -9.06 -7.50 -6.94 -6.79 -6.57 -6.15 -5.81 5.72 4.03 -3.74 3.45
Absolute Effect [K] 12.08 11.74 10.19 9.25 9.06 7.50 6.94 6.79 6.57 6.15 5.81 5.72 4.03 3.74 3.45
Contrast [K] 6.04 -5.87 -5.10 -4.62 -4.53 -3.75 -3.47 -3.39 -3.29 -3.08 -2.91 2.86 2.01 -1.87 1.72
Lenth Statistic [---] 75.29 -73.15 -63.50 -57.62 -56.45 -46.75 -43.25 -42.31 -40.95 -38.35 -36.22 35.62 25.10 -23.32 21.48
p-value [---] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
142 Similar to main effects results, this table includes interaction effect, absolute interaction effect, contrast, Lenth statistic, and resulting p-value. Table 8.5 shows that these interaction effects are also significant based on their small p-values. Consequently, each combination of these factors plays an important part in influencing Tmaxd. For example, moving from the cold-case to hot-case orbit has an effect of increasing Tmaxd by approximately 12 K over the range of external absorptivities (i.e. 0.123 to 0.561). Consequently, it was noted that external absorptivity was the most significant factor impacting Tmaxd; however, this impact is more significant for the hot-case orbit than the cold-case. A pareto chart of the first 15, in magnitude, interaction effects is shown in Figure 8.8 and illustrates the relative impact each factor has on satellite Tmaxd. Additionally, the first four interaction effects are plotted in Figure 8.9. All interaction effect plots are included in Appendix J.
Figure 8.8: Effects Pareto Chart for Tmaxd Responses (First 15 Interaction Effects).
143
a) Interaction Effect EXT_ABS*ORBIT
b) Interaction Effect GLBL_DIS*TOT_PWR
c) Interaction Effect d) Interaction Effect EXT_ABS*EXT_EMS ORBIT*TOT_PWR Figure 8.9: Illustration of First Four Interaction Effects: a) EXT_ABS*ORBIT, b) GLBL_DIS*TOT_PWR, c) EXT_ABS*EXT_EMS, and d) ORBIT*TOT_PWR and Including a Linear Regression Line and Horizontal Jitter for Clarity. 8.3.3 Full-Factorial Screening Discussion The following provides an evaluation of these factors in terms of how they should be approached to reduce Tmaxd.
144 •
ORBIT: Averaged over all simulations, changes in LEO orbital properties will produce a maximum increase of 22.6 K in Tmaxd. Improvements in facesheet thermal conductivity and decreasing external absorptivity can help reduce these values. Additionally, higher power satellites will be more orbit insensitive due to the fact that the ratio of heat loading from components to that of environmental sources becomes greater.
•
TOT_PWR: Total power, on average, has little effect on Tmaxd. On average, increasing total power from 60 W to 600 W has an effect of reducing Tmaxd by 0.33 K.
•
C_DIM: Increasing component dimensions from 0.1 x 0.1 m to 0.2 x 0.2 m decreases localized heat fluxes. Consequently, Tmaxd is reduced, on average, 7.7 K.
•
C_I_CND: Component interface conductance reduces Tmaxd, on average, by 7.7 K when increased from 110 to 700 W/m2-K. Its impact is more effective with increases in overall bus power but diminished by increased component dimensions.
•
F_T_CND: Facesheet transverse thermal conductivity has a significant impact on Tmaxd. Increasing from 170 to 1,000 W/m-K will reduce Tmaxd by 32.9 K averaged over all simulations. This impact is more apparent for the hot-case orbit and is improved with improvements in panel-to-panel conductivity.
•
HT_PIPE: The addition of heat pipes reduces Tmaxd by 8.5 K. As expected, its impact is reduced with improved facesheet conductivity.
•
P2P_CND: Panel-to-panel conductivity will reduce Tmaxd by 6.2 K. Interestingly, the impact is improved with improved facesheet conductivity.
•
EXT_ABS: External absorptivity has the most significant impact on Tmaxd. Increasing its value from 0.123 to 0.561 will increase Tmaxd by 54.6 K. These effects are more significant in the hot-case orbit and reduced for higher-power bus designs. The impact of external absorptivity can be significantly reduced by increasing facesheet conductivity and increasing surface emissivity.
145 •
EXT_EMS: Increasing external emissivity from 0.1 to 0.9 will reduce Tmaxd by 7.9 K, on average. These effects are more prevalent in the hot-case orbit and at higher external absorptivities, but are diminished with improvements in facesheet conductivity.
•
GLBL_DIS: Optimized global distribution will reduce Tmaxd, on average, by 15.6 K. This approach is more effective in the hot case orbit and higher power bus designs. However, the impact is diminished by improved bus designs (e.g. increased component dimensions (i.e. decreased flux), component interface conductance, and facesheet conductivity.
•
LCL_PLC: Local placement optimization routines will reduce Tmaxd, on average, by 4.6 K. This approach is significantly diminished at the high value of facesheet conductivity but works more effectively for higher power designs.
Overall, Tmaxd can be reduced by a combination of approaches. First, coupling with solar loading should be diminished. This is traditionally accomplished by insulating all surfaces and providing a deployable radiator. However, this approach adds extra mass, touch labor, and reliability issues as a result of deployment mechanisms. Future approaches relying on electrochromic surfaces would be attractive, given that these technologies can minimize the impact of solar loading (i.e. low absorptivity electrochromic surfaces). Improving panel transverse conductivity is another approach that can provide significant benefits. Although heat pipes provide significant improvements, they introduce mass penalties. In addition, simulations have shown that heat pipes are outperformed by introducing high transverse thermal conductivity facesheet materials. These materials are recommended for improved satellite thermal bus performance and are enhanced through the use of improved panel-to-panel conductivity. Finally, global and local placement optimization routines provide performance improvements at virtually no cost. 8.4 Reduced-Order Model Development Computer simulations have been targeted as a replacement to physical experiments for many applications because they are more time and cost efficient and can provide valuable insight early in the design stages (Jones and Johnson, 2009). However, computer
146 experiments are often complex and computationally expensive. When built to evaluate several variables, these costs can become unacceptable. Consequently, developing reduced-order models that capture the effects of more complicated computer simulations can have a significant benefit. Reduced-order models are based on computer simulations and therefore, should be designed to capture effects of the original model with the fewest number of runs. When properly developed, these surrogate models can predict responses at untested design points very quickly. Computer simulation experiments are uniquely different from their physical counterparts in that they typically have no results variability. Consequently, different approaches to experimentation are used. The following provides an overview of the development of reduced-order models using a Latin Hypercube spacefilling approach and Gaussian Process model fitting. 8.4.1 Reduced-Order Model Although full-factorial approaches examine all combinations of variables, they do so only at extreme values (i.e. design space boundaries). Consequently, interior points are overlooked and reduced-order models can often fail far from the boundaries. Therefore, space-filling designs were utilized to efficiently identify and evaluate interior points that would provide improvements in the reduced-order model. Space-filling designs attempt to efficiently evaluate a design space for a given number of computer simulations. Design approaches include: sphere packing, Latin Hypercube, uniform design, maximum entropy, and the Gaussian-Process IMSE designs (Jones and Johnson, 2009). However, Latin Hypercube approaches are the most commonly used for computer experiments (Jones and Johnson, 2009) and were used as the basis for the reduced-order models presented in the current work. A reduced-order model should accurately capture the effects of the 11 input factors on Tmaxd. However, 4 of the 11 factors are categorical (i.e. non-continuous factors) which include ORBIT, HT_PIPE, GLBL_DIS, and LCL_PLC. Reduced-order models should then be developed for all combinations of these factors (24 = 16). However, to minimize the number of reduced-order models, the optimization factors (i.e. GLBL_DIS and LCL_PLC) were changed concurrently. Consequently, eight reduced-order models (i.e. 23) were developed as illustrated in Table 8.6. For each categorical model, 128 (27)
147 sampling points were identified using Latin Hypercube approaches. A summary of Latin Hypercube sampling points for the remaining seven continuous variables is included in Appendix L. Table 8.6: Summary of Eight Reduced-Order Models versus Four Categorical Factors. Reduced-Order Model
ORBIT
HT_PIPE
GLBL_DIS
LCL_PLC
LH000 LH001 LH010 LH011 LH100 LH101 LH110 LH111
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
Once the design points were determined using the Latin Hypercube approaches, the Gaussian-Process (GP) regression model was used to fit the data and provided the framework for a reduced-order model. GP regression models, widely-used in computer simulation research, is a model that has only as many parameters as variables plus one parameter for mean and one for variance (Jones and Johnson, 2009). Additionally, GP models exactly fit observed responses, which is especially useful for deterministic computer simulations. Consider an experiment with training data evaluated at n locations each defined by a k-dimensional vector (i.e. k input factors). For training data at the i-th location,
x i = ( xi1 ,..., xik ) , a given response is denoted by yi = y ( x i ) . Consequently, there is a n × k training data matrix, X . Outputs of these trials, Y = y ( X ) = ( y1 ,..., xn )′ , is an n-
dimensional vector. For a single trial, the output is modeled by y (x i ) = μ + z (x i )
i = 1,.., n .
(8.6)
The value μ is the overall mean and z ( x i ) is a Gaussian Process with E ( z (x i )) = 0 , Var ( z ( x i )) = σ z2 , and Cov ( z ( x i ), z ( x j )) = σ z2 Rij ( X, θ) (Ranjan, Haynes, and Karsten,
148 2010). Although several approaches can be utilized for the correlation structure, the approach used was that described by Jones and Johnson (2009) shown here k
Rij ( X, θ) = exp[ − ∑θ s ( xis − x js ) 2 ] .
(8.7)
s =1
As described by Jones and Johnson (2009), values of θ s ≥ 0 . When θ s = 0 , the correlation is 1.0 over the range of the kth factor and the corresponding fitted surface is flat in that direction. Conversely, large θ s indicate lower correlation in the kth factor and the fitted surface will be bumpy in that direction. The parameters, μ , σ , and θ are estimated using maximum likelihood methods (Jones and Johnson, 2009). Representing these values by μˆ , σˆ , and θˆ the prediction model at a non-sample point, x* = ( x1* ,..., xk* ) , becomes (Ranjan, Haynes, and Karsten, 2010) yˆ (x* ) = μˆ + r′( x* , x, θˆ )R −1 ( X, θˆ )( Y − 1n μˆ ) .
(8.8)
The matrix, R ( X , θˆ ) , is an n x n correlation matrix between training data sites while the n-dimensional vector, r ( x* , x, θˆ ) , is given by k ⎡ ⎤ exp[ θ s ( x1s − x*s )2 ]⎥ − ∑ ⎢ s =1 ⎢ ⎥ * ˆ r ( x , x, θ) = ⎢ ⎥. ⎢ ⎥ k ⎢ exp[− ∑θ s ( xns − x * )2 ]⎥ s s =1 ⎣⎢ ⎦⎥
(8.9)
Consequently, the resulting prediction equation contains one model term for each design point in the original experiment (i.e. training data). Introduced for computer experiments by Sacks, Welch, Mitchell, and Wynn (1989), this approach is desirable in computer experiments since they provide an exact fit to the training data and require only k+1 parameters (i.e. μˆ and θˆk ). A full summary of model coefficients is included in
Appendix N.
149 8.4.2 Development of Test Cases Evaluation of reduced-order models was done using 100 test cases found in Appendix L. These test cases were generated using random functions for all categorical and continuous variables. 8.4.2.1 Results Reduced-order (RO) models based on latin-hypercube sampling with Gaussian Process regression fits were developed. These models were evaluated by comparing predicted Tmax, Tmin, and Tmaxd values versus computer simulation (CS) results from the 100 test-case values. The results of this comparison are illustrated in Figure 8.10 through Figure 8.12.
Figure 8.10: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmax Result versus Computer Simulation (CS) Tmax Results.
150
Figure 8.11: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmin Result versus Computer Simulation (CS) Tmin Results.
Figure 8.12: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmaxd Result versus Computer Simulation (CS) Tmaxd Results. To get a better understanding of this relationship, a residual analysis was conducted. Residuals (computer simulation results minus reduced-order model results) were
151 determined for each of 100 test cases. The residuals were then plotted versus computer simulated Tmax, Tmin, and Tmaxd results as shown in Figure 8.13 through Figure 8.15.
Figure 8.13: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmax Results versus CS Tmax.
Figure 8.14: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmin Results versus CS Tmin.
152
Figure 8.15: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmaxd Results versus CS Tmaxd. As this figure shows, residual variance appears to increase slightly with increasing Tmin and Tmaxd values. Additionally, residuals appear to be centrally located near 0 K. This was confirmed by constructing a dotplot diagram of residuals to evaluate their distribution (Figure 8.16 through Figure 8.18).
Figure 8.16: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmax Results.
153
Figure 8.17: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmin Results.
Figure 8.18: Dotplot of Computer Simulation (CS) minus Latin-Hypercube / GaussianProcess Reduced-Order (RO) Model Tmaxd Results. The dotplot diagrams indicate that residual values are normally distributed around a mean of ~0 K. This assumption was tested by constructing normal probability plots as shown in Figure 8.19 through Figure 8.21.
154
Figure 8.19: Normal Probability Plot of Tmax Residual Results with 95% Confidence Intervals and Statistical Indices.
Figure 8.20: Normal Probability Plot of Tmin Residual Results with 95% Confidence Intervals and Statistical Indices.
155
Figure 8.21: Normal Probability Plot of Tmaxd Residual Results with 95% Confidence Intervals and Statistical Indices. These normal probability plots were used to evaluate the normality assumption and include 95% confidence intervals. The Anderson-Darling (AD) statistic was used to measure how well the data follow a particular distribution. The better the distribution fits the data, the smaller this statistic will be. The Anderson-Darling test evaluates the hypotheses shown here: H 0 : The data follow a specified distribution H 1 : The data do not follow a specified distribution
.
(8.10)
If the p-value for the Anderson-Darling test is greater than a typical significance level (usually 0.05 or 0.10), it can be concluded that the data is normally distributed with a given mean and standard deviation. Figure 8.19 through Figure 8.21 show that the Tmaxd residuals are normally distributed. The normality assumption is challenged for the Tmax and Tmin residuals as a result of residual values in the tails of the dotplot figures. Estimated means and standard deviations for all responses are shown in Table 8.7.
156 Table 8.7: Residual Normality Results for Three Response Variables. Response
Mean [K]
Standard Deviation [K]
Tmax Tmin Tmaxd
-0.1448 0.06414 0.08643
1.547 1.077 1.518
8.4.2.1.1 Factor Sweep Analysis The reduced-order modeling approach was further examined by evaluating how it performs over a range of two select factors. Computer-simulation and reduced-order model Tmax, Tmin, and Tmaxd results were plotted versus both external absorptivity (EXT_ABS) and facesheet transverse thermal conductivity (F_T_CND) design factors in both the hot- and cold-case orbits for both nominal and optimized global and local component placements. All other design factors, set to nominal levels, are summarized in Table 8.8. Table 8.8: Summary of Nominal Factors Levels. Factor Total Component Power Component Side Dimension Component Interface Heat Transfer Coefficient Facesheet Material Transverse Thermal Conductivity Heat Pipes Panel to Panel Thermal Conductance Surface Solar Absorptivity Surface Longwave Emissivity
Variable Name TOT_PWR C_DIM
Nominal Value 333 W 0.15 m
C_I_CND
405 W/m2-K
F_T_CND
585 W/m-K
HT_PIPE
none
P2P_CND
24 W/K
EXT_ABS EXT_EMS
0.342 0.500
Plots of these results are shown in Figure 8.22 through Figure 8.27. These figures illustrate that the reduced-order models provide an acceptable representation of computer-simulation results. These figures also indicate that computer simulation results
157 for the optimized cases are not smooth (e.g. Figure 8.23 Hot-Case Results). This noise is the results of the stochastic nature of the genetic-algorithm optimization algorithms.
a) Hot-Case Results b) Cold-Case Results Figure 8.22: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmax versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.
a) Hot-Case Results b) Cold-Case Results Figure 8.23: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmin versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.
158
a) Hot-Case Results b) Cold-Case Results Figure 8.24: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmaxd versus External Absorptivity (EXT_ABS) Results for both a) Hot-Case and b) Cold-Case Orbits.
a) Hot-Case Results b) Cold-Case Results Figure 8.25: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmax versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits.
159
a) Hot-Case Results b) Cold-Case Results Figure 8.26: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmin versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits.
a) Hot-Case Results b) Cold-Case Results Figure 8.27: Nominal Computer Simulation (CS) and Reduced-Order (RO) Tmaxd versus Facesheet Transverse Thermal Conductivity (F_T_CND) Results for both a) Hot-Case and b) Cold-Case Orbits.
160 8.4.2.1.2 Evaluation of Reduced-Order Model for Other Distributions The reduced-order models were based on 36 components with a Case C power distribution (i.e. Figure 6.2). Investigation was conducted to determine how well these models capture results for alternative distributions. The 100 test points used to initially evaluate the reduced-order models (i.e. Appendix L) were tested using 36 components with a Case B power distribution (i.e. Figure 6.2). The effectiveness of the reduced-order models at capturing the effects of alternative component distributions was done by comparing predicted (i.e. RO Model) Tmax, Tmin, and Tmaxd values versus computer simulation (CS) results from the 100 test-case values. The results of this comparison are illustrated in Figure 8.28 through Figure 8.30.
Figure 8.28: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmax Result versus Computer Simulation (CS) Tmax Case B-36 Results.
161
Figure 8.29: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmin Result versus Computer Simulation (CS) Tmin Case B-36 Results.
Figure 8.30: Latin-Hypercube / Gaussian-Process Reduced-Order (RO) Model Tmaxd Result versus Computer Simulation (CS) Tmaxd Case B-36 Results. To get a better understanding of this relationship, a residual analysis was conducted. Residuals (computer simulation results minus reduced-order model results) were
162 determined for each of 100 test cases. The residuals were then plotted versus computer simulated Tmax, Tmin, and Tmaxd results as shown in Figure 8.31 through Figure 8.33.
Figure 8.31: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmax Case B-36 Results versus CS Tmax.
Figure 8.32: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmin Case B-36 Results versus CS Tmin.
163
Figure 8.33: Computer Simulation (CS) minus Latin-Hypercube/Gaussian-Process Reduced-Order (RO) Model Tmaxd Case B-36 Results versus CS Tmaxd. These figures show that the developed reduced-order models tend to underpredict Tmax and Tmaxd results for the Case B distribution. This can be attributed to the shape of the Case B distribution (i.e. Figure 6.2). Most of the Case B component power is contained in a small set of components as compared to the Case C distribution. Consequently, the relative power difference, and consequently temperature difference, between any two components is greater for the Case B over the Case C distribution. Additionally, the results show that as overall thermal performance is improved (e.g. lower Tmaxd), the residuals become smaller (i.e. Figure 8.33). This can be attributed to the fact that improved thermal performance is, in part, due to optimized global and local component placement. These results show that the developed optimization routines are somewhat insensitive to component distributions and trend towards similar optimized results. 8.4.2.1.3 Example of Reduced-Order Model Use A simple example of the reduced-order models is provided here to illustrate their utility. A satellite design was explored with properties summarized in Table 8.9. Note that this satellite design has relatively low-capability technologies for most factors except facesheet transverse thermal conductivity (900 W/m-K) and optimized global/local
164 placement. Additionally, the external emissivity is assumed to be variable (e.g. electrochromic devices applied to the external satellite surfaces) while the external absorptivity is fixed at 0.3. The cold-case was based on a cold-case orbit and an assumed survival power of 60 W. The hot-case was based on a hot-case orbit and full power conditions (i.e. 600 W). Table 8.9: Summary of Example Factor Levels. Factor Orbit Total Component Power Component Side Dimension Component Interface Heat Transfer Coefficient Facesheet Material Transverse Thermal Conductivity Heat Pipes Panel to Panel Thermal Conductance Surface Solar Absorptivity Surface Longwave Emissivity Global Component Distribution Local Component Placement
Variable Name ORBIT TOT_PWR C_DIM
Cold-Case Values Cold 60 W 0.1 m
Hot-Case Values Hot 600 W 0.1 m
C_I_CND
110 W/m2-K 110 W/m2-K
F_T_CND
900 W/m-K
900 W/m-K
HT_PIPE
none
none
P2P_CND
12 W/K
12 W/K
EXT_ABS EXT_EMS GLBL_DIS LCL_PLC
0.3 Varied Optimized Optimized
0.3 Varied Optimized Optimized
Temperature results for both the hot- and cold-cases was plotted versus external emissivity (EXT_EMS) as shown in Figure 8.34. Also shown on this figure are an assumed acceptable component temperature range of 0 to 40°C (i.e. 273 K to 313 K). The maximum temperature difference, Tmaxd, was found to be approximately 40 K for all values of EXT_EMS. This figure illustrates that the proposed design would provide acceptable temperature conditions over the range of assumed conditions (i.e. hot- and cold-case conditions). The proposed design is capable of providing these acceptable temperatures through emissivity variations. In the cold-case situation, an emissivity of 0.23 to 0.28 is required. In the hotcase situation, an emissivity of approximately 0.75 is necessary. This simple analysis illustrates that the proposed design would be an acceptable design strategy. However, it
165 should be mentioned that it is based on several assumptions including the given temperature margins (i.e. 0 to 40°C) and component distribution. In addition, the variability in the reduced-order model results is not shown. However, this variability was shown to be relatively small.
Figure 8.34: Tmax and Tmin Temperature Results versus external emissivity (EXT_EMS) for both Cold- and Hot-Case Conditions. 8.5 Conclusions Reduced-order models to predict satellite temperature responses for an 11-factor computer simulation model were developed. In total, eight reduced-order models were developed (one for each level of the categorical variables) based on Tmax, Tmin, and Tmaxd response data obtained from computer simulations run at evaluation points determined using Latin Hypercube sampling. A Gaussian Process regression model approach was used to fit the subsequent data. The eight reduced-order models were tested using 100 randomly generated test cases. The reduced-order model results were compared to computer simulation results by subtracting the two results at all test points. Residuals, found for Tmax, Tmin, and Tmaxd responses, had means of -0.1448 K, 0.06414 K, and 0.08643 K, respectively.
166 Additionally, the Tmax, Tmin, and Tmaxd responses had standard deviations of 1.547 K, 1.077 K, and 1.518 K, respectively. The approach outlined here illustrates the utility in developing reduced-order models. These surrogate models were shown to provide acceptable results over more computationally expensive computer simulations. Consequently, these models are ideally suited for rapid evaluation of a wide-range of satellite thermal control subsystem design approaches. Further, these models do not require users to have an extensive background in thermal modeling and/or access to expensive thermal simulation software. Consequently, the models developed here could be the beginning of a central database of reduced-order models accessible by the AFRL community. However, the limits of these models should be well understood. These models do not include all approaches that might be of interest to a user (e.g. traditional design approaches including external insulation and deployable radiators). Additionally, these models are not well suited as a replacement for thermal modeling reserved for specific satellite designs (e.g. TacSat-2 described in Appendix A).
167
CHAPTER 9 - CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
Since 1957, satellites have provided tremendous opportunities; however, space presents extraordinary challenges. Consequently, satellites have become exceedingly complex and costly. Current satellites can take from 3 to 7 years to deploy and cost from millions to billions of dollars. To reduce development time and cost while maintaining the United States’ military space authority, a Responsive Space (RS) vision has been initiated to revolutionize the methods in which satellites are designed, developed, and put into use. The objective of RS is to quickly deliver low cost, short-term tactical capabilities to address unmet warfighter and intelligence needs. Development of a robust TCS able to meet the challenges of RS will prove to be one of the most significant obstacles. Although it can rely on relaxed design methodologies to align with the philosophy of RS, it must be robust enough to meet a considerably larger design space over traditional optimized systems. This research presents design and implementation tools supporting the development of Responsive Space thermal control subsystems. An extended literature review revealed topics requiring immediate attention and others providing promising contributions. Consequently, several diverse works were carried out and conclusions provided henceforth. 9.1 Determination of Hot- and Cold-Case Design Orbits for Robust Thermal Control Subsystem Design Single hot- and cold-case design orbits that work well in the design of robust thermal control subsystems over a wide range of satellite surface properties and likely operating environments were determined. A general approach was developed which employed a combination of statistical and historical data such that statistically insignificant orbits were disregarded. From the analysis, a hot-case design orbit was found ( β = 72°,
168 i = 52°, altitude = 350 km) which provided orbital averaged heat fluxes within 0.5% of
the values determined using design orbits specific to individual surface types. A general cold-case design orbit was found ( β = 0°, i = 28°, altitude = 1000 km) which provided orbital averaged heat fluxes within 9.0% of the values determined using design orbits specific to individual surface types. If more accuracy is required, two cold-case design orbits were identified. For spacecraft with predominantly solar reflector surfaces, a coldcase design orbit ( β = 0°, i = 74°, altitude = 1000 km) provides minimum orbital averaged heat fluxes. For all other surface types, a cold-case design orbit was found ( β = 0°, i = 28°, altitude = 350 km) which provided orbital averaged heat fluxes within 3.4% of the values determined using design orbits specific to individual surface types. These results were based on a spherical satellite model and therefore, care should be taken when significant departures from this assumption are encountered. This includes both geometric and attitude deviations such as those for a Sun-pointing solar panel. In addition, the design orbits are intended for LEO missions with altitudes in the range of 350 to 1000 km. Consequently, their use for highly elliptic orbits and those with relatively large altitudes is not recommended. The hot- and cold-case design orbits have many potential applications. First, they provide consistent design criteria for the development and comparison of robust thermal control subsystem approaches. In the absence of clearly defined mission criteria, these design orbits also provide a reasonable surrogate. Finally, they are useful for the design of robust spacecraft over a wide range of both satellite surface properties and low Earth orbit operating environments. 9.2 Optimal Distribution of Electronic Components to Balance Environmental Fluxes A computational tool was developed which provides rapidly optimized component distributions over multi-sided structures each with unique environmental loading. The tool is based on a genetic algorithm utilizing a combination of elitist strategies, reproduction, best- and worst-fit heuristics, and mutation. A tuning study revealed appropriate convergence criteria, population size, and values for evolution parameters. The results showed that a strictly rule-based approach would provide the least
169 computational expense and provide reasonable results for the cases considered here. However, this approach might not be suitable for cases not considered. Consequently, an approach that takes full advantage of the capabilities of the algorithm was used. Computational requirements, on average, are expected to be approximately 12 seconds or less for up to 54 components using a 2.5 gigahertz dual-core computer. The maximum computational requirements are expected to be less than approximately 22 seconds for up to 54 components using the same machine. Optimized, even, and worst-case distributions for Case C-36 in the hot-case orbit were found for total power of 100 W up to 1200 W. On average, optimized component distributions reduced maximum temperatures by 5.4 K over evenly distributed components. Optimizing component distribution increased minimum temperatures by 7.1 K, on average, over evenly distributed components. Finally, optimized component distributions reduced maximum temperature differences by 12.6 K, on average, over evenly distributed components. The largest and smallest maximum temperature difference reductions were 17.8 K (at 400 W) and 9.5 K (at 100 W), respectively. 9.3 Optimal Placement of Electronic Components to Minimize Heat Flux NonUniformities A computational tool was developed which provides rapidly optimized component placement within an individual panel to approach a uniform heat flux distribution. The tool is based on a genetic algorithm utilizing a combination of elitist strategies, reproduction, local gradient searches, and mutation. A tuning study revealed appropriate convergence criteria, population size and values for evolution parameters. Optimized results were obtained for 18 uniform and 11 non-uniform components within 20 s and 7 s, respectively, using a 2.5 gigahertz dual-core processor. Advantages of this method include no need for thermophysical properties and boundary conditions. Optimized results are obtained using only component averaged power and domain size. Consequently, this approach is ideally suited to situations where limited information is readily available. In addition, limiting the required inputs provides for relatively fast solutions. However, care should be taken to ensure that a uniform distribution of fluxes is required for optimized placement. This robust and fast approach
170 can be utilized in a variety of applications including microelectronics and satellite development and is especially suited to those demanding low computational expense. 9.4 Development and Evaluation of Reduced-Order Satellite Thermal Models Reduced-order models to predict maximum satellite temperature difference for an 11 factor computer simulation model was developed. In total, eight reduced-order models were developed (one for each level of the categorical variables) based on Tmaxd data obtained from computer simulations run at evaluation points determined using Latin Hypercube sampling. A Gaussian Process regression model approach was used to fit the subsequent data. The eight reduced-order models were tested using 100 randomly generated test cases. The reduced-order model results were compared to computer simulation results by subtracting the two results at all test points. Residuals, found for maximum temperature, minimum temperature, and maximum temperature difference over all components, had means of -0.1448 K, 0.06414 K, and 0.08643 K, respectively. Additionally, the maximum temperature, minimum temperature, and maximum temperature difference responses had standard deviations of 1.547 K, 1.077 K, and 1.518 K, respectively. The approach outlined here illustrates the utility in developing reduced-order models. These surrogate models were shown to provide acceptable results over more computationally expensive computer simulations. Consequently, these models are ideally suited for rapid evaluation of a wide-range of satellite thermal control subsystem design approaches. Further, these models do not require users to have an extensive background in thermal modeling and/or access to expensive thermal simulation software. Consequently, the models developed here could be the beginning of a central database of reduced-order models accessible by the AFRL community. However, the limits of these models should be well understood. These models do not include all approaches that might be of interest to a user (e.g. traditional design approaches including external insulation and deployable radiators). Additionally, these models are not well suited as a replacement for thermal modeling reserved for specific satellite designs.
171 9.5 Assumptions and Limitations of Work Results presented in this thesis were based on specific assumptions and approaches. Consequently, care should be taken to ensure that these results are applied in an appropriate manner. Although provided throughout this thesis, a concise summary of assumptions and limitations will be presented here for convenience. The following subsections, broken down by research effort, provide a summary of the underlying assumptions and resulting limitations of each work. 9.5.1 Determination of Hot- and Cold-Case Design Orbits for Robust Thermal Control Subsystem Design 9.5.1.1 Assumptions •
Sun coordinates for the determination of temporal beta angle predictions were based on a low-precision method which provides the equation of time to a precision of 0m.1 between 1950 and 2050 due to expansion truncation and coordinates of the Sun to 0.01° (Vallado, 2007; Astronomical Almanac, 2007). Additionally, Sun coordinates were developed under the following assumptions: 1) gravity is the only force, 2) a spherically symmetric Earth, 3) the Earth’s mass is much greater than the satellite’s, and 4) the Earth and the satellite are the only two bodies in the system.
•
Perturbing forces that can influence orbits include: solar radiation pressure, atmospheric drag, gravitational forces of the Sun and Moon, and forces as a result of a non-spherical Earth. Of these, perturbations due to a non-spherical Earth dominate the other sources; therefore, they were the only type of perturbations used in this study (Wertz and Larson, 2005).
•
Man-made plane changes and naturally occurring forces (e.g. luni-solar forces) were neglected. Natural occurring forces have been found to be very small (~1/100° per year) (Gopinath et al., 2004). Consequently, orbit inclination, i , was assumed constant.
•
Historical inclination information, used in the development of weighting matrices, was based on data from 1957 to 2006 (Satcat.txt, 2007) which includes 29,493
172 unclassified orbiting objects (i.e. satellites, debris, and rocket fairings). Classified information was not utilized. •
Orbital-averaged external environmental heat flux values were based on a spherical satellite model in circular orbits at altitudes from 350 to 1000 km. Hotand cold-case heat loads were based on three primary components: direct solar, albedo, and outgoing longwave radiation (OLR) flux (i.e. Earth emitted radiation). Albedo heat flux was based on the assumption of a diffusely reflecting spherical Earth. These results were verified within 3.2% of values obtained from MonteCarlo ray tracing software.
•
The characteristic temperature of space is 3 K and as a result, deep space radiation was neglected as a radiation source (e.g. assumed to be at 0 K) with negligible error (Justus et al., 2001). In addition, collisions with atmospheric gasses can provide an additional heating source (Gilmore, 2002). This free molecular heating occurs only at exceptionally low orbit altitudes (below 180 km) and during fairing separation. Consequently, this heating source was disregarded for the analysis. Finally, charged particle heating is the result of near-Earth trapped charged particles in the Van Allen belts. This weak source of heating is typically disregarded for typical thermal analyses (Gilmore, 2002).
•
The Earth Radiation Budget Experiment (ERBE) was the basis for albedo and Earth emitted radiation values. From this database of information, 90 minute time averaged values were chosen, which were the closest to the period of low Earth orbit (LEO) orbits.
9.5.1.2 Limitations •
The hot- and cold-case design orbits are intended for LEO missions with altitudes in the range of 350 to 1000 km. Consequently, their use for highly elliptic orbits and those with relatively large altitudes is not recommended.
•
Heat flux models used in the development of the design orbits were based on an orbital-averaged value. This was chosen due to the relatively large thermal capacitance of typical satellite architectures. Consequently, these design orbits are not suited for devices with low thermal capacitance (e.g. deployable solar panels).
173 For these devices, an instantaneous heat flux value would be a more suitable approach in determining design orbits. This was not part of the work presented here. •
Orbit-averaged heat flux results are based on a spherical satellite model and therefore, care should be taken when significant departures from this assumption are encountered. The hot- and cold-case design orbits are ideally suited for symmetric satellite shapes (e.g. six-sided architectures). These design orbits should not be applied to structures that significantly deviate from this symmetry (e.g. deployable solar panels).
•
Historical inclination information was based on unclassified data. Classified missions might produce a different distribution of inclinations and therefore would influence the resulting hot- and cold-case design orbits.
9.5.2 Optimal Distribution of Electronic Components to Balance Environmental Fluxes 9.5.2.1 Assumptions •
All satellite panels (i.e. face) are assumed to have the same optical properties and receive a uniformly-distributed orbital-averaged heat load from the environment. Additionally, each panel is assumed to dissipate energy to the surroundings via longwave radiation at the same temperature and emissivity.
•
It is assumed that component heat gains occur on one side of the face while environmental heat loads and dissipation occur on the opposite side.
•
Algorithm tuning was based on three unique component power distributions that bound the range of possible power distributions (Case A, Case C, and Case D).
•
Computational requirement results were based on the use of a 2.5 gigahertz dualcore computer.
•
Optimized, even, and worst-case distribution results for total power of 100 W up to 1200 W were based on a 1.0 m x 1.0 m x 1.0 m six-sided frame-and-panel satellite architecture in the hot-case orbit. Panels were constructed of 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material. Component power was based on a Case C-36 component distribution.
174 9.5.2.2 Limitations •
The global distribution optimization algorithm was developed to improve the thermal performance of satellites by approaching isothermal bus conditions. This is achieved by approaching the same net heat flux due to component and environmental sources on each face. Consequently, this approach is not suited to situations where isothermal conditions are not desired.
9.5.3 Optimal Placement of Electronic Components to Minimize Heat Flux NonUniformities 9.5.3.1 Assumptions •
Case studies were based on a single 1.0 m x 1.0 m honeycomb panel with 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material. Multiple heat-generating components, each with a contact area of 10 cm x 10 cm, were attached to one side of the panel. Heat generation of all components was conducted to the panel only (i.e. not lost through radiation) with a contact conductance of 110 W/m2-K. All heat generation was ultimately dissipated to a deep space environment (0 K) through radiation from the bottom surface of the panel with an emissivity of 1.0.
•
Algorithm tuning was based on the Case D power distribution for 3 to 18 components. The algorithm was demonstrated using 100 W of total component power in two distributions. The first used a Case D-18 distribution (i.e. 18 uniformly powered components) while the second used 11 non-uniformly powered components (i.e. 20.3, 17.3, 14.6, 12.2, 10.0, 8.1, 6.3, 4.7, 3.3, 2.1, and 1.0 W).
•
The local placement optimization algorithm was developed based on the assumption that component heat is spread evenly in all directions (i.e. radial heat spreading).
•
Computational requirement results were based on the use of a 2.5 gigahertz dualcore computer.
175 9.5.3.2 Limitations •
The approach is limited to situations where a uniform distribution of fluxes provides for optimized thermal performance.
•
The local placement optimization algorithm only considers thermal effects. Consequently, it does not consider the impact of component interference (i.e. fit issues).
•
The optimization approach is based on circular effective areas for each component. Consequently, deviations from symmetric shapes (e.g. rectangular components with high aspect ratios) could be problematic.
9.5.4 Development and Evaluation of Reduced-Order Satellite Thermal Models 9.5.4.1 Assumptions •
The reduced-order satellite thermal models were based on a 1.0 m x 1.0 m x 1.0 m six-sided frame-and-panel satellite architecture with honeycomb panel construction having 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material. The satellite is nadir facing and has 36 components with a Case C power distribution (Case C-36) with total power of up to 600 W. The number of components per panel was limited to six to provide a more realistic optimized result (i.e. each panel has limited space to place components). However, global and local optimization routines were applied to not only optimally distribute power throughout the 36 components but also local optimize the component placements on each panel.
•
Reduced-order models were based on specific factor ranges and should only be applied within these limits. Specific factor ranges are described in the limitations section to follow.
9.5.4.2 Limitations •
Evaluations using the reduced-order models are limited to the 11 factors summarized in Table 8.2.
176 •
Output of the reduced-order models are limited to maximum component temperature, minimum component temperature, and maximum temperature difference across all components.
•
The reduced-order models can be used between the following levels: o Total component power from 60 to 600 W o Component footprint from 10 cm x 10 cm to 20 cm x 20 cm o Component interface heat transfer coefficient from 110 W/m2-K to
700 W/m2-K o Facesheet transverse thermal conductivity from 170 W/m-K to
1,000 W/m-K o Embedded heat pipes at one of two levels: 0 and 10 per panel. A
traditional heat pipe construction was modeled with a diameter of 0.9525 cm (0.375 in) and a vapor core of 0.5080 cm (0.200 in) diameter. The evaporator and condenser heat transfer coefficients were set at 11,356 W/m2-K and 5,678 W/m2-K, respectively. o Panel-to-panel conductance from 12 W/K to 36 W/K o Satellite exterior emissivity from 0.100 to 0.900 o Satellite exterior absorptivity from 0.123 to 0.561
•
The reduced-order models were based on 36 components with a Case C power distribution. Use at other power distributions is not recommended.
9.6 Suggestions for Future Work The work summarized in this thesis provides a wide-range of tools necessary and valuable for robust TCS design. Although this work is broad in nature, there still exists opportunity for continuing research efforts in these areas. The following topics, broken down by subjects covered in this thesis, are suggested ideas and approaches for future research. 9.6.1 Research in Design Orbits for Robust Thermal Control Subsystem Design •
The orbit study summarized in this work focused on low-Earth orbits. However, robust and RS missions are not limited to LEO. Consequently, it is recommended
177 that a similar analysis be applied to other orbit types including highly-elliptical and geosynchronous orbits. For orbits other than circular, the analysis approach will need to be expanded to include the expected range of eccentricities and argument of perigee rotations. •
The orbit study was based on a symmetric satellite model (i.e. sphere). This approach is acceptable for many satellite designs such as the six-sided satellite model used throughout this thesis. However, some satellite or component architectures have a significantly different shape factor. Additionally, design orbits will be dependent on pointing. For example, solar panels deviate from the symmetric satellite model and are significantly impacted by pointing. Consequently, the design orbits developed here should be expanded to include a wide-range of shape factors and pointing conditions.
9.6.2 Component Placement Optimization Research •
Global optimization routines were developed to improve the thermal performance of satellites. This approach was based on orbital-averaged thermal environmental factors based on one design orbit (i.e. beta angle). However, due to beta angle drift, satellites will encounter a wide-range of beta angles over their life for a given inclination (refer to Figure 5.15) Values include minimum, maximum, and average beta angles. However, the global optimization routine is based on one specific orbital environment (i.e. beta angle). Therefore, for a given inclination, analysis should be conducted to determine the ideal beta angle that the global optimization algorithms should be applied. For example, if components were globally optimized for the maximum beta angle then how will the optimized power distribution perform under minimum beta angle conditions.
•
An optimization routine was developed to improve satellite thermal performance through intelligent placement of components. This approach was based on optimization over rectangular domains. However, satellite architectures sometimes consist of irregularly shaped faces (e.g. hexagons). Consequently, this optimization approach should be extended to include other satellite panel shapes.
178 •
Global component distribution and local component placement optimization routines were developed to be used in series (i.e. globally optimize distribution then locally optimize placement). Although this approach has been shown to provide significant thermal improvements at acceptable computational cost, it overlooks the interactions between panels. After local optimization of a panel, its rotation/orientation in relationship to surrounding panels is arbitrarily defined. It is anticipated that further improvements could be made by examining these interactions between panels.
9.6.3 Research in Reduced-Order Satellite Thermal Models •
Reduced-order models were developed to evaluate 11 factors over specific ranges. These models were shown to perform well; however, their use is limited to the specific factors and ranges. Consequently, additional computer experiments could be run to expand the number of factors and factor ranges that the reduced-order models could be used for.
•
Reduced-order models were shown to be a useful surrogate to more computationally-expensive computer simulations. However, this approach requires running a large set of computer simulations that can be costly depending on the number of factors of interest. Consequently, development of reduced-order models based on physical phenomenon (e.g. thermal spreading, Fourier’s Law, Stefan-Boltzmann Law) without the need to run large sets of computer simulations would be useful. This approach has the potential to be more robust than models based on computer experiments. However, this approach presents several challenges including non-linear boundary conditions and superposition of heat loads in a non-linear system.
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179
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192 Williams, A., 2005, Robust satellite thermal control using forced air convection thermal switches for operationally responsive space missions, Master’s Thesis, University of Colorado, Department of Aerospace Engineering Sciences. Williams, A. and Palo, S., 2006a, Issues and implications of the thermal control system on responsive space missions, Proceedings of the 20th Annual AIAA/USU Conference on Small Satellites (Publication No. SSC06-3-1), Logan, UT, pp. 112. Williams, A. and Palo, S., 2006b, Modeling and analysis of a robust thermal control system based on forced convection thermal switches, Proceedings of SPIE – Modeling, Simulation and Verification of Space-based Systems III (v. 6221), pp. 622108(1-11). Xu, Y. and Ren-Bin, X., 2006, Hybrid particle swarm algorithm for packing of unequal circles in a larger containing circle, Proceedings of the 6th World Congress on Intelligent Control and Automation, Dalian, China, pp. 3381-3385. Young, Q., 2008, Thermal management for modular satellites (TherMMS), Final Report (Report No. AFRL-RV-PS-TR-2008-1073), Space Dynamics Lab – Utah State University Research Foundation, Logan, Utah. Young, Q., Stucker, B., Gillespie, T. and Williams, A., 2008, Modular thermal control architecture for modular spacecraft, 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Publication No. AIAA 2008-1959), Schaumburg, IL, pp. 1-8. Zeng, W., Shi, Y., and Teng, H., 2006, A co-evolutionary game optimization method for layout design of satellite module, IEEE, pp. 270-273.
APPENDICES
193 Appendix A – Review of Robust Satellite Architectures
Several satellite architectures have been developed to respond to the challenges of RS. The following summarizes some of those efforts. A.1 TacSat The ORS office is proceeding with a demonstration program consisting of a series of space flights used to illustrate some of the advancements in RS (Wegner et al., 2005). These TacSats will be used to test and verify the capability of potential RS technologies including: modularity, plug-and-play-components, low-cost capable payloads, low-cost responsive launch vehicles, semi-autonomous operations, responsive command/control, and data dissemination, innovative science and technology development, and acquisition processes (Wegner et al., 2005). Five TacSat satellites have been or are currently being developed. TacSat-1 was the initial demonstration effort led by the NRL (Figure A.1). The year long development cost approximately $23 million (U.S. GAO, 2008). This 108 kg satellite was designed for a circular 410 km 40° inclination orbit. Although scheduled for a 2004 launch, due to development delays the program was cancelled in 2007 in favor of TacSat-2. There has been renewed interest in this satellite and a launch in the future could be expected (Berger, 2007).
Figure A.1: TacSat-1 (Raymond et al., 2004).
194 TacSat-2 (Figure A.2) was an effort led by the AFRL taking 29 months to develop at a cost of $39 million (U.S. GAO, 2008). This 370 kg satellite was designed for an approximately 410 km circular 40° inclination orbit (Peck, 2006). It was launched in December 2006 on a Minotaur I.
Figure A.2: TacSat-2 (Peck, 2006). TacSat-3 is a $62.7 million effort led by the AFRL to develop the first implementation of bus standards (U.S. GAO, 2008 and Raymond et al., 2005). This 400 kg satellite was designed for an approximately 425 km circular orbit (TacSat-3 Fact Sheet, 2006). TacSat-3 is expected to launch in late 2008 on a Minotaur I. TacSat-4 is a NRL led endeavor estimated to cost $114 million (U.S. GAO, 2008). TacSat-4 will evaluate current and provide the next generation of bus standards (Raymond et al., 2005). It is expected to launch into a highly elliptical orbit for communications purposes in 2009 on a Minotaur IV (Doyne et al., 2006). TacSat-5 is a collaborative effort with the Army Space and Missile Defense Center, the Joint ORS Office, AFRL, and Space and Missile Systems Center (U.S. GAO, 2008). Currently, this satellite is in development. A.2 Plug-and-Play Sat (PnPSat) The AFRL conducted a Responsive Space Advanced Technology Study (RSATS), which determined that plug-and-play (PnP) technologies such as Universal Serial Bus (USB), SpaceWire, Firewire, and Ethernet could be utilized in spacecraft (Wegner and Kiziah, 2006). Ultimately, application of these technologies could provide some of the answers to RS questions. Consequently, a responsive satellite cell was developed with the
195 goal of demonstrating the real-time events from call-up of a new spacecraft to launch (Wegner and Kiziah, 2006). Efforts include the development of a first-generation PnP satellite bus. PnPSat is an approximately 51 x 51 x 61 cm modular frame and panel design (Figure A.3). Each of the six panels are a 6061-T6 aluminum isogrid structure (Fronterhouse, Lyke, and Achramowicz, 2007).
Figure A.3: Plug-and-Play Satellite General Schematic (Spaceworks, Inc., 2008). A.3 HexPak The HexPak satellite architecture, developed by Lockheed Martin, consists of stackable hexagonal bays with a large radius in comparison to its height. The hexagonal bays when stacked easily fit within the fairing of a launch vehicle. These bays are interconnected with hinges and latches and when unfolded provide a large planar structure (Figure A.4) (Hicks, Hashemi, and Capots, 2006). Each bay provides a separate module that allows for parallel AI&T.
Figure A.4: Illustration of HexPak Deployment (Hicks, Enoch and Capots, 2005). Although Lockheed Martin is developing independent PnP avionics for HexPak, they are expected to align with the SPA efforts of AFRL.
196 A.4 SMARTBus AeroAstro’s SMARTBus spacecraft architecture is an attempt to provide a modular, robust satellite design capable of responding to the unique challenges of RS. The SMARTBus concept relies on a modular bus design consisting of individual hexagonal structures stacked on top of one another (McDermott and Jordan, 2005). Individual modules perform specific spacecraft functions such as communication, attitude determination, power storage, and solar coupling. Modules utilize standard designs such that they can be produced ahead of time. When needed, modules performing the needed functions are pulled off the shelf, stacked, tested, and attached to a payload in a few hours.
Figure A.5: Illustration of SMARTBus Concept (McDermott and Jordan, 2005). Intrinsic to the success of this design is smart software architecture. As individual models are connected, they must be able to recognize each other similar to PnP devices currently available in COTS systems. Due to the unique requirements of satellite design, AeroAstro has improved upon PnP and implemented a plug-and-sense (PnS) system. In a traditional PnP system, equipment is able to identify itself and its capabilities on a common bus. All necessary information required to operate that equipment is contained within the equipment itself. In a PnS system, identification of the device’s physical characteristics is included such as mass, size, shape, center of gravity, moments of inertia, and position within the spacevehicle. These details are crucial in satellite design (McDermott and Jordan, 2005). Under the Small, Smart, Spacecraft for Observation and Utility Tasks (SCOUT) program funded by DARPA, AeroAstro has developed and fabricated a prototype of the PnS architecture.
197 The Flexible, Extensible Bus for Small Satellites (FEBSS) program sponsored by AFRL is developing three specific SMARTBus modules: a battery module, a communication module, and a fold-out solar array module (McDermott and Jordan, 2005). A.5 MightySat MightySat is a United States Air Force (USAF) small satellite program with objectives to transition advanced space technologies from the laboratory to the warfighter in a timely manner. MightySat II is a three-axis stabilized 123.7 kg satellite launched in July 2000 at a total cost of $36.4 millions (MightySat II DataSheet, 2005). Maximum battery power is 330 W with nominal levels closer to 100 W.
Figure A.6: Illustration of MightySat II.1 (MightySat II DataSheet, 2005). The Thermal Control Subsystem (TCS) utilizes the satellite’s structure as well as insulation and control. The composite structure of MightySat II distributes thermal loads through fibers within the material that are connected to radiator panels. Exposed surfaces are covered with 15-layer, gold-colored insulation blanketing. Component and payload ‘on-off’ management provides active thermal control (Freeman, Rudder, and Thomas, 2000). Integration and Testing (I&T) was accomplished on components and systems over 15 months during which 173 problems or failures were discovered and corrected. Testing
198 included vibration, separation and pyrotechnic shock, and thermal balance (Freeman, Rudder, and Thomas, 2000). A.6 CubeSat CubeSats were developed jointly by Stanford and California Polytechnic State University and currently consists of collaboration between 40 universities, high schools, and private firms. The objectives of this satellite architecture include: 1) providing a low cost way of learning how to build satellites, 2) providing a test-bed for new space devices and ideas, and 3) providing an architecture that responds to the goals of RS (Ince, 2005). The CubeSat is standardized satellite architecture in the shape of a cube with 10 cm sides and a maximum mass of 1 kg (CubeSat Design Specification, 2008). Because of standardization, they are cheaper and faster to develop than current one-of-a-kind technologies, but do not offer the same level of capabilities as traditional satellites (Ince, 2005). The development of a CubeSat is typically two years where the rapid development time is partly due to the availability of standardized CubeSat parts (e.g. solar panels, microprocessor boards, connectors, etc.). The fact that CubeSats are not as capable as traditional satellite designs along with the fact that they are relatively young (4 to 5 years) has isolated their use to an educational environment.
199 Appendix B – Review of Robust Thermal Control Subsystem (TCS) Approaches
The following is a summary of a review of RS TCS concepts that have been developed. Concepts include the Thermal Management for Modular Satellites, Satellite Modular and Reconfigurable Thermal System, Forced Air Convection Thermal Switch, HexPak, SMARTBus, and Integrated Thermal Energy Management System. B.1 Thermal Management for Modular Satellites (TherMMS) Young (2008) investigated enabling thermal control architectures for fully modular spacecraft necessary for RS (Figure B.1). These included thermal bus concepts from traditional up to modular isothermal approaches. Evaluation revealed that isothermal bus architectures provide the greatest modularity and thermal performance for a reasonable increase in complexity (Young, 2008).
Figure B.1: TherMMS TCS Candidate Solutions (Young, 2008).
200 The thermal control architectures were applied to each of three architectures: frame and panel, shelf and building block. Young (2008) revealed that in general, each of the spacecraft architectures behaved in a similar manner and the isothermal thermal bus architectures were shown to be flexible in the ability to adapt and scale to various spacecraft types (Young, 2008). In addition, varying the heat radiated to space significantly increased thermal performance and reduces or eliminates the need for survival heater power. The evaluation of the modularity and thermal performance showed that isothermal bus architecture with thermal balance modulation on the heat rejection side enables full modularity in the thermal control design (Young, 2008). The key characteristics that enabled this are the decoupling of the thermal control subsystem from the system and sufficient control authority, or dynamic range. B.2 Satellite Modular and Reconfigurable Thermal System (SMARTS) The SMARTS concept adheres to four design rules: 1) modest radiator oversizing, 2) maximum external insulation, 3) internal isothermalization and 4) radiator heat flow modulation (Bugby, Zimbeck, and Kroliczek, 2008). The SMARTS TCS is based on a frame and panel satellite architecture consisting of multiple Al-isogrid or Al-honeycomb panels bolted together along common edges. An embedded heat pipe is placed around the edges of each panel, while C-shaped heat pipes are distributed throughout the center (Figure B.2). The combination of these two heat pipes allows for heat spreading across each panel and attempts to approach isothermal conditions.
Figure B.2: SMARTS TCS Concept (Bugby, Zimbeck, and Kroliczek, 2008).
201 Several RS SMARTS TCS configurations were modeled in SINDA based on many of the modeling requirements of Williams (2005). The one-meter cube shaped model include several different panel constructions from Al isogrid and no heat pipes up to AlAPG isogrid with one circumferential and six spreader heat pipes. B.3 Forced Air Convection Thermal Switch (FACTS) This thermal design was based on a concept of a hermetically sealed enclosure allowing the use of convection heat transfer (Williams, 2005). During the hot case, convection was utilized to increase heat transfer of the system. The fan was switched off for cold case conditions thus heat transfer is solely through conduction throughout the satellite. It was shown a conductance ratio on the order of 69:1 was achievable resulting in a more robust TCS and a significant reduction in survival heater power (Williams, 2005). The advantage of this system is using a simple DC axial fan as a thermal switch and reduced clean room requirements due to the sealed enclosure (Williams and Palo, 2006b). Disadvantages include added mass and introduced vibrations due to the fan that could be problematic for the attitude control subsystem (Williams and Palo, 2006b). B.4 HexPak The HexPak satellite architecture provides a unique method of responding to the challenges of RS by introducing a deployable structure aimed at maximizing the available radiating area (Hicks, Hashemi and Capots, 2006). It consists of stackable hexagonal bays that when unfolded provide a thermally advantageous large radiating area (Hicks, Hashemi and Capots, 2006). In effect, it provides for a linear relationship between radiating area and mass (Hicks, Enoch and Capots, 2005). During the hot case, the temperature ranged from -12°C to 39°C and -38°C to 73°C with an applied uniform heat flux of 400 W/m2 to 800 W/m2, respectively (Hicks, Hashemi and Capots, 2006). These temperatures are out of the acceptable range and therefore additional strategies must be implemented. These include passive thermal spreading to neighboring bays and embedded heat pipes. An additional concept involves the use of thermal footprints adjacent to components. These radiative areas are sized such
202 that the maximum heat dissipation requirements of the components can be achieved (Hicks, Hashemi and Capots, 2006). During the orbit cold case, temperatures ranged from -126°C up to -14°C (Hicks, Hashemi and Capots, 2006). Since this is well out of the acceptable range, survival heaters must be installed. Variable conductance joints could also be utilized between the components and the structure to reduce the capacity of these heaters (Hicks, Hashemi and Capots, 2006). B.5 SMARTBus SMARTBus is a shelf architecture consisting of several stacked modules. Shelf designs appear to be more strongly influenced by electrical engineering concepts and usually have well-defined electrical and mechanical interfaces between shelves. The removal of heat and the fixed interface (particularly where the shelf stack can only grow in one dimension) seem to be the greatest drawback for this architecture (Young, 2008). Under the SCOUT program, modules were interconnected via an external ‘backbone’ providing a thermally conductive path between modules. This aids in isothermalizing the satellite both transferring excess heat from hot modules and allowing the outside surface to radiate excess heat (Rogers, Cameron, and Jordan, 2003). Under the FEBSS program, a thermal system was developed allowing a SMARTBus vehicle of any configuration to fly in any LEO orbit at any orientation thus eliminating custom thermal analysis and design unnecessary (McDermott and Jordan, 2005). This proprietary system allows each module to maintain its thermal balance without regard to those above or below it. B.6 Integrated Thermal Energy Management System (ITEMS) The Integrated Thermal Energy Management System (ITEMS) developed by JPL is an attempt to provide universal spacecraft thermal control enabling faster and cheaper design cycles (Birur and O’Donnell, 2001). Although developed for deep space exploration (e.g. planet, moon, comet, and sun exploration) and microsatellites (i.e. 0.5 m cube at 50 W), the overall development is consistent with those for RS TCS including heat load sharing, heat rejection modulation, and minimized survival heater power.
203 The ITEMS approach utilizes a cooling loop to thermally integrate all spacecraft subsystems in conjunction with thermal switches and valves (Figure B.3). The heat rejected from one subsystem is transferred to another subsystem where the heat is needed to maintain its minimum temperature. Any excess heat generated in the spacecraft above what is needed is rejected at a deployable radiator system that uses variable emittance devices on its surface. In addition, a phase change material is used for battery thermal control. This type of architecture provides the needed flexibility and accommodates lowcost overall design and implementation.
Figure B.3: ITEMS TCS Concept (Birur and O’Donnell, 2001).
204 Appendix C - Astrodynamics
Astrodynamics is the study of a satellite’s trajectory or orbit. Although a full understanding is not necessary for the scope of work, it is important to have a grasp of basic concepts. A discussion of equations of motion, sun coordinates, classical orbital elements, thermal orbital elements, perturbations, and orbit classifications is presented. C.1 Equations of Motion The motion of a satellite is largely defined by Kepler’s three laws. The first states that the orbit of each satellite is an ellipse with the Earth at one focus. The second reads that the line joining the satellite to the Earth sweeps out equal areas in equal times. One of the consequences of this is that a satellite is slowest at apogee (furthest point from Earth) and fastest at perigee (closet point to Earth). Finally, the third law states that the square of the period of a satellite is proportional to the cube of its mean distance from the Earth. Mathematically, Newton’s Law of Gravitation says that any two bodies attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. In vector form this becomes: Fg =
−G ⋅ M ⊕ ⋅ m r − μ ⋅ m r ⋅ = ⋅ . r2 r r2 r
(C.1)
Fg is the force caused by gravity, G is the universal gravitational constant, M ⊕ and m
are the masses of Earth and satellite, respectively and r is the distance from the center of the Earth to the center of the satellite. Earth’s gravitational constant, μ , a product of G and M ⊕ , equals 398,600.5 km3/s2. Combining Equation (C.1) with Newton’s 2nd Law of Motion gives the equation of motion describing the position of a Earth orbiting satellite r + ( μ ⋅ r −3 ) ⋅ r = 0 .
(C.2)
This equation assumes gravity as the only force, a spherically symmetric Earth, the Earth’s mass much greater than the satellite’s and the Earth and the satellite are the only two bodies in the system. This expression provides a tool describing a satellite’s location.
205 Another tool useful in describing the motion of a satellite is specific mechanical energy, ε , or sum of the kinetic and potential energy per unit mass (Sellers, 2005)
ε=
V2 μ − . 2 r
(C.3)
Since this is a conservative term, the sum of potential and kinetic energy must always be constant and as a result, knowing one provides enough information to determine the other if specific mechanical energy is known. A further derivation simplification of this equation was determined as shown in (Sellers, 2005) V 2 μ −μ − = . ε= 2 r 2⋅a
(C.4)
Where, V , is the satellite speed and a is the semi-major axis. For a circular orbit, a is equal to the orbital radius, r , so that we can directly calculate the velocity of a satellite, Vcir (Wertz and Larson, 2005) by, 1
1
− ⎛ μ ⎞2 ⎛ R ⎞2 Vcir = ⎜ ⎟ ≅ 7.905366 ⋅ ⎜ E ⎟ ≅ 631.3481 ⋅ r 2 . ⎝r⎠ ⎝ r ⎠ 1
(C.5)
RE is the Earth’s radius (6378 km). Therefore, satellite velocity for a circular orbit is
dependent only upon orbital radius. Mean motion, n , or average angular velocity, of a satellite becomes the following 1
⎛ μ ⎞2 n=⎜ 3⎟ ⎝a ⎠
(C.6)
which is utilized to determine orbital perturbations (Wertz and Larson, 2005). C.2 Sun Coordinates The coordinate system, mean equinox of date (MOD), is based on the intersection of the ecliptic and equatorial planes (i.e. γ direction) on a given date (Vallado, 2007). The MOD reference frame accounts for motion of the coordinate system as a result of
206 gravitational forces (Astronomical Almanac, 2007). All pertinent Sun coordinate variables illustrated in the following figure are based on this reference.
Figure C.1: Illustration of Sun Coordinate Variables. The right ascension of the Sun, Ω sun , is the angular distance measured eastward along Earth’s equator from the equinox to a plane perpendicular to the Earth’s equator and passing through the Sun (i.e. hour circle) (Vallado, 2007; Astronomical Almanac, 2007; and Wiesel, 1997). The declination of the Sun, δ , is the angular distance north or south of Earth’s equator measured along the hour circle (Astronomical Almanac, 2007). Ecliptic longitude, λ , is defined as the angular distance measured eastward along the ecliptic (i.e. the mean plane of the Earth’s orbit around the Sun) from the equinox to a plane passing through the poles of the ecliptic and the celestial object. The obliquity of ecliptic, ε , is the angle between the planes of the equator and the ecliptic and varies with Julian Day due to gravitational perturbations (Astronomical Almanac, 2007). Mean motion is a measure of the average angular velocity of an object. Therefore, mean longitude of the Sun, L , approximates the ecliptic longitude. The mean anomaly, g , describes the angle from the perihelion (i.e. the point where Earth is closet to the Sun) of an Earth moving with constant angular speed equal to the mean motion. This approximates the true anomaly, ν .
207 C.3 Classical Orbital Elements Traditionally, the description of an orbit is described with five constants and one quantity that varies with time. Together these are referred to as the six classical orbital elements (Figure C.2).
Figure C.2: Diagram of Six Classical Orbital Elements. The first element is semi-major axis which describes the size of an orbit while the second, eccentricity describes the shape (Figure C.3). In combination, these two elements fully specify the orbit (i.e. size and shape) in a single plane.
Figure C.3: Semi-major Axis and Eccentricity Diagram. Once the shape and size of an orbit is determined, it can be placed in relation to Earth. For elliptical orbits, Earth is at one of the foci. Before describing the remaining
208 orbital elements, it is important to understand two planes: the ecliptic and equatorial planes. The ecliptic plane defines the plane about which the Earth remains throughout its orbit around the sun. The equatorial plane, as its name implies, defines the plane that passes through Earth’s equator. The previously defined orbit is now placed on the equatorial plane. Further, the Vernal Equinox, also known as the Ram direction, is put in place as shown in Figure C.4.
Figure C.4: Ecliptic and Equatorial Planes along with Vernal Equinox and Orbit Normals. The third classical orbital element, inclination, is the angle between the orbital and equatorial planes. Further, it is the angle between the angular momentum vector of the orbit and the vector parallel to the Earth’s spin axis and is shown in Figure C.5.
Figure C.5: Diagram of Orbital Inclination.
209 The fourth classical orbital element is the right ascension of the ascending node (RAAN). It is the angle from the vernal equinox to the ascending node. The ascending node specifies the point at which the orbit crosses the equatorial plane traveling from south to north. This is shown in Figure C.6.
Figure C.6: Diagram of the Right Ascension of the Ascending Node (RAAN). With these references defined, the fifth classical orbital element can be defined. The argument of perigee is the angle measured from the ascending node to perigee. Perigee is the point where a satellite is closest to Earth and is shown in Figure C.7.
Figure C.7: Diagram of Argument of Perigee. The sixth classical orbital element can be defined. True anomaly is the angle measured from the argument of perigee to the satellite and is shown in Figure C.8.
210
Figure C.8: Diagram of True Anomaly. Figure C.2 provides a summary of the six classical orbital elements. For equatorial ( i = 0° ) orbits, a single angle, Π the Longitude of Perigee replaces the right ascension of the ascending node and argument of perigee. It is defined as the algebraic sum of Ω and ω . For circular (i.e. e = 0 ) orbits a single angle, u the Argument of Latitude replaces the argument of perigee and true anomaly where u = ω + ν as shown in Figure C.9. If the orbit is circular and equatorial a single angle, , true longitude specifies the angle between the vernal equinox direction and the satellite position vector.
Figure C.9: Diagram of Orbital Elements for Circular Orbits.
211 C.4 Thermal Orbital Elements Although the classical orbital elements are useful to describe the motion of satellites in orbit, they are inefficient when dealing with the satellite thermal environment. Consequently, several additional parameters will be presented. Beta angle, β , is the minimum angle between the solar vector and the orbit plane and can vary from -90° to 90° where the sign describes the direction of orbit as viewed from the Sun. Satellites with positive β appear to be going counterclockwise while negative β indicate a clockwise direction. Figure C.10 and Figure C.11 illustrate orbits with β = 0°. In Figure C.10, the inclination is 113.4° while the RAAN could be either 0° or 180° (depending upon satellite direction). In Figure C.11 the inclination is 0°.
Figure C.10: Diagram of β = 0° and i = 113.4°.
Figure C.11: Diagram of β = 0° and i = 0°.
212 Figure C.12 illustrates an orbit with a β of 90°. In this particular orientation, the inclination is also 113.4° with a RAAN of 90° or 270°.
Figure C.12: Diagram of β = 90° and i = 113.4°. As the previous figures illustrate, a β of 0° ensures that a satellite in this orbit will find itself in eclipse. A β of 90° ensures that a satellite will not be in eclipse. Further complicating the thermal environment of a satellite are β changes over time. The simplest example occurs for a satellite in an equatorial orbit (Figure C.13). This figure shows that over one year, the β changes from 0° to an absolute value of 23.4° due to the tilt of the Earth.
Figure C.13: Illustration of β Relationship Over One Year for a Circular Orbit at an Inclination of 0° and a RAAN of 0°.
213 The maximum beta angle, β max , varies as the declination of the Sun changes throughout the year and therefore changes with inclination by ± ( i + 23.4 ) . The absolute value of the maximum β varies with inclination by 1
β max = 23.4 + ⋅ ⎡⎣180 − ( i + 23.4 ) − 90 − ( i − 23.4 ) − 90 ⎤⎦ 2
(C.7)
and is shown graphically in Figure C.14.
Figure C.14: Absolute Value of the Maximum β versus i (0 to 180°). This relationship sets the limiting values of β and inclination. Further analysis was necessary to understand their interaction at interior points. Specifically, the distribution of
β for an arbitrary inclination was considered. The solar zenith angle, θ , describes the angle between the solar vector and the Earth satellite vector. For an arbitrary satellite, the minimum θ is equal to the β such that
θ≥β .
(C.8)
This relationship implies that orbits with large β will only have relatively large solar zenith angles. This is an important fact when considering environmental heat loads. For
214 satellites, the θ varies quickly throughout the orbit and can reach extremes when the Sun is at zenith ( θ = 0° ) to the Sun at nadir ( θ = 180° ). When describing the thermal environment of a satellite, it is necessary to describe the Earth/Atmosphere environment. The top of atmosphere (TOA) is described as a sphere that is 30 km greater than the equatorial radius, such that, RTOA = 6378 + 30 = 6408 km. C.5 Perturbations Although, the two-body equation of motion is quite powerful it does make several simplifying assumptions including: gravity is the only force, a spherically symmetric Earth, the Earth’s mass much greater than the Satellite’s and the Earth and the satellite are the only two bodies in the system. In particular, gravity is not the only force acting upon a satellite. Therefore, additional perturbing forces are present resulting in time varying classical orbital elements. Perturbing forces include solar radiation pressure, atmospheric drag, third-body perturbations such as gravitational forces of the Sun and the Moon, and perturbations as a result of a non-spherical Earth. Of these, perturbations due to a non-spherical Earth dominate other sources (Wertz and Larson, 2005). Earth is not spherically symmetric. In fact, it has a bulge at the equator (oblateness) of approximately 22 km (Figure C.15) because of its rotation (Sellers, 2005).
Figure C.15: Earth’s Oblateness.
215 Because of this, periodic variations in the orbital elements are present which are dominated by variations in the RAAN and the argument of perigee. The variation in RAAN is described by the following in deg/day (Wertz and Larson, 2005): 2
−2 ⎛R ⎞ Ω J 2 = −1.5 ⋅ n ⋅ J 2 ⋅ ⎜ E ⎟ ⋅ cos ( i ) ⋅ (1 − e 2 ) ⎝ a ⎠ .
≅ −2.06474 × 1014 ⋅ a
−
7 2
⋅ cos ( i ) ⋅ (1 − e
(C.9)
)
2 −2
The factor, J 2 , is the geopotential coefficient that dominates variations in RAAN and argument of perigee. As Equation (C.9) illustrates, the variation in RAAN is greatest at low inclinations and altitudes and decreases to zero at an inclination of 90°. The variation in the argument of perigee is described by the following in deg/day (Wertz and Larson, 2005): 2
−2 ⎛R ⎞ ωJ 2 = 0.75 ⋅ n ⋅ J 2 ⋅ ⎜ E ⎟ ⋅ ( 4 − 5 ⋅ sin 2 ( i ) ) ⋅ (1 − e2 ) ⎝ a ⎠ .
≅ 1.03237 × 1014 ⋅ a
−
7 2
⋅ ( 4 − 5 ⋅ sin 2 ( i ) ) ⋅ (1 − e
(C.10)
)
2 −2
Equation (C.10) becomes zero at an inclination of 63.4° and 116.6°. C.6 Orbital Classifications Orbits can be categorized by their position relative to the radiation environment, more specifically the Van Allen belts. This environment undergoes a significant change at an altitude of approximately 1000 km (Wertz and Larson, 2005). Because of this, Low Earth Orbits (LEO) are those defined above an altitude with significant atmospheric drag and below the severe radiation environment. This occurs in the range of 200 to 1000 km. Medium Earth Orbits (MEO) must contend with the harsh radiation environment and therefore require additional protection. Geosynchronous Orbits (GEO) occur well above the Van Allen belts and are maintained at an altitude such that the corresponding orbit period is similar to Earth’s. Because of this, GEO maintain relatively stationary positions with respect to Earth. Geostationary orbits occur with inclinations of 0° and therefore have fixed positions over the equator. Other specialized orbits are often encountered.
216 At unique combinations of altitude and inclination, orbital perturbations due to Earth’s oblateness equal Earth’s natural rotational rate. This is achieved by matching the variations of the RAAN with Earth’s rotation rate which occurs when Ω J 2 ≅ 0.9856 deg/day (Wertz and Larson, 2005). Under these circumstances, β becomes relatively constant thus providing for a Sun synchronous orbit (Figure C.16).
Figure C.16: Sun Synchronous Orbit with β = 0°. A Sun synchronous orbit has an orbital plane that remains fixed with respect to the Sun; consequently, the corresponding solar zenith angle changes significantly. Figure C.17 illustrates a Sun synchronous orbit at an altitude of 650 km, RAAN of 0° and inclination of 98°. As shown, at an altitude of 650 km and inclination of 98°, the β history remains relatively constant. As a result, a satellite on this orbit would see the same relative angle with respect to the Sun. Molniya Orbits are those with a high eccentricity ( e ~ 0.75) with periods of approximately two revolutions per day and inclinations of 63.4° and possibly 116.6° (Wertz and Larson, 2005). As discussed previously, these inclinations result in a zero rate of change of the argument of perigee ( ωJ 2 ≅ 0 ). Perigee is placed in the Southern Hemisphere thus causing a satellite to remain in the northern hemisphere for approximately 11 hours per orbit.
217
Figure C.17: β versus Time for Select Inclinations at an Altitude of 650 km (Beginning on March 21st 2007 and RAAN = 0°). Several orbits have been identified with application to RS. These include highlyelliptical orbits (HEO) like the cobra and magic orbits, a medium Earth orbit (MEO) and circular orbits including the LEO Sun synchronous, LEO Fast Access, and LEO Repeat Coverage orbits. Key characteristics including typical inclinations, orbit sizes, mean response times and available times for these are shown in Table C.1. RS has identified that missions will be either LEO or HEO. LEO missions will range from equatorial to Sun synchronous orbits. In addition, altitude requirements were impacted by drag requirements. Drag at low altitudes must be overcome by propulsion lending to a minimum altitude of 350 km. To meet 25-year atmospheric drag re-entry requirements, a maximum altitude of 705 km is required (ORSBS-001, 2007). The reference HEO mission is a 63.4° inclination elliptical orbit (i.e. Magic) to allow for long dwell times necessary for communications missions.
218 Table C.1: Summary of Key Characteristics of RS Appropriate Orbits (Wertz, 2005; Wertz, 2007). Orbit Type
Inclination
Orbit Size
[degrees]
[km]
Mean Response Time [hrs]
Cobra
63.4
800 x 27,000
10.0
Magic
116.6 (retrograde) 63.4 (prograde)
525 x 7,800
11.5
Medium Earth Orbit (MEO)
Varies
5,000 to 15,000
---
LEO Sun Synchronous
96.7
300 (typical)
6.0
300 (typical)
0.7
300 (typical)
9.0
90 to 180 (retrograde) LEO Fast Access 0 to 90 (prograde) 3 to 5 above LEO Repeat Coverage latitude of interest
Available Mass [lbs] 360 (prograde) 350 (retrograde) 510 (prograde) --650 (retrograde) 480 to 680 (retrograde) 680 to 980 (prograde) 680 to 980 (prograde)
219 Appendix D – Viable Weighting Matrices at Various Thresholds
a) Threshold of 0.00
b) Threshold of 0.01
c) Threshold of 0.02 d) Threshold of 0.03 Figure D.1: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.00, b) 0.01, c) 0.02, and d) 0.03.
220
a) Threshold of 0.04
b) Threshold of 0.05
c) Threshold of 0.10 d) Threshold of 0.20 Figure D.2: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.04, b) 0.05, c) 0.10, and d) 0.20.
221
a) Threshold of 0.25
b) Threshold of 0.26
c) Threshold of 0.30
c) Threshold of 0.50
Figure D.3: Contour Plot of Viable Weighting Matrices for Combinations of β and i at Thresholds of a) 0.25, b) 0.26, c) 0.30, and d) 0.50.
222 Appendix E – Critical Weighted Orbital Averaged Hot- and Cold-Case Orbits at Various Thresholds
Table E.1: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.01 Threshold. Orbit Type
Heat Earth IR Altitude β i Flux and Albedo1 [degrees] [---] [km] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 51,52 3 350 Cold Case 0.95 0.87 275.7 0 23,25,28,29 2 1,000 Flat Reflector Hot Case 0.15 0.05 60.9 72 51,52 2 350 Cold Case 0.15 0.05 37.3 0 23,25,28,29 1 350 Solar Absorber Hot Case 0.90 0.10 348.5 72 51,52 1 350 Cold Case 0.90 0.10 206.5 0 23,25,28,29 1 350 Solar Reflector Hot Case 0.10 0.80 112.2 72 51,52 3 350 65-67,70,71,73-76,81Cold Case 0.10 0.80 63.5 0 3 1,000 83,85-87,90,97-100,108 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
223 Table E.2: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.02 Threshold. Orbit Type
Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 51,52 Cold Case 0.95 0.87 275.7 0 28,29 Flat Reflector Hot Case 0.15 0.05 60.9 72 51,52 Cold Case 0.15 0.05 37.3 0 28,29 Solar Absorber Hot Case 0.90 0.10 348.5 72 51,52 Cold Case 0.90 0.10 206.5 0 28,29 Solar Reflector Hot Case 0.10 0.80 112.2 72 51,52 65,66,70,71,73-75, Cold Case 0.10 0.80 63.5 0 81-83,86,90,97-99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 2
350 1,000
2 1
350 350
1 1
350 350
3
350
3
1,000
Table E.3: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.03 Threshold. Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 52 Cold Case 0.95 0.87 275.7 0 28 Flat Reflector Hot Case 0.15 0.05 60.9 72 52 Cold Case 0.15 0.05 37.3 0 28 Solar Absorber Hot Case 0.90 0.10 348.5 72 52 Cold Case 0.90 0.10 206.5 0 28 Solar Reflector Hot Case 0.10 0.80 112.2 72 52 65,66,70,71,73,74, Cold Case 0.10 0.80 63.5 0 81-83,86,97-99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) Orbit Type
α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 2
350 1,000
2 1
350 350
1 1
350 350
3
350
3
1,000
224 Table E.4: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.04 Threshold. Orbit Type
Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 52 Cold Case 0.95 0.87 275.7 0 28 Flat Reflector Hot Case 0.15 0.05 60.9 72 52 Cold Case 0.15 0.05 37.3 0 28 Solar Absorber Hot Case 0.90 0.10 348.5 72 52 Cold Case 0.90 0.10 206.5 0 28 Solar Reflector Hot Case 0.10 0.80 112.2 72 52 65,66,70,71,73,74, Cold Case 0.10 0.80 63.5 0 81-83,86,97-99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 2
350 1,000
2 1
350 350
1 1
350 350
3
350
3
1,000
Table E.5: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.05 Threshold. Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 52 Cold Case 0.95 0.87 275.7 0 28 Flat Reflector Hot Case 0.15 0.05 60.9 72 52 Cold Case 0.15 0.05 37.3 0 28 Solar Absorber Hot Case 0.90 0.10 348.5 72 52 Cold Case 0.90 0.10 206.5 0 28 Solar Reflector Hot Case 0.10 0.80 112.2 72 52 65,66,70,71,73,74, Cold Case 0.10 0.80 63.5 0 81,83,86,97-99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) Orbit Type
α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 2
350 1,000
2 1
350 350
1 1
350 350
3
350
3
1,000
225 Table E.6: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.10 Threshold. Orbit Type
Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 52 Cold Case 0.95 0.87 275.7 1 28 Flat Reflector Hot Case 0.15 0.05 60.9 72 52 Cold Case 0.15 0.05 37.4 1 28 Solar Absorber Hot Case 0.90 0.10 348.5 72 52 Cold Case 0.90 0.10 206.5 1 28 Solar Reflector Hot Case 0.10 0.80 112.2 72 52 Cold Case 0.10 0.80 63.5 0 65,73,74,81,83,98,99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 2
350 1,000
2 1
350 350
1 1
350 350
3 3
350 1,000
Table E.7: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.20 Threshold. Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 437.4 72 52 Cold Case 0.95 0.87 278.3 0 74,81,99 Flat Reflector Hot Case 0.15 0.05 60.9 72 52 Cold Case 0.15 0.05 37.9 0 74,81,99 Solar Absorber Hot Case 0.90 0.10 348.5 72 52 Cold Case 0.90 0.10 211.9 0 74,81,99 Solar Reflector Hot Case 0.10 0.80 112.2 72 52 Cold Case 0.10 0.80 63.5 0 74,81,99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) Orbit Type
α
ε
Earth IR Altitude and Albedo1 [---] [km] 3 1
350 1,000
2 1
350 350
1 1
350 350
3 3
350 1,000
226 Table E.8: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.25 Threshold. Orbit Type
Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 430.4 72 74,81,83,98,99 Cold Case 0.95 0.87 278.3 0 74,99 Flat Reflector Hot Case 0.15 0.05 60.7 72 74,81,83,98,99 Cold Case 0.15 0.05 37.9 0 74,99 Solar Absorber Hot Case 0.90 0.10 347.6 67 52 Cold Case 0.90 0.10 211.9 0 74,99 Solar Reflector Hot Case 0.10 0.80 105.7 72 74,81,83,98,99 Cold Case 0.10 0.80 63.5 0 74,99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
Earth IR Altitude and Albedo1 [---] [km] 2 1
350 1,000
2 1
350 350
1 1
600 350
3 3
350 1,000
Table E.9: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.26 Threshold. Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 430.4 72 74,81,83,98,99 Cold Case 0.95 0.87 278.3 0 74 Flat Reflector Hot Case 0.15 0.05 60.7 72 74,81,83,98,99 Cold Case 0.15 0.05 37.9 0 74 Solar Absorber Hot Case 0.90 0.10 347.6 72 74,81,83,98,99 Cold Case 0.90 0.10 211.9 0 74 Solar Reflector Hot Case 0.10 0.80 105.7 72 74,81,83,98,99 Cold Case 0.10 0.80 63.5 0 74 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) Orbit Type
α
ε
Earth IR Altitude and Albedo1 [---] [km] 2 1
350 1,000
2 1
350 350
2 1
350 350
3 3
350 1,000
227 Table E.10: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.30 Threshold. Orbit Type
Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 430.4 72 74,81,83,99 Cold Case 0.95 0.87 278.3 1 65,74,81,83,98,99 Flat Reflector Hot Case 0.15 0.05 60.7 72 74,81,83,99 Cold Case 0.15 0.05 38.0 1 65,74,81,83,98,99 Solar Absorber Hot Case 0.90 0.10 347.6 72 74,81,83,99 Cold Case 0.90 0.10 212.0 1 65,74,81,83,98,99 Solar Reflector Hot Case 0.10 0.80 105.7 72 74,81,83,99 Cold Case 0.10 0.80 63.5 1 65,74,81,83,98,99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) α
ε
Earth IR Altitude and Albedo1 [---] [km] 2 1
350 1,000
2 1
350 350
2 1
350 350
3 3
350 1,000
Table E.11: Critical Weighted Orbital Averaged Hot and Cold Case Orbits for Each of Four Surface Categories at a 0.50 Threshold. Heat β i Flux [degrees] [---] [---] [W/m2] [degrees] Flat Absorber Hot Case 0.95 0.87 416.7 64 99 Cold Case 0.95 0.87 278.3 1 74,99 Flat Reflector Hot Case 0.15 0.05 60.0 64 99 Cold Case 0.15 0.05 38.0 1 74,99 Solar Absorber Hot Case 0.90 0.10 346.0 64 99 Cold Case 0.90 0.10 212.0 1 74,99 Solar Reflector Hot Case 0.10 0.80 100.7 40 52 Cold Case 0.10 0.80 63.5 1 74,99 1 Value Correspondence (1 = ALB, 2 = CMB, 3 = OLR) Orbit Type
α
ε
Earth IR Altitude and Albedo1 [---] [km] 2 1
750 1,000
2 1
750 350
1 1
750 350
3 3
350 1,000
228 Appendix F – Numerical Model Investigation
Creation of a satellite numerical model was necessary to illustrate other works including component placement optimization and thermal control approach comparisons. Building confidence in the accuracy of the numerical results can be accomplished with a thorough investigation of various numerical modeling parameters. These include number of orbital simulations, nodal resolution, Monte-Carlo Rays per Node, and number of orbital positions. Before these are investigated, an understanding of the software tool is developed. F.1 Description of Software Tool The numerical model was created using Thermal Desktop® as a CAD based interface to the SINDA/FLUINT equation solver. Thermal Desktop® and SINDA/FLUINT are widely accepted tools for spacecraft thermal design. All computational analysis was done using a 2.5 GHz dual-core processor. Originally released in 1996, Thermal Desktop® and its integrated modules, RadCAD and FloCAD allow users to quickly build, analyze, and post-process thermal models. RadCAD® is a module used to generate radiation exchange factors and orbital heating rates. It uses an oct-tree accelerated Monte-Carlo Ray-Tracing algorithm to compute radiation exchange and view factors. FloCAD® generates flow networks and calculates convective heat transfer factors. The output of Thermal Desktop®, RadCAD, and FloCAD is processed in order to create inputs for SINDA/FLUINT. Since its introduction in 1986 SINDA/FLUINT has become an industry standard thermal/fluid analyzer and is intended primarily for designing and analyzing thermal systems represented in electrical analog, lumped parameter form. Given properly prepared inputs, SINDA/FLUINT can solve both finite-difference and finite-element equations. Upon running an individual simulation, Thermal Desktop® will first calculate radiation conductors and heating rates. Nodes and conductors are then computed and output in a format useable for SINDA/FLUINT. A SINDA/FLUINT model is then built
229 and run. Two typical solution routines - a transient solver and a steady-state solution routine - were utilized for simulations. F.1.1 Transient Solver A generic transient solution routine (neglecting applied heat rates) seeks to solve Equation (F.1) where Tnew and Told are temperatures at the next and current times, respectively. The weighting factor, λ , determines what type of solution method is utilized (e.g. explicit or implicit) as shown here Tnew = Told + f (λ ⋅ Tnew ,(1 − λ ) ⋅ Told ) .
(F.1)
The subroutine TRANSIENT (a.k.a. FWDBCK) performs a transient thermal analysis by implicit forward-backward differencing (λ ≠ 0) . A heat balance equation is written about a diffusion node as a forward finite difference equation and a second as a backward finite difference equation. The resulting sum of these two equations is (Cullimore, Ring, and Johnson, 2008):
(
) . ⎤ ⋅ ( (T ) − (T ) ) ⎦⎥
N 4 4 2 ⋅ Ci ⋅ (Ti n +1 − Ti n ) = 2 ⋅ Q i + ∑ ⎡G ji ⋅ (T jn +1 − Ti n +1 ) + Gˆ ji ⋅ (T jn +1 ) − (Ti n +1 ) ⎤ ⎢ ⎦⎥ Δt j =1 ⎣ N
+ ∑ ⎡G ji ⋅ (T jn − Ti n ) + Gˆ ji ⎢ j =1 ⎣
where, Ci
= thermal capacitance of node i
Δt
= time step equal to t n +1 − t n
Ti n +1
= temperature of node i at t n +1
Ti n
= temperature of node i at t n
Qi
= source/sink for node i
T jn +1
= temperature of node j at t n +1
T jn
= temperature of node j at t n
G ji
= linear conductor attaching node j to node i
n 4 j
n 4
i
(F.2)
230 Gˆ ji
= radiation conductor attaching node j to node i
N
= number of nodes
This equation uses the average of the temperature derivatives at the current and next times to predict the overall temperature change (i.e. trapezoidal method). It is 2nd order accurate in time and 1st order in space. The set of equations are solved by iterative relaxation or simultaneous matrix methods until diffusion node temperatures change less than the control parameter DRLXCA and arithmetic nodes change less than the control parameter ARLXCA, and until energy imbalances are tolerable. F.1.2 Steady-State Solver The steady-state solution routine treats diffusion nodes like arithmetic (i.e. zero capacitance) nodes and seeks to find the root of Equation (F.3) where Q are the heating rates and T are node temperatures Q = f ( T) .
(F.3)
The steady-state solution algorithm STEADY (a.k.a. FASTIC) iteratively solves for the temperatures. For the k + 1 iteration, the energy balance on node i becomes (Cullimore, Ring, and Johnson, 2008):
(
i −1
) . ⎤ ) )⎦⎥
4 4 0 = Q i + ∑ ⎡G ji ⋅ (T jk +1 − Ti k +1 ) + Gˆ ji ⋅ ( T jk +1 ) − (Ti k +1 ) ⎤ ⎢ ⎦⎥ j =1 ⎣ N
(
4 + ∑ ⎡G ji ⋅ (T jk − Ti k +1 ) + Gˆ ji ⋅ (T jk ) − (Ti k +1 ⎢ j =i ⎣
4
where, T jk +1
= temperature of node j at iteration k + 1
Ti k +1
= temperature of node i at iteration k + 1
T jk
= temperature of node j at iteration k
The default solution routine for both steady-state and transient solutions is an algebraic multi-grid method in combination with a conjugate gradient method. This
(F.4)
231 AMG-CG method, referred to as MATMET=12 in Thermal Desktop®, improves speed and memory utilization especially for large problems and/or dense matrices (Cullimore, Ring, and Johnson, 2008). In this approach, all thermal submodels are combined into a single matrix and solved simultaneously. Internal iterations are utilized for solving and therefore, this approach is subject to speed/accuracy tradeoffs dictated by the AMGERR control constant with a default value of 1.0E-10. F.2 Description of Numerical Testing Model A model was developed consistent with current responsive satellite architectures (Arritt et al., 2008). A 1.0 m x 1.0 m x 1.0 m six-sided frame-and-panel construction satellite was used with a honeycomb panel construction having 0.00127 m thick Al 6061-T6 facesheets and a 0.0254 m thick Al 5052 honeycomb material. Material properties were based on Gilmore (2002). A uniform heat-flux source was centered on the interior of each panel to represent an electrical component. Each component was modeled as a square surface utilizing an arithmetic (i.e. zero capacitance) node. Component size varied for each analysis and is described later. Heat generation of these six internal components totaled 300 W and was conducted to the panel with a contact conductance of 110 W/m2-K (Gilmore, 2002). Panel to panel conductance was 12 W/K to simulate panelto-panel longerons and bolted joints (Bugby, Zimbeck, and Kroliczek, 2008). All heat generation was ultimately dissipated to a deep space environment (0 K) through radiation from the exterior of the panel surfaces. Exterior panel surfaces had a solar absorptivity of
α = 0.1 while surface infrared emissivity was ε = 0.8. F.3 Number of Orbital Simulations An investigation was carried out to understand the effect of orbit number on simulation accuracy. For each trial, a steady-state routine was completed first to initialize all node temperatures. Next, several transient orbital simulations were completed. Both hot- and cold-case orbits were simulated for three types of panel constructions described in the following table. These cover a broad range of panel thermal capacitance and conductivity.
232 Table F.1: Three Panel Construction Types Used for Orbital Simulation Analysis. Panel Type
Panel Construction
Panel Mass [kg]
Nominal
Al 6061-T6 Facesheet with Honeycomb Core
54.91
High-Conductivity
APG Facesheet with Honeycomb Core
50.41
High-Capacitance
Al 6061-T6 Facesheet and Core
464.36
The following table summarizes simulation parameters that were utilized for this study. Note that many of these parameters were based on analysis discussed later. Table F.2: Numerical Model Parameters for Orbital Simulation Analysis. Numerical
Range
Model
of
Parameter
Nodal Resolution
Values 1 to 10 (typical) Up to 30 orbits utilized for highcapacitance panel construction 51 x 51
Component Size
0.1 m x 0.1 m
Monte-Carlo Rays per Node
10,000
Number of Orbital Positions
36
Control Parameter Setting
0.001
Number of Orbital Simulations
The following figures illustrate the results of this analysis. Each figure shows the resulting temporal variation of temperature for each subsystem. This was done for hotand cold-case orbits for the three panel constructions.
233
Figure F.1: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on Nominal Panel Construction (54.91 kg).
234
Figure F.2: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Conductivity Panel Construction (50.41 kg).
235
Figure F.3: Cold-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Capacitance Panel Construction (464.36 kg).
236
Figure F.4: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on Nominal Panel Construction (54.91 kg).
237
Figure F.5: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Conductivity Panel Construction (50.41 kg).
238
Figure F.6: Hot-Case Transient Temperature Results (First 10 Orbits) for Each of Six Subsystems on High-Capacitance Panel Construction (464.36 kg).
239 From these figures, it appears that the 2nd orbit simulation provides quasi steady-state conditions. In addition, it appears that an orbit simulation exits where temporal temperature results do not change for any subsequent simulations. This true steady-state simulation condition was found for each case and subsystem. The 2nd orbit result were compared these steady-state results and the maximum percent difference (absolute value) was retrieved. The results are summarized in the following tables. Table F.3: Orbital Simulation Analysis Hot-Case Summary. Subsystem Orientation
-X
+X
-Y
+Y
-Z
+Z
Nominal Panel Construction Maximum % Difference 0.025 0.003 0.004 0.005 0.019 0.003 True steady-state orbit 7 7 7 9 7 7 High-Conductivity Panel Construction Maximum % Difference 0.005 0.005 0.005 0.005 0.005 0.005 True steady-state orbit 6 6 6 5 5 6 High-Capacitance Panel Construction Maximum % Difference 0.061 0.060 0.001 0.000 0.005 0.002 True steady-state orbit 28 28 22 25 29 20 Table F.4: Orbital Simulation Analysis Cold-Case Summary. Subsystem Orientation
-X
+X
-Y
+Y
-Z
+Z
Nominal Panel Construction Maximum % Difference 0.062 0.114 0.028 0.028 0.059 0.010 True steady-state orbit 8 8 9 9 9 9 High-Conductivity Panel Construction Maximum % Difference 0.036 0.035 0.036 0.036 0.036 0.034 True steady-state orbit 7 7 7 8 8 8 High-Capacitance Panel Construction Maximum % Difference 0.153 0.146 0.004 0.005 0.110 0.002 True steady-state orbit 29 29 29 27 29 29 These tables demonstrate that using 2nd orbit results generates small error. The –X subsystem placed on the high-capacitance panel construction yielded the largest maximum percent difference value of 0.153%. This value occurred during one time step and therefore, orbital averaged percent difference would be much smaller. This is a
240 substantial improvement over using 1st orbit results. Although not presented in tabular form, the largest maximum percent difference value for 1st orbit results was 1.5%. This occurred for the cold-case +X subsystem using nominal panel construction. Therefore, 2nd orbit simulation results provide a reasonable estimate of true steady-state orbit simulation conditions without excess computational expense. F.4 Nodal Resolution Analysis The transient solution routine is 1st order in space. Consequently, nodal resolution is significant and must be addressed. A study was done using the case study satellite structure parameters previously described with nominal panel construction (i.e. aluminum facesheets with honeycomb core). Several simulations were run over a wide-range of inplane nodal resolution values and subsystem sizes. Edge nodes were utilized for panel construction. Table F.5 summarizes simulation parameters that were utilized for this study. Note that many of these parameters were based on analysis discussed later. Table F.5: Numerical Model Parameters for Nodal Resolution Analysis. Numerical
Range
Model
of
Parameter
Values nd
Number of Orbital Simulations
2 (2 Orbit Results Used)
Nodal Resolution
1 x 1 up to 61 x 61
Component Size
0.1 m x 0.1 m to 1.0 m x 1.0 m
Monte-Carlo Rays per Node
10,000
Number of Orbital Positions
36
Control Parameter Setting
0.001
The results are shown in Figure F.7. These figures show that utilizing a nodal resolution of 51 x 51 provides good results for component sizes down to 0.1 x 0.1 m.
241
a) Maximum Subsystem Temperature versus b) Minimum Subsystem Temperature versus Nodal Resolution for Various Component Nodal Resolution for Various Component Sizes Sizes
c) Maximum Subsystem Temperature d) Total and SINDA/FLUINT Difference versus Nodal Resolution for Computational Time versus Nodal Various Component Sizes Resolution for 0.1 m x 0.1 m Subsystems Figure F.7: Subsystem Temperature and Computational Time Results versus Nodal Resolution for the Hot-Case Orbit. F.5 Monte-Carlo Rays per Node Analysis RadCAD® uses a Monte-Carlo stochastic integration technique for computing radiation conductors ( Gˆ ji ) and heating rates. Rays are emitted from each node and traced
242 around the geometry simulating the effect of a bundle of photons. When a ray strikes another surface, energy is decremented from the ray and absorbed by the struck surface. The ray is then reflected or transmitted, according to surface optical properties. A Gˆ ji for a given node i and j is computed by keeping track of these ray firings and calculated as shown in Gˆ ij = ε ⋅ A ⋅ Bij
(F.5)
where ε and A are the surface emissivity and area, respectively. Bij is the fraction of energy that leaves node i and is absorbed by node j by all
possible reflection paths (i.e. direct and indirect). Hence, the sum of
∑B
ij
for any given
j
node i is equal to unity. For a single trial, Bij is calculated by shooting a single ray from node i and recording the fraction of energy absorbed at node j . This process is repeated N rays times and the average of these trials gives the final result. The error for a given Gˆ ji
is based on the variability in the sample of ray firings and estimated based on a confidence interval of 90% and shown in (Panczak et al., 2008) Errorij = 1.65 ⋅
1 − Bij N rays ⋅ Bij
⋅ 100 .
(F.6)
This equation illustrates that error associated with the Monte-Carlo Ray-Tracing process is inversely proportional to the square root of the number of rays shot. Therefore, to reduce the error by half, the number of rays fired need to be quadrupled. Additionally, error depends on Bij . A relatively large Bij implies a relatively higher number of emitted rays that were captured (for a given i and j ). This yields a lower sample variance and therefore error estimate. Hence, larger Bij have correspondingly lower associated errors for a given N rays . Rays shot from both nodes i and j , result in two estimates of the Gˆ ji between them. RadCAD takes this factor into account when combining Bij and B ji to compute the
243 final Gˆ ji for SINDA/FLUINT. The ray tallies are weighted by their areas and error estimates, and always produce a Gˆ ji with less error than either individual estimate. For example, if the areas and emissivities of the two nodes are the same, then Bij equals B ji and combining the ray tallies yields a result that has an error reduced by the square root of two. As this Equation (F.6) shows, N rays impacts simulation error for given values of Bij . Consequently, a study was conducted to address the impact of this issue. Panczak et al. (2008) recommend that a series of cases with an increasing number of rays be conducted to demonstrate accuracy. A study was conducted utilizing the case study previously described. Several simulation were run with increasing levels of N rays (i.e. 2000, 3000, 4000, 5000, 10000, 20000, 100000). For each, the maximum and minimum 2nd orbit temperatures were recorded along with the total computational time. The following table summarizes simulation parameters that were utilized for this study. Note that some of these parameters were based on analysis discussed later. Table F.6: Numerical Model Parameters for Monte-Carlo Rays per Node Analysis. Numerical
Range
Model
of
Parameter
Values nd
Number of Orbital Simulations
2 (2 Orbit Results Used)
Nodal Resolution
51 x 51
Component Size
0.1 m x 0.1 m
Monte-Carlo Rays per Node
2,000 to 100,000
Number of Orbital Positions
36
Control Parameter Setting
0.001
The results are shown in the following figure. These results show that increasing rays per node changes the results slightly, but significantly impacts the computational time required.
244
b) Maximum Subsystem Temperature Difference and Total Computational Time versus Rays per Node Figure F.8: Hot- and Cold-Case Monte-Carlo Rays per Node Analysis Results for Five Levels of Rays per Node (2000, 3000, 4000, 5000, 10000, 20000, and 100000). a) Subsystem Maximum and Minimum Temperature versus Rays per Node
From these figures, it appears that the decreasing rays per node does not significantly affect maximum and minimum temperature results. It appears to slightly decrease maximum temperatures and increase minimum temperatures. In combination, these two trends imply a more pronounced effect on maximum subsystem temperature difference. Therefore, maximum temperature difference versus rays per node results were analyzed in more detail in the following tables. Table F.7: Rays per Node Analysis Hot-Case Summary. Rays per Maximum Node Temperature Difference [---] [K] 100000 44.727 20000 44.711 10000 44.684 5000 44.595 4000 44.542 3000 44.338 2000 43.561
Percent Difference [%] --0.036 0.096 0.295 0.414 0.870 2.607
Total Computational Time [seconds] 50,580 11,520 10,008 7,632 3,672 3,266 3,040
Percent Reduction [%] --77.224 80.214 84.911 92.740 93.542 93.991
245 Table F.8: Rays per Node Analysis Cold-Case Summary. Rays per Maximum Node Temperature Difference [---] [K] 100000 33.947 20000 33.94 10000 33.923 5000 33.857 4000 33.823 3000 33.681 2000 33.182
Percent Difference [%] --0.021 0.071 0.265 0.365 0.784 2.254
Total Computational Time [seconds] 31,824 7,380 3,708 2,843 2,573 2,382 2,384
Percent Reduction [%] --76.810 88.348 91.067 91.914 92.515 92.509
These tables demonstrate that decreasing rays per node down to 3,000 has little impact on maximum temperature difference results. However, it has a significant impact on computational time. Consequently, 5,000 rays per node was selected to provide reasonable results without excess computational expense. Rays per node are used to determine internal and external Bij along with external heating rates. Analysis revealed that further reductions in computational time could be achieved by tuning individual rays per node values (i.e. internal Bij , external Bij , and external heating rates) without sacrificing results. Further analysis will be conducted if this is deemed necessary. F.6 Analysis of Number of Orbital Positions The orbital heating rates are output to SINDA/FLUINT in two forms: average and position dependant. When a steady-state solution is called in SINDA/FLUINT, the average heating rate (time-weighted average of all positions in the heating rate case) is used in the solution. When a transient solution is called, the position-dependent values are interpolated using a cyclical, linear interpolation based on the solution time (Cullimore, Ring, and Johnson, 2008). Computing orbital heating rates using the MonteCarlo Ray-Tracing method, rays are shot from the node to the source. If the ray is not absorbed by intervening surfaces, the ray is reversed, direct energy is deposited into the
246 node and the ray is reflected. The traversal of the ray for the reflected energy is the same as traversing the ray for Gˆ ji calculations. It seemed likely that the number of orbital positions would affect simulation results. Therefore, a study was conducted to determine the impact of number of orbital positions on resulting maximum subsystem temperature, minimum subsystem temperature, and maximum subsystem temperature difference. Table F.9 summarizes simulation parameters that were utilized for this study. The results are shown in Figure F.9. These results show that decreasing number of orbital positions changes the results slightly, but significantly impacts the computational time required. It should be noted that an additional two orbital positions are included for each analysis to account for shadow crossings. From this figure, it appears that the decreasing orbital positions does affect maximum and minimum temperature results along with maximum subsystem temperature difference. Maximum temperature difference versus orbital position results were analyzed in more detail in Table F.10 and Table F.11.
a) Subsystem Maximum and Minimum b) Maximum Subsystem Temperature Temperature versus Number of Orbital Difference and Total Computational Time Positions versus Number of Orbital Positions Figure F.9: Hot- and Cold-Case Orbital Position Analysis Results for Five Levels of Orbital Positions (3, 6, 9, 12, and 36).
247 Table F.9: Numerical Model Parameters for Orbital Position Analysis. Numerical
Range
Model
of
Parameter
Values nd
Number of Orbital Simulations
2 (2 Orbit Results Used)
Nodal Resolution
51 x 51
Component Size
0.1 m x 0.1 m
Monte-Carlo Rays per Node
5,000
Number of Orbital Positions
3 to 36
Control Parameter Setting
0.001
Table F.10: Orbital Position Analysis Hot-Case Summary. Orbital Maximum Positions Temperature Difference [---] [K] 36 44.595 12 44.535 9 44.45 6 44.269 3 43.26
Percent Difference [%] --0.135 0.325 0.731 2.994
Total Computational Time [s] 7,632 1,975 1,729 1,080 1,199
Percent Reduction [%] --74.127 77.351 85.849 84.292
Table F.11: Orbital Position Analysis Cold-Case Summary. Orbital Maximum Positions Temperature Difference [---] [K] 36 33.862 12 33.948 9 34.017 6 34.202 3 35.366
Percent Difference [%] --0.254 0.458 1.004 4.442
Total Computational Time [s] 2,491 1,279 1,486 1,319 1,148
Percent Reduction [%] --48.651 40.366 47.038 53.902
248 These tables demonstrate that decreasing rays per node down to 12 significantly reduces the required computational time with little effect on results (i.e. ~0.3% difference over 36 orbital positions). Consequently, 12 orbital positions were selected to provide reasonable results without excess computational expense.
249 Appendix G – Characteristic Component Design Cases
Testing component placement optimization algorithms, evaluating quasi-isothermal conditions, and investigating required control authority necessitates the development of a comprehensive set of characteristic component design cases consistent with RS missions. Each design case is characterized by a total orbital averaged power, Ptotal , distributed over N components. Allocation of power over the n components is defined by a set of
distribution curves consistent with typical power allocations found in the literature. These design cases along with design orbits will lead to a systematic method of evaluating future RS TCS design and development efforts. G.1 Reference Missions Reference missions based on the works of Williams’ (2005) low-capability (LCB) and high-capability bus (HCB) FACTS approach, a representative military satellite, and Young’s (2008) low-capability (LCB), medium-capability (MCB) and high-capability (HCB) TherMMS concept provided the basis for the development of the design cases. For each reference mission, components were organized by subsystem including Attitude Determination and Control (ADCS), Orbit Determination and Control (ODCS), Telemetry Tracking and Command (TTCS), Command and Data Handling (CDHS), and Electrical Power (EPS). Refer to G.3 for a detailed explanation of each of these. All components were included in the overall mass and power budget but those not meeting the requirements of reconfigurable placement were not included in the overall component count. This includes solar arrays and wiring harnesses to name a few. Further, Thermal Control (TCS) and Structure and Mechanisms (SMS) subsystems were included to appropriately define total mass and do not add to the component numbers. Table G.1 through Table G.6 shows subsystem components for each of six reference missions. Included in each table are number of components along with individual and total component mass and orbital averaged power.
250 Table G.1: FACTS LCB Distribution (Williams, 2005). Component Description ADCS and ODCS Momentum Wheel Electromagnet Sun Sensor Suite Star Sensor GPS Receiver Magnetometer MEMS IMU Controller Card TTCS S-Band Transmitter S-Band Receiver Hemi Antenna Miscellaneous CDHS CDH Unit EPS Li-Ion Batteries Power Management and Distribution Miscellaneous TCS SMS TOTALS
Number Mass Total Mass [kg] [kg] 10 --- 10.5 1 6.4 6.4 3 1.1 3.3 1 0.2 0.2 1 0.3 0.3 1 0.0 0.0 1 0.1 0.1 1 0.0 0.0 1 0.2 0.2 4 --2.4 1 0.2 0.2 1 0.2 0.2 2 0.3 0.5 0 1.5 1.5 1 --- 15.2 1 15.2 15.2 3 --- 17.9 2 3.8 7.6
Orbital Averaged Power [W] --12.0 0.5 0.0 1.0 0.8 0.7 0.3 3.0 --6.0 0.8 0.0 0.6 --50.0 --5.0
Total Orbital Averaged Power [W] 19.3 12.0 1.5 0.0 1.0 0.8 0.7 0.3 3.0 7.4 6.0 0.8 0.0 0.6 50.0 50.0 70.3 10.0
1
3.1
3.1
50.3
50.3
0 0 0 18
7.2 -------
7.2 0.0 21.5 67.5
10.0 -------
10.0 0.0 0.0 147.0
251
Table G.2: FACTS HCB Distribution (Williams, 2005). Component Description ADCS and ODCS Reaction Wheel Electromagnet Sun Sensor Suite Star Sensor GPS Receiver Magnetometer MEMS IMU Controller Card TTCS S-Band Transmitter S-Band Receiver Hemi Antenna Ku-Band Transponder TWT Amplifier Parabolic Antenna Miscellaneous CDHS CDH Unit EPS Li-Ion Batteries Power Management and Distribution Miscellaneous TCS SMS TOTALS
No. Mass Total Mass [kg] [kg] 16 --- 24.2 3 6.4 19.2 3 1.1 3.3 2 0.2 0.4 2 0.3 0.6 1 0.0 0.0 1 0.1 0.1 1 0.0 0.0 3 0.2 0.6 7 --8.0 1 0.2 0.2 1 0.2 0.2 2 0.3 0.5 1 1.4 1.4 1 1.7 1.7 1 1.5 1.5 0 2.5 2.5 1 --- 15.2 1 15.2 15.2 9 --- 67.7 8 3.8 30.4 1
9.8
9.8
0 11.9 27.5 0 --0.0 0 --- 38.6 33 --- 153.7
Orbital Averaged Power [W] --12.0 0.5 0.0 1.0 0.8 0.7 0.3 3.0 --6.0 0.8 0.0 15.0 47.0 0.0 0.6 --50.0 --5.0
Total Orbital Averaged Power [W] 50.3 36.0 1.5 0.0 2.0 0.8 0.7 0.3 9.0 69.4 6.0 0.8 0.0 15.0 47.0 0.0 0.6 50.0 50.0 253.0 40.0
198.0
198.0
15.0 -------
15.0 0.0 0.0 422.7
252
Table G.3: Representative Military Satellite Distribution. Component Description
No. Mass Total Orbital Total Orbital Mass Averaged Power Averaged Power [kg] [kg] [W] [W] ADCS and ODCS 13 --- 19.8 --35.2 Magnetometer 1 0.5 0.5 3.5 3.5 Coarse Sun Sensor 2 0.5 1.0 1.3 2.6 Star Tracker 1 0.8 0.8 3.3 3.3 GPS Receiver 2 0.7 1.4 3.3 6.6 Reaction Wheel Assembly 3 4.5 13.6 4.8 14.4 Torque Rod 3 0.7 2.2 1.6 4.8 IMU 1 0.4 0.4 0.0 0.0 TTCS 6 --- 9.3 --18.5 Micro Black Box Transponder 1 1.0 1.0 6.0 6.0 Tactical Antenna 1 1.9 1.9 0.0 0.0 S-Band Radio 1 4.1 4.1 8.3 8.3 S-Band Antenna Assembly 2 1.1 2.3 0.0 0.0 MCU-110 1 ----4.2 4.2 CDHS 10 --- 3.1 --43.3 Intelligent Data Store (IDS) 3 0.9 2.7 5.0 15.0 Spacewire Router 6 0.0 0.0 4.5 27.0 Spacecraft Clock 1 0.4 0.4 1.3 1.3 EPS 9 --- 22.1 --24.6 Battery 1 10.4 10.4 0.0 0.0 ESM 1 2.6 2.6 4.4 4.4 Power Hub 6 0.0 0.0 3.0 18.0 Solar Array Controller 1 1.6 1.6 2.2 2.2 Miscellaneous 0 7.4 7.4 0.0 0.0 TCS 0 --- 1.7 --0.0 SMS 0 --- 49.6 --0.0 TOTALS 38 --- 105.6 --121.6
253
Table G.4: TherMMS LCB Component Distribution (Young, 2008). Component Description
No. Mass Total Orbital Total Orbital Mass Averaged Power Averaged Power [kg] [kg] [W] [W] ADCS and ODCS 16 --- 14.3 --60.6 Attitude Control Processor 1 3.0 3.0 15.0 15.0 Reaction Wheels 3 1.8 5.3 7.5 22.5 Torque Rods 3 0.3 0.9 0.3 0.9 GPS Receiver 0 0.0 0.0 0.0 0.0 Inertial Measurement Unit 1 1.1 1.1 15.0 15.0 Magnetometer 1 0.1 0.1 0.9 0.9 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Star Tracker 1 0.9 0.9 3.8 3.8 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Miscellaneous 0 1.8 1.8 0.0 0.0 TTCS 7 --- 7.5 --24.9 SGLS Transponder 1 4.5 4.5 19.9 19.9 Adapter 1 0.4 0.4 2.5 2.5 Diplexer 1 0.3 0.3 0.0 0.0 Antenna Switch 1 0.0 0.0 0.0 0.0 Omni-Directional Antenna 2 0.4 0.8 0.0 0.0 Adapter 1 0.4 0.4 2.5 2.5 Miscellaneous 0 1.2 1.2 0.0 0.0 CDHS 1 --- 3.6 --20.0 Processor – Standard 1 3.0 3.0 20.0 20.0 Miscellaneous 0 0.6 0.6 0.0 0.0 EPS 8 --- 32.6 --14.8 Power Management Card 1 0.4 0.4 3.8 3.8 8.6 Amp-hour Lithium-Ion Battery 1 3.0 3.0 0.0 0.0 SADA Controller Card 2 0.4 0.8 2.5 5.0 Sun Sensor 2 0.0 0.1 0.0 0.0 Adapter 2 0.4 0.8 2.5 5.0 Miscellaneous 0 17.5 27.6 0.0 1.0 TCS 0 --- 0.0 --0.0 SMS 0 --- 16.9 --0.0 TOTALS 32 --- 74.9 --120.2
254 Table G.5: TherMMS MCB Component Distribution (Young, 2008). Component Description
No. Mass Total Orbital Total Orbital Mass Averaged Power Averaged Power [kg] [kg] [W] [W] ADCS and ODCS 16 --- 14.3 --60.6 Attitude Control Processor 1 3.0 3.0 15.0 15.0 Reaction Wheels 3 1.8 5.3 7.5 22.5 Torque Rods 3 0.3 0.9 0.3 0.9 GPS Receiver 0 0.0 0.0 0.0 0.0 Inertial Measurement Unit 1 1.1 1.1 15.0 15.0 Magnetometer 1 0.1 0.1 0.9 0.9 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Star Tracker 1 0.9 0.9 3.8 3.8 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Miscellaneous 0 1.8 1.8 0.0 0.0 TTCS 7 --- 7.5 --36.9 SGLS Transponder 1 4.5 4.5 31.9 31.9 Adapter 1 0.4 0.4 2.5 2.5 Diplexer 1 0.3 0.3 0.0 0.0 Antenna Switch 1 0.0 0.0 0.0 0.0 Omni-Directional Antenna 2 0.4 0.8 0.0 0.0 Adapter 1 0.4 0.4 2.5 2.5 Miscellaneous 0 1.2 1.2 0.0 0.0 CDHS 1 --- 3.6 --20.0 Processor – Standard 1 3.0 3.0 20.0 20.0 Miscellaneous 0 0.6 0.6 0.0 0.0 EPS 9 --- 42.3 --18.5 Power Management Card 1 0.4 0.4 5.0 5.0 17.2 Amp-hour Lithium-Ion Battery 1 6.0 6.0 0.0 0.0 Adapter 1 0.4 0.4 2.5 2.5 SADA Controller Card 2 0.4 0.8 2.5 5.0 Sun Sensor 2 0.0 0.1 0.0 0.0 Adapter 2 0.4 0.8 2.5 5.0 Miscellaneous 0 17.5 34.0 0.0 1.0 TCS 0 --- 0.0 --0.0 SMS 0 --- 16.9 --0.0 TOTALS 33 --- 84.6 --135.9
255 Table G.6: TherMMS HCB Component Distribution (Young, 2008). Component Description
No. Mass Total Orbital Total Orbital Mass Averaged Power Averaged Power [kg] [kg] [W] [W] ADCS and ODCS 17 --- 15.8 --66.4 Attitude Control Processor 1 3.0 3.0 15.0 15.0 Reaction Wheels 3 1.8 5.3 7.5 22.5 Torque Rods 3 0.3 0.9 0.3 0.9 GPS Receiver 1 1.5 1.5 5.9 5.9 Inertial Measurement Unit 1 1.1 1.1 15.0 15.0 Magnetometer 1 0.1 0.1 0.9 0.9 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Star Tracker 1 0.9 0.9 3.8 3.8 Sun Sensor 1 0.0 0.0 0.0 0.0 GPS Antenna 1 0.2 0.2 0.0 0.0 Adapter 1 0.4 0.4 1.3 1.3 Miscellaneous 0 1.8 1.8 0.0 0.0 TTCS 8 --- 8.0 --68.1 SGLS Transponder 1 4.5 4.5 31.9 31.9 Adapter 1 0.4 0.4 2.5 2.5 Power Amplifier 1 0.5 0.5 31.3 31.3 Diplexer 1 0.3 0.3 0.0 0.0 Antenna Switch 1 0.0 0.0 0.0 0.0 Omni-Directional Antenna 2 0.4 0.8 0.0 0.0 Adapter 1 0.4 0.4 2.5 2.5 Miscellaneous 0 1.2 1.2 0.0 0.0 CDHS 1 --- 8.1 --45.0 Processor - Triple CPU 1 7.5 7.5 45.0 45.0 Miscellaneous 0 0.6 0.6 0.0 0.0 EPS 9 --- 49.0 --18.5 Power Management Card 1 0.4 0.4 5.0 5.0 17.2 Amp-hour Lithium-Ion Battery 1 6.0 6.0 0.0 0.0 Adapter 1 0.4 0.4 2.5 2.5 SADA Controller Card 2 0.4 0.8 2.5 5.0 Sun Sensor 2 0.0 0.1 0.0 0.0 Adapter 2 0.4 0.8 2.5 5.0 Miscellaneous 0 20.8 40.7 0.0 1.0 TCS 0 --- 0.0 --0.0 SMS 0 --- 16.9 --0.0 TOTALS 35 --- 97.8 --198.1
256 A summary of reference missions (Table G.7) includes number of components, total bus mass, and total bus power. Total power attributed to reconfigurable components accounts for over 92% of total bus power for all of the cases. The difference in power can be attributed to devices which provide a heat source but will not be able to be repositioned (e.g. wiring). Future analyses were limited to reconfigurable components. Table G.7: Summary of Reference Missions. FACTS FACTS Military TherMMS TherMMS TherMMS LCB HCB Satellite LCB MCB HCB Number of Components 18 33 38 32 33 35 Total Mass [kg] 67.5 153.7 105.6 74.9 84.6 97.8 Total Component Power [W] 136.4 407.1 121.6 119.2 135.0 197.1 Total Bus Power [W] 147.0 422.7 121.6 120.2 135.9 198.1 Reference mission components were sorted and numbered from one to the maximum number of components, n . This information was plotted in Figure G.1.
Figure G.1: Component Power versus Component Number for Each Reference Mission. G.2 Design Case Distribution Development Each reference mission has a maximum component power, P1 (i.e. power of the first component after sorting), and maximum number of components, N . The component
257 distributions from Figure G.1 were normalized in both power and component number for all components Pi (1 < j < N ) using y=
Pj P1
(1 < j < N )
(G.1)
and x=
j −1 N −1
(1 < j < N ) .
(G.2)
Both normalized power and component number then ranged from 0 to 1. Normalization provided a way of comparing reference missions on an equal basis and shows a generalized shape or distribution of power versus component number (Figure G.2).
Figure G.2: Normalized Component Power versus Normalized Component Number for Each of Six Reference Missions. From this plot, four design cases were developed which capture a broad range of potential component distributions. Case A was based on the FACTS HCB. Case B is an averaged model based on all six reference missions. Case C is based on the representative military satellite. Finally, Case D is a uniform distribution such that all components will have equal power. The mathematical model used to curve fit each of these cases was based on Equation (G.3).
258 y = (1 − x ) ⋅ e s⋅ x
(G.3)
This model has an exponential base with an additional term which forces the curve to one for x equal to zero (first component) and zero for x of one (number of components plus one). This was consistent with the normalization process. A Mathcad routine was developed using the ‘genfit’ function that utilizes the Levenberg-Marquardt method for minimization to determine the exponential value s , which describes the shape of each curve fit (Table G.8). For each, the Pearson coefficient of correlation, ρ , was calculated to provide a measure of goodness of fit of each model. The model performed well when fitting through individual reference mission data (e.g. Case A). However, it did not perform as well for Case B due to inherent variability between reference missions. Table G.8: Summary of s Values for Each Design Case .
s = ρ =
Case A
Case B
Case C
Case D
-33.415
-5.670
-0.694
Not Applicable
0.984
0.814
0.936
Not Applicable
The representative design and bounding cases were plotted versus normalized component power and number in Figure G.3.
Figure G.3: Mathematical Models for Each of Four Design Cases.
259 The four design cases provide insight into the shape of component distribution but this information must be scaled for various total power and component numbers. A magnitude factor, m , was used to scale power and is defined as the fraction of total power, Ptotal , of the first (i.e. largest) component, P1 as shown here m=
P1 . Ptotal
(G.4)
Additionally, the actual component number, j (1 < j < N ) , was scaled as shown in x=
j −1 . N −1
(G.5)
Therefore, the power of the j th component, Pj , for each design case is based on the total power as shown in j −1 s⋅ j −1 N −1 )⋅e . Pj = m ⋅ Ptotal ⋅ (1 − N −1
(G.6)
Summing all individual powers gives total component power, Ptotal , illustrated by N
N
j =1
j =1
Ptotal = ∑ Pj = ∑ m ⋅Ptotal ⋅ (1 −
j −1 s⋅ j −1 ) ⋅ e N −1 . N −1
(G.7)
Therefore, values of m are a function of total component number and s values only as follows: m=
1 . j −1 s⋅ j −1 N −1 (1 − )⋅e ∑ N −1 j =1 N
(G.8)
Variation of m for various combinations of s and N are shown for the first three design cases (Table G.9). Case D is not based on these values. The N values provide a range consistent with RS spacecraft.
260 Table G.9: m Values at Various Levels of s and N . s
N =18
36
54
-33.415
0.868
0.626
0.478
-5.670
0.333
0.178
0.122
-0.694
0.136
0.069
0.046
For various levels of each component number, normalization values shown in Pj Ptotal
= m ⋅ (1 −
j −1 s⋅ j −1 ) ⋅ e N −1 N −1
(G.9)
were calculated for each design case. Presenting the results in this manner allow distributions to be generated for any Ptotal simply by multiplication. Normalization values for each case with 18 components are shown in Table G.10 and plotted in Figure G.4. Table G.10: Normalization Values for 18 Components for Each of Four Design Cases. Component Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 TOTAL
Case A-18
Case B-18
Case C-18
Case D-18
0.868 0.114 0.015 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
0.333 0.225 0.151 0.101 0.067 0.044 0.029 0.019 0.012 0.008 0.005 0.003 0.002 0.001 0.001 0.000 0.000 0.000 1.000
0.136 0.123 0.111 0.099 0.088 0.078 0.069 0.060 0.052 0.044 0.037 0.031 0.025 0.019 0.014 0.009 0.004 0.000 0.998
0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056 1.000
261
a) Case A-18
b) Case B-18
c) Case C-18 d) Case D-18 Figure G.4: Normalization Values for 18 Components for Each of Four Design Cases. G.3 Subsystems G.3.1 Attitude Determination and Control Subsystem (ADCS) Critical to the success of a satellite’s mission is its proper orientation in orbit. Without it, payload components may not be adequately positioned to take full advantage of their capabilities. Specific to the mission is the level of accuracy required for orientation.
262 Responsible for these tasks is the attitude determination and control subsystem (ADCS). The ADCS is responsible for determining the current attitude of the satellite along with providing the controls and equipment necessary to make any changes. In essence, it measures and maintains a spacecraft’s position orientation about its center of mass (Wertz and Larson, 2005). Several methods of determining the current orientation of a satellite are available each providing different levels of capability. These include sun sensors, star sensors, horizon sensors, magnetometers, GPS receivers, and gyroscopes. Sun sensors are relatively accurate (< 0.01°) and reliable devices and as a result are utilized often (Wertz and Larson, 2005). They are basically light sensors that provide information regarding the location of the sun with respect to a fixed reference on the satellite. As expected, for certain orbits such as LEO, eclipse periods make these devices unusable at times. Alternatives include star sensors. As their name implies, star sensors operate by sensing star locations and either utilize these as reference points or compare them to a database of star patterns. As with sun sensors, star sensors are susceptible to interference by the Sun, Moon, and Earth and as a result are often utilized with other sensing technologies. The Earth provides another point of reference for attitude determination. Since it occupies a relatively large portion of viewable area especially for LEO, a more refined and specific point on Earth must be located. Horizon or Earth sensors locate the transition from deep space to that of Earth’s atmosphere. In some cases, Earth sensors detect a narrow band of radiation emitted by CO2 in the atmosphere (Sellers, 2005). Accuracies of these devices are typically between 0.1 and 0.25° (Wertz and Larson, 2005). Magnetometers measure the magnetic field caused by Earth and compare those readings to a known map thus providing attitude information. These devices are commonly used in conjunction with other sensor types to improve accuracy. Similar to their use for terrestrial applications, GPS receivers can be used to determine a spacecraft’s attitude. Two receivers are required to provide a differential signal upon which attitude can be determined. These devices offer low cost and weight although currently do not have the accuracy of other technologies. Gyroscopes provide the last method of attitude determination. They provide an inertial reference and measure any deviations from this. This information is provided at a relatively high rate but information
263 regarding an absolute position is not provided. Because of this, they are utilized in combination with other sensor types to fill in the gaps. Gyroscopes by themselves provide only one or two axes of information. An inertial reference unit (IRU) is a grouping of gyroscopes that provide three-axis information while inertial measurement units (IMU) provide the same information in addition to velocity sensing via accelerometers. Without any way to control a satellite, attitude determination would be useless. Several concepts are available to provide appropriate steering or control of the spacecraft. These include gravity gradient, spin control, and three-axis control. Gravity gradient control is a passive method that utilizes the natural gravity field of Earth. Since gravitational forces vary with the cube of the orbital radius, elongated spacecrafts or objects have different forces applied to different ends. Because of this, elongated spacecraft are naturally aligned. Spin controlled satellites are another passive control method in which a satellite is spun. Based on the conservation of momentum, the spin axis then remains relatively fixed in inertial space. Another advantage of this control method is that most components see an average thermal environment making thermal management more tractable. The final control method is three-axis that is also the most popular. As its name implies, control is maintained about each of three orthogonal axes. As a result, this method is the most accurate and stable but also the most complex. This control method is accomplished utilizing one of two methods. The first utilizes momentum bias while the second utilizes zero momentum. Momentum bias operates on the same principle as spin-stabilized satellites. That is, a momentum wheel is mounted and spun to provide stiffness to the satellite. This single wheel is typically mounted normal to the orbital plane. Zero momentum systems utilize three momentum wheels each aligned along an axis. In a normal state, these wheels are not spun, but in reaction to disturbances, they spin up to provide the necessary reaction to provide overall stability of the satellite. If enough disturbances are present, the wheels may become saturated. That is, they cannot be spun any faster. When this occurs, external torques must be applied to allow the momentum wheels to be slowed or desaturated. This is commonly accomplished utilizing magnetic torque rods. These devices are basically
264 coiled wires which when energized react to Earth’s natural magnetic field and thus provide the required external torques. An alternative to three axis momentum wheels are control moment gyros (CMG). They operate similarly to three-axis momentum wheels but they are mounted on gimbals giving them further flexibility to not only change spin rates but also momentum direction. They are high weight and cost devices but provide relatively fast maneuvering (Wertz and Larson, 2005). G.3.2 Orbit Determination and Control Subsystem (ODCS) As the ADCS provides information regarding the orientation of a spacecraft about its center of mass, the ODCS provides information regarding the position of the center of mass. In combination, the ADCS and ODCS provide information necessary to describe the position and orientation of a spacecraft. The traditional method to determine spacecraft orbit is through a ground tracking station. A disadvantage of this system is that several passes are required to provide accurate orbit determination (Wertz and Larson, 2005). A modern method of obtaining the same information is with the Tracking and Data Relay Satellite (TDRS). Two TDRS satellites are required to provide 85 to 100% of orbit determination for LEO. In addition, this approach is more accurate than most ground tracking systems (Wertz and Larson, 2005). G.3.3 Telemetry, Tracking, and Command Subsystem (TTCS) Without a way to communicate with a satellite, they would be virtually useless. The telemetry, tracking, and command subsystem (TTCS) provides a necessary connection between ground systems and a spacecraft. It provides spacecraft payload, health and status information to ground equipment and sends commands up to the spacecraft. These subsystem lock onto ground station signals, receive and process uplink signals, transmit subsystem data including health and status along with providing satellite position information. Considerations for any TTCS include access, frequency, and data characteristics (Wertz and Larson, 2005). Access describes the ability to provide a clear path to the spacecraft antenna and supply the required antenna gain. Frequency discusses the
265 internationally approved band utilized. Standard types include S-Band at 2 GHz, X-Band at 8 GHz and Ku-Band at 12 GHz. Data characteristics include bandwidth, error rates, and required RF power levels. The Space Ground Link Subsystem (SGLS) provides uplink and downlink on S-Band frequencies at 1.75 to 1.85 GHz and 2.20 to 2.30 GHz, respectively. The Tracking and Data Relay Satellite System (TDRSS) is a cross-link system providing uplink and downlink on S-Band, K-Band and Ku-Band frequencies. G.3.4 Command and Data Handling Subsystem (CDHS) The CDHS provides the capability necessary to receive, process, store, and distribute commands to and from other spacecraft subsystems. As such, it includes devices such as computer processors and hard drives. Typical activities of the CDHS include telemetry processing, time maintenance, and failure detection among others (Wertz and Larson, 2005). G.3.5 Electrical Power Subsystem (EPS) The EPS includes four general categories of equipment. These include power sources, energy storage, power distribution, power regulation, and control. Power sources include photovoltaic solar cells; static power sources that use heat sources for direct thermal to electric conversion; dynamic power sources that utilize Brayton, Stirling or Rankine cycles; and fuel cells (Wertz and Larson, 2005). Of these, photovoltaic solar cells are the most common for Earth orbiting spacecraft. Inner thermal performance includes such thermally sensitive devices as batteries, optic instrumentation, and individual payloads. The most sensitive of these are batteries that typically must be maintained in a range of -5°C to 15°C 200 W (Baturkin, 2005). G.3.6 Structure and Mechanisms Subsystem (SMS) The SMS includes all devices necessary to support and fasten all other subsystems. A variety of technologies is available including honeycomb panels and advanced composites among others.
266 G.3.7 Payload The payload includes the devices that all other subsystems are serving and are the reason for spacecraft. These devices are based heavily on satellite missions and design. G.3.8 Thermal Control Subsystem (TCS) The task of the TCS is to provide a comfortable environment for all spacecraft components. This requires that all components be maintained within required temperature limits. Operational temperature limits are those imposed when components are operating while survival temperature limits must be met at all times. Two types of thermal control are available. Passive thermal control utilizes technologies that do not require any external control. This includes items such as materials, coatings, and surface finishes. Active thermal control is more complex and includes devices that require an active external control. This includes such devices as heaters and two-phase systems.
267 Appendix H – Hot- and Cold-Case Temporal Flux Characteristics
The following provides a detailed account of the hot- and cold-case orbits. Figure H.1 shows orbital positions for both hot- and cold-case orbits. These images were generated using Thermal Desktop®.
a) Hot-Case Orbit b) Cold-Case Orbit Figure H.1: Hot- and Cold-Case Orbit Illustrations. The following figures show total, direct solar, albedo, and OLR incident fluxes for both design orbits. As shown from these figures, the most significant difference between these orbits is the absence of eclipse for the hot case resulting in overall higher averaged heat fluxes. In addition, the most significant incident flux source is due to direct solar.
268
a) Hot-Case Total Heat Flux b) Cold-Case Total Heat Flux Figure H.2: Hot- and Cold-Case Total Heat Flux Over One Orbital Period for Each of Six Surfaces.
a) Hot-Case Direct Solar Heat Flux b) Cold-Case Direct Solar Heat Flux Figure H.3: Hot- and Cold-Case Direct Solar Heat Flux Over One Orbital Period for Each of Six Surfaces.
269
a) Hot-Case Albedo Heat Flux b) Cold-Case Albedo Heat Flux Figure H.4: Hot- and Cold-Case Albedo Heat Flux Over One Orbital Period for Each of Six Surfaces.
a) Hot-Case OLR Heat Flux b) Cold-Case OLR Heat Flux Figure H.5: Hot- and Cold-Case OLR Heat Flux Over One Orbital Period for Each of Six Surfaces.
270 Appendix I – Resulting Component Placement Values
271 Table I.1: Summary of Hot- and Cold-Case Orbit (Case C-36) – Nominal Global Distribution / Nominal Local Placement. Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
41.18 22.71 15.12 14.00 3.97 3.25 39.22 24.12 16.28 12.92 4.73 2.55 37.32 25.59 17.48 11.87 6.32 1.22 35.48 27.11 18.72 10.86 8.04 0.00 33.69 28.67 20.00 9.89 7.17 0.60 31.97 30.29 21.33 8.95 5.51 1.87
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
TOTAL
600.00
---
---
---
272 Table I.2: Summary of Hot- and Cold-Case Orbit (Case C-36) – Nominal Global Distribution / Optimized Local Placement. Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
41.18 22.71 15.12 14.00 3.97 3.25 39.22 24.12 16.28 12.92 4.73 2.55 37.32 25.59 17.48 11.87 6.32 1.22 35.48 27.11 18.72 10.86 8.04 0.00 33.69 28.67 20.00 9.89 7.17 0.60 31.97 30.29 21.33 8.95 5.51 1.87
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.32 0.73 0.22 0.86 0.78 0.62 0.26 0.78 0.81 0.41 0.11 0.59 0.32 0.72 0.22 0.81 0.90 0.70 0.29 0.78 0.78 0.15 0.43 0.50 0.24 0.77 0.50 0.86 0.13 0.22 0.23 0.75 0.46 0.87 0.11 0.75
0.34 0.77 0.83 0.20 0.45 0.10 0.34 0.27 0.78 0.82 0.87 0.59 0.30 0.76 0.81 0.16 0.43 0.41 0.33 0.27 0.77 0.82 0.84 0.50 0.31 0.30 0.78 0.82 0.85 0.68 0.30 0.30 0.78 0.75 0.73 0.90
TOTAL
600.00
---
---
---
273 Table I.3: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 31.97 24.12 14.00 9.89 4.73 0.60 33.69 21.33 16.28 8.04 5.51 0.00 41.18 30.29 25.59
Panel [---]
Placement x [m]
y [m]
12.92 11.87 2.55 39.22 35.48 20.00 17.48 7.17 6.32 28.67 22.71 15.12 8.95 3.97 1.22 37.32 27.11 18.72 10.86 3.25 1.87
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
600.00
---
---
---
274 Table I.4: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
31.97 25.59 15.12 14.00 6.32 3.25 33.69 24.12 17.48 11.87 7.17 2.55 41.18 30.29 27.11 16.28 12.92 5.51 39.22 35.48 22.71 18.72 10.86 8.04 37.32 28.67 21.33 20.00 9.89 8.95 4.73 3.97 1.87 1.22 0.60 0.00
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
TOTAL
600.00
---
---
---
275 Table I.5: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 14.00 11.87 9.89 7.17 4.73 2.55 15.12 10.86
Panel [---]
Placement x [m]
y [m]
3.97 3.25 1.22 0.60 0.00 30.29 28.67 21.33 20.00 16.28 12.92
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
600.00
---
---
---
8.95 8.04 5.51 1.87 41.18 35.48 31.97 27.11 24.12 17.48 39.22 37.32 33.69 25.59 22.71 18.72 6.32
276 Table I.6: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
18.72 15.12 8.95 6.32 3.97 1.87 20.00 12.92 10.86 5.51 4.73 0.00 41.18 35.48 33.69 28.67 27.11 22.71 39.22 37.32 31.97 30.29 25.59 24.12 17.48 14.00 9.89 8.04 2.55 0.60 21.33 16.28 11.87 7.17 3.25 1.22
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
TOTAL
600.00
---
---
---
277 Table I.7: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 35.48 28.67 21.33 14.00
Panel [---]
Placement x [m]
y [m]
3.25 2.55 1.87 1.22 0.60 0.00 41.18 31.97 24.12 18.72 10.86 7.17 39.22 30.29 22.71 16.28 11.87 4.73 33.69 25.59 17.48 12.92 5.51 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
600.00
---
---
---
8.95 8.04 37.32 27.11 20.00 15.12 9.89 6.32
278 Table I.8: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.123 / ε = 0.900). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 33.69 28.67 20.00 17.48 11.87 8.95 35.48 27.11 21.33 15.12 12.92 9.89 8.04 6.32
Panel [---]
Placement x [m]
y [m]
5.51 4.73 2.55 1.87 0.00
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
600.00
---
---
---
3.97 3.25 1.22 0.60 37.32 30.29 24.12 18.72 14.00 10.86 41.18 39.22 31.97 25.59 22.71 16.28 7.17
279 Table I.9: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.100). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
31.97 28.67 17.48 15.12 8.95 5.51 33.69 27.11 18.72 14.00 8.04 6.32 3.25 2.55 1.87 1.22 0.60 0.00 41.18 39.22 35.48 25.59 20.00 11.87 37.32 30.29 22.71 16.28 10.86 7.17 24.12 21.33 12.92 9.89 4.73 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
TOTAL
600.00
---
---
---
280 Table I.10: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Nominal Local Placement (α = 0.561 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
28.67 24.12 20.00 14.00 10.86 9.89 30.29 22.71 21.33 12.92 11.87 8.95 3.25 2.55 1.87 1.22 0.60 0.00 39.22 35.48 31.97 27.11 17.48 16.28 41.18 37.32 33.69 25.59 18.72 15.12 8.04 7.17 6.32 5.51 4.73 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83 0.17 0.17 0.50 0.50 0.83 0.83
TOTAL
600.00
---
---
---
281 Table I.11: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 31.97 24.12 14.00 9.89 4.73 0.60 33.69 21.33 16.28 8.04 5.51 0.00 41.18 30.29 25.59
Panel [---]
Placement x [m]
y [m]
12.92 11.87 2.55 39.22 35.48 20.00 17.48 7.17 6.32 28.67 22.71 15.12 8.95 3.97 1.22 37.32 27.11 18.72 10.86 3.25 1.87
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.31 0.72 0.21 0.85 0.73 0.66 0.27 0.62 0.80 0.17 0.90 0.50 0.32 0.74 0.23 0.81 0.83 0.49 0.30 0.71 0.80 0.20 0.12 0.37 0.33 0.74 0.24 0.83 0.10 0.71 0.27 0.73 0.82 0.17 0.41 0.56
0.30 0.76 0.81 0.31 0.10 0.43 0.32 0.74 0.24 0.83 0.87 0.50 0.27 0.77 0.79 0.14 0.40 0.59 0.26 0.71 0.21 0.65 0.90 0.90 0.26 0.72 0.82 0.18 0.55 0.38 0.34 0.73 0.23 0.82 0.90 0.10
600.00
---
---
---
282 Table I.12: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
31.97 25.59 15.12 14.00 6.32 3.25 33.69 24.12 17.48 11.87 7.17 2.55 41.18 30.29 27.11 16.28 12.92 5.51 39.22 35.48 22.71 18.72 10.86 8.04 37.32 28.67 21.33 20.00 9.89 8.95 4.73 3.97 1.87 1.22 0.60 0.00
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.31 0.76 0.78 0.21 0.14 0.41 0.31 0.72 0.22 0.82 0.86 0.59 0.31 0.25 0.76 0.80 0.83 0.56 0.26 0.64 0.80 0.19 0.90 0.55 0.27 0.73 0.48 0.79 0.15 0.86 0.27 0.70 0.82 0.18 0.58 0.50
0.50 0.27 0.78 0.87 0.12 0.10 0.31 0.76 0.80 0.17 0.44 0.10 0.25 0.77 0.81 0.16 0.45 0.54 0.30 0.71 0.23 0.80 0.86 0.10 0.50 0.81 0.13 0.42 0.88 0.11 0.33 0.73 0.22 0.83 0.10 0.50
TOTAL
600.00
---
---
---
283 Table I.13: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 14.00 11.87 9.89 7.17 4.73 2.55 15.12 10.86
Panel [---]
Placement x [m]
y [m]
3.97 3.25 1.22 0.60 0.00 30.29 28.67 21.33 20.00 16.28 12.92
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.25 0.73 0.23 0.62 0.88 0.90 0.29 0.26 0.76 0.78 0.83 0.55 0.25 0.54 0.20 0.74 0.81 0.90 0.26 0.74 0.81 0.48 0.20 0.13 0.34 0.28 0.78 0.85 0.89 0.50 0.46 0.71 0.22 0.15 0.88 0.83
0.29 0.74 0.78 0.20 0.40 0.11 0.27 0.78 0.82 0.17 0.48 0.56 0.25 0.61 0.77 0.18 0.85 0.49 0.45 0.25 0.75 0.82 0.13 0.84 0.25 0.79 0.59 0.16 0.90 0.50 0.34 0.80 0.77 0.21 0.45 0.12
600.00
---
---
---
8.95 8.04 5.51 1.87 41.18 35.48 31.97 27.11 24.12 17.48 39.22 37.32 33.69 25.59 22.71 18.72 6.32
284 Table I.14: Summary of Cold-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
18.72 15.12 8.95 6.32 3.97 1.87 20.00 12.92 10.86 5.51 4.73 0.00 41.18 35.48 33.69 28.67 27.11 22.71 39.22 37.32 31.97 30.29 25.59 24.12 17.48 14.00 9.89 8.04 2.55 0.60 21.33 16.28 11.87 7.17 3.25 1.22
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.31 0.73 0.21 0.81 0.53 0.90 0.25 0.77 0.55 0.88 0.16 0.50 0.21 0.26 0.65 0.52 0.80 0.87 0.39 0.75 0.62 0.21 0.19 0.83 0.27 0.77 0.77 0.36 0.10 0.29 0.25 0.76 0.79 0.39 0.10 0.54
0.27 0.61 0.78 0.17 0.90 0.90 0.33 0.27 0.78 0.71 0.83 0.50 0.31 0.79 0.53 0.12 0.83 0.19 0.49 0.19 0.87 0.18 0.82 0.59 0.31 0.27 0.77 0.82 0.71 0.41 0.35 0.27 0.78 0.84 0.79 0.61
TOTAL
600.00
---
---
---
285 Table I.15: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.100). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 35.48 28.67 21.33 14.00
Panel [---]
Placement x [m]
y [m]
3.25 2.55 1.87 1.22 0.60 0.00 41.18 31.97 24.12 18.72 10.86 7.17 39.22 30.29 22.71 16.28 11.87 4.73 33.69 25.59 17.48 12.92 5.51 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.27 0.28 0.75 0.87 0.62 0.86 0.32 0.74 0.20 0.83 0.74 0.46 0.46 0.19 0.63 0.90 0.87 0.50 0.28 0.78 0.51 0.15 0.87 0.87 0.26 0.53 0.79 0.88 0.14 0.63 0.30 0.27 0.81 0.83 0.63 0.59
0.28 0.78 0.56 0.18 0.14 0.88 0.30 0.74 0.78 0.35 0.10 0.90 0.24 0.73 0.77 0.80 0.46 0.50 0.29 0.43 0.79 0.79 0.85 0.10 0.31 0.81 0.21 0.63 0.78 0.47 0.26 0.76 0.23 0.80 0.56 0.90
600.00
---
---
---
8.95 8.04 37.32 27.11 20.00 15.12 9.89 6.32
286 Table I.16: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.123 / ε = 0.900). Component Number [---] SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6 TOTAL
Component Power [W] 33.69 28.67 20.00 17.48 11.87 8.95 35.48 27.11 21.33 15.12 12.92 9.89 8.04 6.32
Panel [---]
Placement x [m]
y [m]
5.51 4.73 2.55 1.87 0.00
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.30 0.29 0.79 0.80 0.63 0.10 0.28 0.57 0.79 0.19 0.87 0.15 0.21 0.74 0.48 0.84 0.37 0.15 0.26 0.58 0.22 0.81 0.82 0.90 0.35 0.34 0.76 0.80 0.81 0.10 0.29 0.74 0.79 0.14 0.40 0.50
0.22 0.78 0.80 0.20 0.50 0.52 0.23 0.69 0.24 0.87 0.83 0.60 0.50 0.71 0.18 0.21 0.90 0.10 0.25 0.53 0.79 0.84 0.16 0.48 0.23 0.77 0.50 0.85 0.15 0.50 0.32 0.73 0.23 0.81 0.85 0.50
600.00
---
---
---
3.97 3.25 1.22 0.60 37.32 30.29 24.12 18.72 14.00 10.86 41.18 39.22 31.97 25.59 22.71 16.28 7.17
287 Table I.17: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.100). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
31.97 28.67 17.48 15.12 8.95 5.51 33.69 27.11 18.72 14.00 8.04 6.32 3.25 2.55 1.87 1.22 0.60 0.00 41.18 39.22 35.48 25.59 20.00 11.87 37.32 30.29 22.71 16.28 10.86 7.17 24.12 21.33 12.92 9.89 4.73 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.24 0.76 0.81 0.18 0.49 0.51 0.30 0.23 0.79 0.62 0.86 0.90 0.32 0.72 0.22 0.80 0.88 0.50 0.25 0.23 0.76 0.57 0.82 0.90 0.26 0.75 0.55 0.17 0.88 0.88 0.30 0.27 0.78 0.83 0.59 0.73
0.29 0.28 0.78 0.79 0.88 0.61 0.27 0.76 0.22 0.80 0.58 0.87 0.29 0.74 0.79 0.16 0.42 0.50 0.25 0.77 0.20 0.60 0.86 0.55 0.30 0.25 0.76 0.80 0.62 0.89 0.24 0.73 0.43 0.81 0.89 0.11
TOTAL
600.00
---
---
---
288 Table I.18: Summary of Hot-Case Orbit (Case C-36) – Optimized Global Distribution / Optimized Local Placement (α = 0.561 / ε = 0.900). Component Number [---]
Component Power [W]
Panel [---]
Placement x [m]
y [m]
SXN_1 SXN_2 SXN_3 SXN_4 SXN_5 SXN_6 SXP_1 SXP_2 SXP_3 SXP_4 SXP_5 SXP_6 SYN_1 SYN_2 SYN_3 SYN_4 SYN_5 SYN_6 SYP_1 SYP_2 SYP_3 SYP_4 SYP_5 SYP_6 SZN_1 SZN_2 SZN_3 SZN_4 SZN_5 SZN_6 SZP_1 SZP_2 SZP_3 SZP_4 SZP_5 SZP_6
28.67 24.12 20.00 14.00 10.86 9.89 30.29 22.71 21.33 12.92 11.87 8.95 3.25 2.55 1.87 1.22 0.60 0.00 39.22 35.48 31.97 27.11 17.48 16.28 41.18 37.32 33.69 25.59 18.72 15.12 8.04 7.17 6.32 5.51 4.73 3.97
XXXXXXX+ X+ X+ X+ X+ X+ YYYYYYY+ Y+ Y+ Y+ Y+ Y+ ZZZZZZZ+ Z+ Z+ Z+ Z+ Z+
0.28 0.54 0.78 0.87 0.17 0.13 0.29 0.75 0.79 0.45 0.18 0.12 0.31 0.29 0.77 0.80 0.86 0.50 0.27 0.76 0.26 0.79 0.50 0.14 0.25 0.77 0.78 0.34 0.13 0.47 0.27 0.79 0.77 0.49 0.20 0.11
0.24 0.71 0.25 0.80 0.88 0.59 0.50 0.76 0.26 0.10 0.87 0.14 0.24 0.76 0.48 0.84 0.12 0.50 0.17 0.75 0.81 0.22 0.48 0.50 0.22 0.25 0.76 0.87 0.63 0.54 0.18 0.76 0.24 0.58 0.86 0.54
TOTAL
600.00
---
---
---
289 Table I.19: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.123 / ε = 0.100). Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface 27.57 17.12 +X Surface 27.48 17.12 -Y Surface 177.04 165.99 +Y Surface 9.81 0.00 -Z Surface 17.12 17.12 +Z Surface 47.81 17.12 AVERAGES 51.14 39.08
2.07 2.05 2.83 1.38 0.00 6.13 2.41
8.38 8.31 8.21 8.43 0.00 24.56 9.65
46.79 46.71 6.19 6.24 51.43 33.08 31.74
40.52 40.52 0.00 0.00 51.43 9.14 23.60
1.79 1.79 1.78 1.78 0.00 6.89 2.34
4.48 4.39 4.42 4.46 0.00 17.05 5.80
Table I.20: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.561 / ε = 0.100). Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface 95.91 78.10 +X Surface 95.75 78.10 -Y Surface 778.22 757.10 +Y Surface 14.71 0.00 -Z Surface 78.10 78.10 +Z Surface 130.63 78.10 AVERAGES 198.89 178.25
9.42 9.34 12.91 6.28 0.00 27.97 10.99
8.38 8.31 8.21 8.43 0.00 24.56 9.65
197.45 197.37 12.53 12.56 234.58 90.16 124.11
184.82 184.82 0.00 0.00 234.58 41.69 107.65
8.15 8.16 8.11 8.10 0.00 31.42 10.66
4.48 4.39 4.42 4.46 0.00 17.05 5.80
290 Table I.21: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.123 / ε = 0.900). Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface 94.60 17.12 +X Surface 93.92 17.12 -Y Surface 242.73 165.99 +Y Surface 77.27 0.00 -Z Surface 17.12 17.12 +Z Surface 244.26 17.12 AVERAGES 128.32 39.08
2.07 2.05 2.83 1.38 0.00 6.13 2.41
75.41 82.60 74.75 81.86 73.91 41.52 75.89 41.94 0.00 51.43 221.01 169.48 86.83 78.14
40.52 40.52 0.00 0.00 51.43 9.14 23.60
1.79 1.79 1.78 1.78 0.00 6.89 2.34
40.29 39.55 39.74 40.17 0.00 153.45 52.20
Table I.22: Orbital Averaged Hot- and Cold-Case Total, Direct Solar, Albedo, and OLR Incident Heat Flux Values for an Optical Solar Reflector Surface (α = 0.561 / ε = 0.900). Hot-Case Orbital Averaged Cold-Case Orbital Averaged Total Solar Albedo OLR Total Solar Albedo OLR Flux Flux Flux Flux Flux Flux Flux Flux 2 2 2 2 2 2 2 [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m ] [W/m2] -X Surface +X Surface -Y Surface +Y Surface -Z Surface +Z Surface AVERAGES
162.94 78.10 162.19 78.10 843.91 757.10 82.17 0.00 78.10 78.10 327.08 78.10 276.07 178.25
9.42 9.34 12.91 6.28 0.00 27.97 10.99
75.41 74.75 73.91 75.89 0.00 221.01 86.83
233.27 232.53 47.85 48.26 234.58 226.56 170.51
184.82 184.82 0.00 0.00 234.58 41.69 107.65
8.15 8.16 8.11 8.10 0.00 31.42 10.66
40.29 39.55 39.74 40.17 0.00 153.45 52.20
291 Appendix J – Full Factorial Main and Interaction Effect Plots
292
Figure J.1: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus Each of Eleven Main Effects.
293
Figure J.2: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus ORBIT for Each of Ten Interaction Effects.
294
Figure J.3: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus TOT_PWR for Each of Ten Interaction Effects.
295
Figure J.4: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus C_DIM for Each of Ten Interaction Effects.
296
Figure J.5: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus C_I_CND for Each of Ten Interaction Effects.
297
Figure J.6: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus F_T_CND for Each of Ten Interaction Effects.
298
Figure J.7: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus HT_PIPE for Each of Ten Interaction Effects.
299
Figure J.8: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus P2P_CND for Each of Ten Interaction Effects.
300
Figure J.9: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus EXT_ABS for Each of Ten Interaction Effects.
301
Figure J.10: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus EXT_EMS for Each of Ten Interaction Effects.
302
Figure J.11: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus GLBL_DIS for Each of Ten Interaction Effects.
303
Figure J.12: Full Factorial Scatterplot and Linear Regression Fit of Maximum Temperature Difference versus LCL_PLC for Each of Ten Interaction Effects.
304 Appendix K – Temperature Versus Time Results for Increasing Levels of Satellite Conductance
x = 100
x = 1,000
x = 10,000 x =100,000 Figure K.1: Cold-Case Component Temperatures (18 of 36 components) versus Second Orbit Orbital Time for Increasing Magnitudes of Thermal Bus Conductivity Values (C_I_CND = F_T_CND = P2P_CND = x).
305
x = 100
x = 1,000
x = 10,000 x =100,000 Figure K.2: Hot-Case Component Temperatures (18 of 36 components) versus Second Orbit Orbital Time for Increasing Magnitudes of Thermal Bus Conductivity Values (C_I_CND (W/m2-K) = F_T_CND (W/m-K) = P2P_CND (W/K) = x).
306 Appendix L – Latin Hypercube Sampling Points
C_PWR
C_DIM
C_I_CND
F_T_CND
P2P_CND
EXT_ABS
EXT_EMS
Table L.1: Latin Hypercube Test Points for Space Filling Approach (1-32 of 128).
[W]
[m]
[W/m2-K]
[W/m-K]
[W/K]
[---]
[---]
514.961 149.291 366.142 217.323 404.409 306.614 242.835 497.953 264.094 68.504 506.457 536.220 293.858 446.929 315.118 600.000 234.331 349.134 106.772 327.874 332.126 582.992 476.693 310.866 174.803 60.000 289.606 183.307 162.047 191.811 238.583 221.575
0.182 0.172 0.145 0.186 0.196 0.139 0.106 0.176 0.160 0.117 0.174 0.132 0.129 0.197 0.148 0.135 0.198 0.157 0.175 0.193 0.143 0.115 0.168 0.110 0.134 0.136 0.154 0.187 0.106 0.128 0.130 0.143
630.315 611.732 374.803 351.575 272.598 444.488 653.543 119.291 625.669 597.795 402.677 230.787 161.102 514.173 133.228 500.236 365.512 700.000 300.472 398.031 509.528 681.417 588.504 411.969 379.449 435.197 667.480 263.307 342.283 240.079 309.764 690.709
823.543 424.882 934.646 653.622 1000.000 464.094 692.835 352.992 202.677 477.165 947.717 921.575 699.370 366.063 339.921 718.976 281.102 333.386 189.606 875.827 398.740 529.449 954.252 882.362 359.528 549.055 993.465 522.913 771.260 392.205 862.756 941.181
22.961 15.402 25.417 35.244 33.732 25.228 20.504 24.094 34.488 31.465 12.945 34.110 34.299 29.764 23.528 33.165 23.717 30.709 31.843 24.661 24.283 19.181 15.591 35.055 34.866 26.173 26.929 24.472 23.339 12.567 12.189 16.535
0.147 0.240 0.340 0.354 0.320 0.533 0.433 0.157 0.230 0.485 0.292 0.206 0.358 0.175 0.347 0.330 0.351 0.385 0.471 0.509 0.161 0.506 0.409 0.430 0.261 0.313 0.499 0.189 0.202 0.182 0.223 0.254
0.743 0.623 0.440 0.661 0.396 0.138 0.352 0.761 0.528 0.869 0.308 0.257 0.434 0.686 0.591 0.843 0.522 0.176 0.201 0.610 0.818 0.705 0.724 0.711 0.188 0.516 0.642 0.900 0.894 0.887 0.579 0.509
307
C_PWR
C_DIM
C_I_CND
F_T_CND
P2P_CND
EXT_ABS
EXT_EMS
Table L.2: Latin Hypercube Test Points for Space Filling Approach (33-64 of 128).
[W]
[m]
[W/m2-K]
[W/m-K]
[W/K]
[---]
[---]
561.732 319.370 361.890 353.386 302.362 374.646 208.819 132.283 255.591 400.157 489.449 587.244 540.472 187.559 408.661 276.850 485.197 383.150 438.425 251.339 548.976 468.189 323.622 179.055 268.346 128.031 570.236 442.677 578.740 119.528 72.756 421.417
0.127 0.119 0.146 0.120 0.118 0.150 0.172 0.151 0.137 0.116 0.139 0.162 0.200 0.126 0.155 0.105 0.189 0.199 0.149 0.191 0.185 0.183 0.133 0.159 0.173 0.113 0.121 0.108 0.131 0.165 0.171 0.102
565.276 621.024 202.913 430.551 114.646 551.339 207.559 486.299 634.961 165.748 672.126 560.630 323.701 346.929 128.583 235.433 639.606 425.906 221.496 695.354 267.953 477.008 648.898 170.394 179.685 472.362 542.047 555.984 332.992 244.724 319.055 314.409
457.559 183.071 490.236 496.772 568.661 686.299 725.512 960.787 673.228 294.173 803.937 248.425 555.591 647.087 888.898 836.614 732.047 411.811 843.150 797.402 620.945 849.685 575.197 967.323 437.953 980.394 287.638 901.969 320.315 856.220 241.890 705.906
15.213 21.260 34.677 25.984 18.236 17.102 29.575 18.047 30.520 30.898 24.850 21.071 15.024 20.693 20.315 14.457 30.142 28.630 25.039 32.409 32.220 27.307 29.008 28.819 14.646 31.276 31.087 20.126 28.441 17.858 32.976 32.031
0.233 0.444 0.216 0.554 0.523 0.154 0.482 0.489 0.199 0.458 0.375 0.406 0.540 0.289 0.140 0.468 0.399 0.395 0.389 0.344 0.302 0.544 0.361 0.195 0.513 0.244 0.454 0.258 0.226 0.402 0.209 0.213
0.465 0.453 0.856 0.850 0.125 0.169 0.213 0.377 0.270 0.182 0.239 0.780 0.617 0.100 0.390 0.768 0.717 0.119 0.881 0.327 0.667 0.144 0.654 0.680 0.774 0.497 0.535 0.749 0.629 0.755 0.585 0.673
308
C_PWR
C_DIM
C_I_CND
F_T_CND
P2P_CND
EXT_ABS
EXT_EMS
Table L.3: Latin Hypercube Test Points for Space Filling Approach (65-96 of 128).
[W]
[m]
[W/m2-K]
[W/m-K]
[W/K]
[---]
[---]
247.087 387.402 115.276 225.827 480.945 591.496 285.354 259.843 123.780 77.008 204.567 519.213 340.630 434.173 281.102 111.024 272.598 493.701 170.551 574.488 81.260 463.937 527.717 544.724 336.378 565.984 531.969 230.079 344.882 64.252 136.535 459.685
0.161 0.164 0.177 0.124 0.135 0.157 0.187 0.140 0.113 0.124 0.181 0.142 0.156 0.154 0.165 0.184 0.194 0.123 0.109 0.122 0.152 0.112 0.128 0.178 0.146 0.161 0.141 0.195 0.179 0.169 0.120 0.114
523.465 281.890 193.622 481.654 216.850 258.661 662.835 518.819 254.016 249.370 184.331 286.535 463.071 151.811 142.520 569.921 393.386 295.827 537.402 546.693 658.189 458.425 328.346 175.039 226.142 616.378 212.205 407.323 291.181 110.000 123.937 439.843
928.110 196.142 431.417 385.669 274.567 640.551 170.000 215.748 209.213 594.803 973.858 758.189 588.268 895.433 634.016 607.874 869.291 307.244 176.535 784.331 418.346 908.504 313.780 483.701 470.630 326.850 254.961 516.378 542.520 535.984 601.339 817.008
27.118 35.811 13.512 35.622 29.953 22.394 22.583 19.559 23.150 12.756 17.669 22.016 13.323 14.079 29.386 27.874 19.370 17.480 28.063 28.252 20.882 21.827 14.268 32.787 13.134 26.740 29.197 15.969 27.685 26.551 26.362 13.701
0.275 0.309 0.282 0.502 0.185 0.271 0.382 0.133 0.413 0.423 0.326 0.126 0.420 0.492 0.558 0.144 0.168 0.371 0.285 0.137 0.378 0.527 0.520 0.464 0.178 0.151 0.451 0.247 0.192 0.268 0.416 0.447
0.837 0.478 0.554 0.459 0.150 0.106 0.812 0.339 0.333 0.428 0.289 0.799 0.472 0.409 0.698 0.484 0.572 0.226 0.692 0.371 0.157 0.566 0.787 0.276 0.283 0.245 0.875 0.194 0.346 0.314 0.806 0.131
309
C_PWR
C_DIM
C_I_CND
F_T_CND
P2P_CND
EXT_ABS
EXT_EMS
Table L.4: Latin Hypercube Test Points for Space Filling Approach (97-128 of 128).
[W]
[m]
[W/m2-K]
[W/m-K]
[W/K]
[---]
[---]
157.795 98.268 370.394 94.016 395.906 523.465 378.898 153.543 196.063 89.764 425.669 85.512 553.228 391.654 412.913 213.071 429.921 102.520 595.748 166.299 200.315 145.039 140.787 357.638 502.205 451.181 472.441 557.480 298.110 455.433 417.165 510.709
0.102 0.147 0.100 0.117 0.101 0.176 0.180 0.191 0.109 0.125 0.107 0.163 0.138 0.158 0.153 0.198 0.194 0.150 0.169 0.167 0.104 0.190 0.166 0.131 0.144 0.188 0.192 0.111 0.180 0.103 0.170 0.183
305.118 532.756 337.638 495.591 504.882 528.110 370.157 579.213 198.268 574.567 449.134 602.441 188.976 467.717 676.772 388.740 137.874 421.260 490.945 356.220 583.858 607.087 360.866 593.150 453.780 384.094 686.063 147.165 644.252 277.244 156.457 416.614
790.866 830.079 444.488 261.496 379.134 777.795 268.031 764.724 451.024 745.118 810.472 509.843 614.409 751.654 346.457 666.693 581.732 235.354 405.276 986.929 627.480 915.039 228.819 300.709 712.441 562.126 679.764 660.157 503.307 738.583 372.598 222.283
25.795 36.000 19.748 13.890 30.331 33.543 31.654 16.346 25.606 18.425 33.921 32.598 12.000 35.433 15.780 17.291 16.157 21.449 22.772 27.496 21.638 19.937 14.835 12.378 33.354 16.913 18.614 22.205 23.906 18.992 18.803 16.724
0.478 0.437 0.333 0.306 0.237 0.251 0.561 0.440 0.171 0.426 0.392 0.278 0.295 0.123 0.368 0.551 0.323 0.495 0.516 0.164 0.130 0.316 0.461 0.364 0.537 0.299 0.337 0.475 0.530 0.220 0.547 0.264
0.383 0.415 0.648 0.547 0.365 0.220 0.541 0.730 0.446 0.793 0.207 0.862 0.598 0.635 0.163 0.251 0.421 0.736 0.320 0.232 0.560 0.113 0.264 0.824 0.503 0.831 0.358 0.604 0.491 0.302 0.295 0.402
310 Appendix M – Summary of Simulation Test Cases
572.107 282.951 994.644 721.398 184.836 220.019 228.786 831.931 768.991 874.765 279.146 630.766 660.581 450.085 305.668 253.858 879.415 677.083 675.783 877.004 757.086 561.863 763.522 254.666 286.190 290.585 552.827 269.610 553.098 594.721 389.358 752.166 592.403
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
25.988 16.854 21.352 18.852 19.390 28.171 20.902 18.018 25.111 20.990 16.903 29.513 35.499 18.867 20.734 19.675 16.876 15.916 28.608 28.874 17.563 14.781 20.939 17.173 20.361 12.710 19.918 33.665 19.057 33.670 20.403 19.821 17.398
0.457 0.512 0.224 0.341 0.214 0.326 0.347 0.296 0.317 0.285 0.221 0.268 0.327 0.479 0.310 0.230 0.410 0.387 0.209 0.417 0.536 0.276 0.557 0.509 0.355 0.299 0.222 0.149 0.228 0.238 0.344 0.365 0.499
GLBL_DIS
EXT_EMS
EXT_ABS
P2P_CND
HT_PIPE
F_T_CND
C_I_CND 676.345 128.350 503.126 506.922 206.554 635.926 677.891 275.315 166.708 659.179 486.550 522.775 626.330 240.092 298.303 492.702 418.016 628.612 115.823 299.461 461.186 575.277 350.290 436.276 469.457 252.303 447.972 562.458 421.355 491.213 556.176 177.932 584.732
Tmaxd
0.177 0.146 0.169 0.101 0.171 0.173 0.103 0.125 0.171 0.173 0.149 0.159 0.108 0.105 0.150 0.139 0.170 0.121 0.185 0.136 0.189 0.166 0.172 0.179 0.103 0.168 0.164 0.159 0.184 0.131 0.142 0.181 0.126
Tmin
265.2 213.0 479.4 433.2 384.6 196.2 469.2 355.8 207.6 390.0 139.8 427.8 73.8 479.4 235.2 196.2 115.8 419.4 408.6 508.2 332.4 166.8 328.2 573.6 208.8 239.4 180.0 246.6 390.0 241.2 576.0 369.0 220.2
Tmax
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[m] [W/m2-K] [W/m-K] [---] [W/K] [---]
LCL_PLC
[---] [W]
C_DIM
TOT_PWR
ORBIT
Table M.1: Summary of Test Cases used to Evaluate Reduced-Order Models (1-33 of 100).
[---] [---] [---]
[K]
[K]
[K]
323.943 309.181 271.398 295.594 363.900 301.860 316.738 319.417 353.878 279.592 321.394 294.332 341.708 339.644 285.643 372.138 294.153 369.169 295.923 316.614 317.488 274.651 333.828 348.705 312.680 301.027 254.909 268.652 400.885 257.032 452.487 438.125 328.578
251.624 219.801 229.209 232.566 295.487 236.855 230.739 265.108 299.633 232.071 270.817 244.866 287.406 247.387 224.716 314.841 230.928 298.644 250.867 250.865 238.849 226.930 252.480 256.722 244.296 243.776 214.650 234.962 357.294 215.139 384.132 380.000 255.008
64.736 84.286 35.769 52.913 66.583 63.207 77.019 49.378 49.611 37.859 49.127 42.515 49.129 85.757 56.287 55.876 53.555 65.633 38.491 56.891 67.699 37.026 59.750 74.039 53.509 46.278 29.946 27.211 35.378 32.177 60.691 50.282 64.760
0.530 0.883 0.875 0.833 0.207 0.522 0.777 0.419 0.206 0.771 0.218 0.618 0.197 0.699 0.697 0.129 0.613 0.288 0.516 0.698 0.750 0.601 0.566 0.564 0.483 0.490 0.760 0.474 0.112 0.876 0.113 0.103 0.502
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0
311
483.050 470.365 656.547 935.071 603.745 287.898 550.001 788.777 701.441 992.996 613.341 836.027 864.926 898.323 257.487 496.022 207.527 817.984 751.707 227.523 225.944 410.922 894.731 206.035 957.371 795.604 452.309 234.734 966.248 891.912 584.227 216.268 617.637
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14.125 33.555 13.402 23.755 19.948 13.319 27.732 15.866 34.346 13.406 14.545 13.798 35.769 28.273 24.262 23.533 28.940 19.448 16.959 33.316 12.239 12.066 32.791 12.268 26.700 31.140 25.145 29.416 26.109 12.442 21.921 19.157 28.195
0.325 0.126 0.206 0.374 0.243 0.207 0.134 0.361 0.401 0.528 0.127 0.362 0.129 0.528 0.316 0.323 0.463 0.278 0.457 0.310 0.347 0.529 0.558 0.509 0.431 0.148 0.207 0.498 0.261 0.245 0.129 0.502 0.169
GLBL_DIS
EXT_EMS
EXT_ABS
P2P_CND
HT_PIPE
F_T_CND
C_I_CND 559.390 656.510 625.691 567.453 253.346 409.746 674.541 366.797 366.675 264.126 516.189 208.843 399.434 460.171 537.263 676.699 465.914 337.925 561.488 471.582 398.322 170.078 402.193 404.228 405.250 684.108 186.842 678.347 209.538 172.800 317.510 496.721 205.652
Tmaxd
0.144 0.172 0.125 0.145 0.138 0.163 0.126 0.110 0.172 0.174 0.113 0.136 0.114 0.115 0.126 0.110 0.159 0.199 0.117 0.122 0.165 0.141 0.116 0.170 0.119 0.134 0.170 0.116 0.123 0.128 0.140 0.146 0.200
Tmin
552.0 195.0 517.2 127.2 532.2 319.8 218.4 279.6 462.6 454.8 521.4 576.6 457.2 83.4 292.8 366.6 445.8 285.0 342.6 63.6 226.8 284.4 183.6 166.2 139.2 519.6 270.0 553.8 504.6 291.6 337.2 596.4 226.2
Tmax
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[m] [W/m2-K] [W/m-K] [---] [W/K] [---]
LCL_PLC
[---] [W]
C_DIM
TOT_PWR
ORBIT
Table M.2: Summary of Test Cases used to Evaluate Reduced-Order Models (34-66 of 100).
[---] [---] [---]
[K]
[K]
[K]
317.375 249.602 277.632 299.315 293.178 350.136 257.150 286.710 292.312 384.685 279.185 336.651 258.665 361.691 306.241 341.497 394.655 361.457 331.511 416.587 330.473 399.167 340.645 399.484 436.028 345.244 341.395 370.078 458.950 358.911 281.255 419.276 282.866
257.920 221.496 231.939 249.580 244.084 311.020 234.326 233.233 240.896 311.213 254.392 277.808 236.911 296.108 251.917 290.202 317.906 295.954 240.548 298.430 223.586 273.116 247.892 258.623 348.573 300.128 275.085 241.224 387.996 288.804 231.529 281.548 233.532
53.255 23.976 41.339 41.967 41.411 32.821 18.944 40.122 37.122 55.575 19.882 46.192 18.844 52.462 43.063 40.139 64.755 64.453 88.058 116.770 104.703 124.863 89.974 139.706 85.351 43.600 64.403 124.504 68.314 67.980 48.274 134.540 48.241
0.570 0.566 0.825 0.452 0.698 0.174 0.471 0.722 0.798 0.275 0.549 0.437 0.734 0.248 0.482 0.291 0.232 0.283 0.853 0.161 0.719 0.438 0.771 0.421 0.153 0.293 0.315 0.781 0.114 0.295 0.697 0.428 0.631
0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
312
259.729 329.849 748.588 343.929 396.786 491.109 776.367 608.950 285.707 274.509 988.380 675.991 464.075 916.865 948.776 251.607 659.014 212.472 254.741 696.030 173.302 343.390 997.420 470.166 330.031 373.523 302.595 756.034 286.063 239.694 470.926 851.263 974.209 343.518
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
14.132 28.703 25.623 33.404 24.156 26.688 22.980 13.257 22.043 12.264 33.714 20.878 27.437 16.752 26.946 23.727 32.014 12.318 29.238 26.135 27.338 24.826 34.432 33.847 26.120 23.190 31.299 17.862 31.205 30.791 13.755 34.098 12.641 12.155
0.559 0.470 0.318 0.274 0.205 0.344 0.467 0.558 0.187 0.561 0.197 0.517 0.294 0.460 0.277 0.433 0.430 0.142 0.514 0.174 0.340 0.395 0.503 0.480 0.194 0.475 0.144 0.230 0.202 0.285 0.312 0.211 0.284 0.458
GLBL_DIS
EXT_EMS
EXT_ABS
P2P_CND
HT_PIPE
F_T_CND
C_I_CND 299.260 552.746 254.650 657.152 129.179 631.321 208.513 256.953 576.559 636.107 466.596 463.582 573.516 231.533 638.413 340.120 580.854 399.129 164.758 145.264 560.729 350.849 181.865 166.718 433.448 480.768 382.314 129.069 252.066 184.593 370.587 680.568 374.218 398.815
Tmaxd
0.164 0.111 0.132 0.168 0.190 0.173 0.179 0.164 0.156 0.180 0.117 0.106 0.164 0.134 0.179 0.111 0.199 0.154 0.177 0.178 0.148 0.162 0.134 0.133 0.164 0.142 0.124 0.162 0.179 0.178 0.123 0.100 0.190 0.193
Tmin
502.2 420.6 66.0 165.6 512.4 444.6 539.4 430.2 436.2 91.8 500.4 582.0 131.4 435.0 575.4 517.8 390.6 189.6 597.6 323.4 420.6 500.4 65.4 349.2 600.0 169.2 418.2 460.2 531.0 255.0 112.2 236.4 123.6 147.0
Tmax
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[m] [W/m2-K] [W/m-K] [---] [W/K] [---]
LCL_PLC
[---] [W]
C_DIM
TOT_PWR
ORBIT
Table M.3: Summary of Test Cases used to Evaluate Reduced-Order Models (67-100 of 100).
[---] [---] [---]
[K]
[K]
[K]
357.512 382.309 306.298 292.322 282.865 332.105 390.505 497.730 410.243 372.212 318.506 395.314 347.191 342.111 352.875 400.199 340.504 328.488 366.986 350.048 316.719 318.428 394.158 342.577 355.265 325.259 271.687 287.842 357.058 315.434 319.174 372.273 322.451 351.888
266.066 293.883 238.323 225.863 253.888 275.357 322.650 400.982 338.401 245.623 273.515 292.983 271.956 257.159 298.324 280.644 264.221 263.616 254.040 301.404 251.977 259.471 311.557 270.995 328.837 237.781 251.089 257.824 324.769 251.639 246.808 337.212 263.258 253.207
89.690 86.739 66.036 64.972 28.199 54.646 64.312 94.536 69.723 125.171 42.769 98.041 74.071 80.782 52.347 116.217 74.623 64.475 109.785 46.456 62.543 56.580 79.989 69.382 25.625 84.929 19.535 28.484 31.419 61.803 70.900 34.211 57.751 97.106
0.784 0.435 0.623 0.830 0.848 0.572 0.352 0.139 0.145 0.591 0.497 0.462 0.328 0.760 0.385 0.416 0.639 0.263 0.758 0.229 0.818 0.866 0.265 0.677 0.260 0.887 0.748 0.788 0.253 0.616 0.561 0.149 0.421 0.581
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
313 Appendix N – Summary of Reduced-Order Model Coefficients
The following section provides the model form and coefficients necessary to replicate the reduced-order models developed and evaluated in CHAPTER 8. The form of the model (i.e. Equation (8.8) and (8.9)) was reduced to the following. 128
k
i =1
s =1
yˆ ( x* ) = μˆ + ∑ Ci ⋅ exp[ − ∑θ s ( xis − x *s ) 2 ]
(N.1)
The form of this model has one coefficient, Ci , for each training data point (i.e. Appendix L) and therefore 128 coefficients are necessary. The μˆ and θ s parameters were estimated using maximum likelihood methods as described in CHAPTER 8. Resulting coefficients (i.e. θ s , μˆ , and Ci ) for the eight categorical models (refer to Table 8.6) and three responses (i.e. Tmax, Tmin, and Tmaxd) are summarized in Table N.1 through Table N.24. As an example, the Tmax response for model LH000 was based on the training data of Appendix L and coefficients of Table N.1. The resulting Tmax reduced-order LH000 model is summarized as follows yˆ(x * ) = 706.618 + 6,557.818 ⋅ e −73,335.005 ⋅ e
[ − [6.123E-08⋅(514.961− x*s )2 +...+1.909E+00⋅(0.743− x*s )2 ]]
[ − [6.123E-08⋅(510.709 − x*s )2 +...+1.909E+00⋅(0.402 − x*s )2 ]]
+ ... +
.
(N.2)
314
Table N.1: Summary of Reduced-Order (LH000 Model) Tmax Coefficients. θ1 =
6.123E-08
θ3 =
2.034E-08
θ5 =
2.931E-07
θ7 =
1.909E+00
θ2 =
3.214E-01
θ4 =
1.618E-08
θ6 =
1.594E-01
μ=
706.618
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
6,557.818 -30,463.759 -97,368.501 90,540.963 2,592.459 -76,496.566 74,258.217 91,145.523 8,479.614 5,530.447 23,222.859 -59,935.648 5,576.901 19,589.998 -115,050.584 -78,889.976 34,990.597 22,294.825 114,614.215 -19,529.920 6,538.501 -47,105.298 98,850.006 -24,704.689 -97,050.297 64,886.177 -42,316.248 39,505.880 30,845.566 -44,249.952 14,429.396 -70,662.449
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-87,533.896 80,125.910 18,982.431 -48,472.187 35,255.842 -102,596.570 -153,037.548 73,120.277 142,437.178 69,237.760 -31,869.880 94,782.801 26,765.251 71,318.893 128,088.163 24,203.258 74,850.337 82,589.135 36,045.363 84,386.954 -212,876.768 -36,305.960 2,679.056 -40,194.203 -47,342.153 -74,721.251 -34,041.146 43,499.195 36,549.803 20,239.769 -14,526.641 74,551.599
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-13,853.914 74,966.504 17,397.784 -33,756.058 -482.398 23,614.048 -117,259.544 23,928.967 -56.764 17,144.183 92,275.990 -107,573.782 25,372.023 57,369.429 -68,634.402 -88,556.382 19,585.301 -136,050.197 -10,586.107 -79,841.726 -87,984.313 -76,111.477 -73,883.323 -102,196.779 -7,081.269 79,881.214 60,726.289 -57,124.142 16,231.768 8,649.518 -55,885.571 61,835.164
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-98,326.172 49,701.288 -63,794.586 -88,948.146 66,742.581 3,469.885 2,333.089 63,933.971 -29,065.606 9,843.568 -17,685.205 -48,601.466 -131,838.245 115,300.351 -57,159.638 -7,329.831 -27,196.349 148,962.772 101,198.893 -102,509.590 38,358.802 53,100.574 207.845 89,231.928 -46,957.863 -49,316.320 33,096.811 307,320.745 -102,741.541 112,450.064 675.515 -73,335.005
315
Table N.2: Summary of Reduced-Order (LH000 Model) Tmin Coefficients. θ1 =
2.263E-07
θ3 =
3.540E-13
θ5 =
1.236E-08
θ7 =
4.876E+00
θ2 =
5.483E-03
θ4 =
1.246E-07
θ6 =
2.246E-01
μ=
215.763
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
104,465.931 19,482.956 -38,086.509 -129,915.653 -43,442.179 -4,825.119 -56,636.833 -62,703.837 141,205.008 -15,137.623 -32,953.762 99,441.552 228,745.952 14,322.986 80,217.085 10,996.738 224,137.226 114,474.435 -9,269.945 23,253.052 97,037.143 34,518.221 -86,291.794 68,986.552 69,547.723 27,963.592 -47,753.092 28,085.386 -11,079.156 -66,464.689 -14,239.583 -62,326.055
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
3,406.243 -245,449.022 60,206.144 -16,504.406 69,774.598 -61,680.742 -84,311.746 4,146.969 206,265.685 -106,195.181 74,628.824 -9,642.125 -50,836.596 26,023.506 53,369.655 80,721.124 3,294.364 27,252.798 33,411.799 -111,696.216 151,914.581 46,672.008 -244,664.432 23,136.827 -74,068.637 29,643.184 -11,310.550 -24,852.817 -22,668.346 57,759.662 -51,555.344 -19,010.084
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-54,049.014 -213,669.898 36,053.611 70,346.372 -28,878.113 62,292.926 47,607.103 -54,225.983 -27,850.519 -15,961.332 196,428.666 -53,925.730 -60,360.663 -85,324.398 -47,518.470 -215,153.658 143,366.580 28,583.892 -112,623.222 -38,853.884 -83,844.850 87,756.360 44,283.058 -170,517.354 -93.349 21,760.141 -17,411.988 -31,995.382 -22,992.066 8,299.096 133,587.440 -138,649.620
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-9,040.888 -21,383.915 150,009.822 29,181.310 12,581.249 -78,086.610 67,708.106 47,612.930 64,481.442 -117,270.232 45,506.462 18,281.192 -47,014.639 -111,428.381 -80,229.524 -8,374.646 -113,244.990 28,758.086 98,551.122 -110,389.315 52,454.299 23,382.433 82,221.501 -22,597.696 119,668.994 -83,697.217 57,236.161 -79,042.865 -85,705.656 -95,748.953 102,472.858 133,744.371
316
Table N.3: Summary of Reduced-Order (LH000 Model) Tmaxd Coefficients. θ1 =
3.787E-07
θ3 =
1.475E-07
θ5 =
1.386E-05
θ7 =
6.037E-01
θ2 =
3.409E+00
θ4 =
6.624E-07
θ6 =
4.015E+00
μ=
98.681
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
445.871 -1,580.315 -1,136.114 1,346.108 -209.527 -909.011 270.529 1,052.937 629.110 -82.539 600.995 1,017.624 654.415 -1,116.786 -633.732 -253.414 -503.026 -385.386 1,236.651 -823.939 632.608 -757.936 741.005 639.110 -150.976 637.783 -87.397 24.014 119.430 1,109.472 -482.168 -224.823
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
565.500 1,842.113 -515.139 525.874 988.724 -84.495 -1,201.019 960.254 627.593 1,665.847 -735.897 659.513 -273.208 -407.930 -300.559 680.493 987.778 -752.234 -453.394 -540.296 -685.011 788.598 562.863 298.016 51.613 -83.371 -915.586 -399.013 -463.406 -174.512 -720.363 -828.379
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-351.501 1,836.091 1,050.582 -628.225 1,134.014 -1,884.134 -200.128 -882.792 -1,517.111 452.090 250.647 -74.245 139.836 -791.564 60.724 -17.190 -533.344 -1,273.704 -453.146 -1,827.380 -248.166 -1,138.195 -535.022 -1,783.827 -254.231 158.791 142.715 2,009.699 -1,075.588 347.882 -919.202 142.045
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-1,589.091 623.452 -1,539.682 706.379 530.330 1,086.643 -74.240 225.103 -1,837.759 -179.305 102.464 113.405 316.722 1,218.155 212.466 1,105.677 604.317 168.859 1,105.813 -496.690 934.413 -425.224 -982.631 170.435 -402.131 -1,321.968 612.717 3,229.082 -492.176 2,223.410 -322.199 -454.704
317
Table N.4: Summary of Reduced-Order (LH001 Model) Tmax Coefficients. θ1 =
1.177E-07
θ3 =
4.802E-09
θ5 =
9.175E-08
θ7 =
2.359E+00
θ2 =
4.216E-02
θ4 =
1.288E-08
θ6 =
2.602E-01
μ=
475.352
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
34,458.427 -4,341.108 -51,189.283 -1.078 -44,440.286 -31,154.588 -30,249.835 114,565.456 32,784.781 20,959.707 -1,792.830 9,873.407 22,443.164 -50,416.382 -35,925.468 -33,017.453 19,718.643 6,003.133 72,996.948 14,749.766 -32,673.757 -62,199.514 -19,467.305 -54,641.194 -46,117.689 -43,956.075 546.021 7,437.920 -2,564.516 -17,375.756 -16,958.102 -10,508.820
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-8,450.603 -24,137.929 26,030.058 -16,040.378 -11,152.048 -21,338.545 -85,804.686 44,568.000 57,130.403 38,602.094 -38,032.019 76,889.932 5,861.783 3,473.005 3,705.521 33,013.667 39,858.739 11,147.410 39,078.603 55,065.609 -78,599.114 13,123.203 14,899.417 3,916.610 -73,220.555 -20,976.611 27,635.402 69,058.446 -113,007.564 13,424.013 -5,756.226 31,456.784
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
22,149.278 1,271.147 2,687.437 3,486.505 42,596.608 -5,829.248 18,203.892 -12,441.360 13,291.307 20,336.227 25,677.373 -47,860.300 -55,985.435 46,853.788 -5,421.967 16,738.543 4,081.421 -6,745.651 29,804.628 35,100.324 -42,817.782 -34,674.116 12,675.614 -52,439.174 -14,102.699 -44,671.303 1,077.005 -18,872.284 -18,261.375 610.607 -27,678.123 42,858.682
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-38,851.222 26,987.983 71,186.317 -62,682.908 23,496.749 -23,780.333 -27,055.025 49,212.871 6,464.500 -36,723.534 -4,172.315 -11,329.319 2,648.285 -25,496.330 -7,646.388 15,484.186 31,359.678 42,739.555 13,512.495 -30,890.343 25,752.127 39,506.728 9,774.411 -20,721.423 3,230.639 -83,807.882 20,615.254 94,266.537 10,020.423 39,284.623 -31,303.818 48,249.144
318
Table N.5: Summary of Reduced-Order (LH001 Model) Tmin Coefficients. θ1 =
1.265E-07
θ3 =
1.334E-13
θ5 =
2.320E-08
θ7 =
5.990E+00
θ2 =
1.618E-05
θ4 =
9.016E-10
θ6 =
1.272E-01
μ=
229.288
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-89,147.600 -28,384.152 -8,658.977 -8,157.025 4,589.951 -26,350.743 -17,489.134 -18,210.028 -23,831.129 31,392.912 -58,395.773 -6,255.676 51,716.115 60,378.867 -82,845.706 30,605.997 7,511.990 6,681.282 8,891.603 -28,040.477 103,629.835 70,110.597 -23,917.007 30,722.554 -535.780 16,535.726 -14,757.682 14,354.095 -11,252.179 -56,484.577 19,576.190 -11,791.311
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
22,460.309 -69,700.603 25,590.183 -31,134.517 44,579.306 -24,333.690 41,422.908 -57,302.101 84,642.622 -62,422.520 15,552.044 -92,490.558 -27,441.734 31,513.564 6,431.002 -17,960.434 -1,804.248 2,029.071 23,050.392 -8,938.066 -20,166.948 34,557.692 21,435.807 -8,193.699 -25,530.330 14,604.575 -85,139.954 -54,252.253 81,000.186 1,934.419 -34,224.343 59,130.662
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
20,393.029 29,877.443 59,574.573 -10,070.132 3,162.248 22,375.595 -54,186.975 -32,897.778 35,846.739 13,261.141 28,239.214 -7,165.897 -17,568.626 -8,424.291 -26,694.200 -61,941.918 34,410.343 -25,384.422 -51,858.800 -24,863.864 -21,572.939 25,166.008 81,941.260 -10,908.379 28,377.644 15,439.730 -37,358.791 37,906.706 76,794.144 -21,898.264 30,472.252 -37,557.633
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
11,717.575 -28,864.480 61,344.311 79,746.912 -49,765.993 11,262.360 -9,444.208 -7,671.309 -4,766.587 -9,925.022 21,097.309 -22,922.034 -45,922.085 -9,947.067 -55,303.955 -23,361.295 34,046.551 35,301.465 52,705.202 -83,435.379 -21,441.906 -21,523.131 38,335.884 23,390.355 68,042.426 19,737.569 -2,281.819 1,456.040 46,716.485 37,330.008 -28,717.031 -40,913.816
319
Table N.6: Summary of Reduced-Order (LH001 Model) Tmaxd Coefficients. θ1 =
1.958E-06
θ3 =
3.330E-08
θ5 =
8.068E-06
θ7 =
1.204E-01
θ2 =
2.577E+00
θ4 =
2.394E-07
θ6 =
1.935E+00
μ=
71.803
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
209.919 162.960 -265.900 113.907 -356.751 549.632 -560.308 1,184.891 332.932 11.322 123.683 -42.300 224.041 -636.116 -1,117.172 -748.250 -1,232.236 -306.292 1,600.684 -260.758 -129.341 -1,355.909 -269.365 -149.518 -322.003 -418.896 -0.852 809.606 135.284 103.074 163.878 174.008
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
1,663.319 -1,139.340 -225.506 1,489.062 -320.367 -621.715 -835.563 658.469 226.096 1,243.122 -866.410 901.698 21.850 -138.902 907.883 635.169 927.583 224.389 -235.787 357.900 -207.155 203.156 381.440 -333.018 385.741 85.344 373.374 -869.504 -1,011.061 -129.095 -1,039.596 552.637
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
371.724 1,326.817 -285.125 -751.318 564.961 -1,142.847 230.570 158.530 -104.102 496.809 264.874 -1,542.051 128.624 588.902 -722.772 422.172 -360.253 -256.032 693.278 1,025.295 294.885 11.381 -371.750 -1,724.794 -884.600 -1,508.470 31.064 101.930 -747.945 245.390 176.084 264.077
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-910.938 -42.712 764.623 -118.838 -771.104 629.642 -192.860 -74.545 173.039 -362.763 -80.580 -62.742 -112.951 586.251 282.689 186.337 836.265 -170.918 898.676 -704.817 156.425 252.545 206.864 -1,198.221 -220.306 -607.723 649.804 1,670.015 82.252 696.748 -1,335.488 906.955
320
Table N.7: Summary of Reduced-Order (LH010 Model) Tmax Coefficients. θ1 =
4.797E-08
θ3 =
2.511E-08
θ5 =
1.259E-06
θ7 =
1.487E+00
θ2 =
2.648E-01
θ4 =
1.379E-08
θ6 =
3.239E-01
μ=
718.788
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
5,817.874 -27,006.918 -55,565.873 64,627.362 19,921.989 -53,127.923 47,449.699 73,983.198 -38,426.409 18,137.499 -22,733.454 -30,058.899 21,867.379 32,763.078 -57,079.917 -53,670.734 11,227.568 18,280.745 120,050.064 -10,861.805 63,097.301 2,483.315 70,306.668 -18,096.828 -106,125.424 29,067.387 -59,746.235 -23,087.335 4,857.653 -15,386.760 -7,718.877 -51,642.890
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-40,831.804 3,958.527 39,815.809 -44,898.664 33,012.213 -32,637.503 -159,368.118 76,985.919 118,067.299 67,938.768 -41,791.214 35,969.960 -5,371.090 60,846.778 64,758.631 15,790.246 48,441.798 44,835.959 23,120.896 83,169.225 -138,309.000 14,872.651 -52,542.141 -21,822.346 -52,879.655 -49,903.819 19,853.048 24,564.220 -92,915.181 52,360.659 -34,821.191 22,342.559
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
16,460.092 74,836.299 31,865.827 -29,945.361 -7,411.575 52,688.724 1,265.973 15,498.998 19,209.664 -51,026.924 59,300.565 -54,089.084 10,788.692 55,998.620 -62,145.858 -59,484.036 4,646.935 -117,751.266 -10,488.889 -62,270.052 -95,932.690 -76,037.634 -37,446.753 -98,811.701 -11,292.958 21,591.342 49,257.521 -53,393.927 65,179.706 5,409.471 -39,162.485 23,805.452
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-59,541.522 50,044.617 -28,035.075 -13,696.920 116,812.905 -43,791.303 5,223.776 37,495.731 -41,003.383 30,575.436 -34,805.538 -65,256.810 -71,993.127 76,597.757 -27,937.251 12,989.679 -28,108.684 99,072.310 55,938.475 -79,003.090 46,658.269 42,817.710 13,614.444 -7,777.803 5,660.409 -53,138.566 40,325.816 241,056.357 -76,043.313 92,824.067 -32,154.367 -2,749.631
321
Table N.8: Summary of Reduced-Order (LH010 Model) Tmin Coefficients. θ1 =
1.875E-07
θ3 =
0.000E+00
θ5 =
3.275E-08
θ7 =
4.673E+00
θ2 =
6.058E-04
θ4 =
3.033E-08
θ6 =
2.242E-01
μ=
172.212
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-11,965.171 127,778.688 89,054.155 -136,857.841 3,590.625 -13,025.944 -1,441.362 -53,371.660 56,372.107 -33,774.354 -32,656.311 138,900.329 -108,033.363 161,700.838 17,386.115 75,293.926 318,620.349 128,828.943 -51,859.033 -123,646.901 67,866.005 15,596.011 9,893.138 14,588.796 71,207.810 131,761.998 111,151.136 134,702.540 -35,771.669 -100,037.376 -302,394.035 92,403.680
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
232,867.089 -323,335.447 -4,016.423 -8,133.563 63,358.533 -32,761.275 25,630.169 -2,865.454 36,940.325 -226,106.241 68,395.187 29,592.066 37,357.959 30,225.826 96,615.140 -49,810.992 40,465.817 111,773.940 2,278.268 -61,486.153 -90,734.160 82,719.433 -75,901.127 97,885.929 44,546.839 185,290.725 7,858.633 211.494 -119,215.011 -14,181.452 34,641.702 72,239.920
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-36,603.760 -262,799.189 45,206.287 23,587.947 -120,529.865 98,004.956 41,613.179 -85,238.227 77,011.297 -279,291.304 152,051.720 -10,741.754 -81,640.414 -178,870.869 -64,834.546 -385,305.701 21,873.993 -19,180.921 -216,614.031 -184,958.352 -77,966.580 28,008.008 -25,590.767 -68,902.781 127,018.628 89,776.449 -33,477.343 -15,005.902 -59,392.584 -31,866.473 219,004.320 -224,654.660
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
67,901.540 -18,224.603 91,451.918 13,862.835 82,747.229 -154,732.902 20,708.145 -95,386.151 313,441.710 29,112.239 35,299.505 -56,855.722 62,653.305 -2,649.282 -15,358.286 -53,844.279 -64,774.917 -21,631.067 133,210.237 -78,617.538 -64,721.974 -1,894.598 123,128.232 78,007.308 42,193.527 -153,858.986 -23,956.368 -65,008.672 252,048.032 -29,141.113 66,360.015 -25,369.950
322
Table N.9: Summary of Reduced-Order (LH010 Model) Tmaxd Coefficients. θ1 =
2.763E-07
θ3 =
1.866E-07
θ5 =
5.652E-05
θ7 =
4.009E-01
θ2 =
4.038E+00
θ4 =
7.564E-07
θ6 =
3.081E+00
μ=
85.377
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
640.983 -942.010 -634.335 1,496.410 -95.461 -936.613 59.375 679.062 -65.514 528.781 561.735 429.827 442.443 -317.543 -292.521 -698.944 -188.775 -227.943 1,018.168 -1,070.471 1,172.331 -243.000 741.340 356.916 46.423 973.069 -210.723 213.274 322.418 770.833 -344.226 110.135
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-86.417 1,605.601 -367.662 -129.826 1,028.976 137.896 -949.276 908.793 783.809 1,046.284 -294.221 316.435 -309.163 -534.211 -402.304 562.413 1,018.844 -1,153.622 -66.042 -414.202 -718.552 698.093 266.018 63.030 158.404 -0.250 -611.930 -751.022 -464.099 47.329 -485.241 -93.817
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-528.994 852.192 330.639 -807.854 832.458 -1,377.904 -60.242 -688.918 -1,227.697 131.239 345.822 331.604 185.589 -773.432 -98.101 -663.304 -673.110 -531.124 -315.540 -1,341.184 -221.713 -893.084 -392.653 -979.280 -288.930 -233.736 265.140 1,969.075 -343.750 208.835 -1,010.995 -389.697
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-1,546.751 691.005 -1,172.730 535.353 813.436 814.545 264.324 228.403 -1,587.166 -74.265 313.675 -521.526 -426.538 529.262 262.632 1,161.426 454.803 -64.360 450.023 -45.212 400.372 -852.600 -484.002 -126.154 -130.872 -1,355.368 482.140 3,273.052 -414.509 1,790.783 -542.176 162.131
323
Table N.10: Summary of Reduced-Order (LH011 Model) Tmax Coefficients. θ1 =
5.299E-08
θ3 =
1.075E-09
θ5 =
5.760E-07
θ7 =
1.914E+00
θ2 =
2.974E-03
θ4 =
9.129E-09
θ6 =
1.415E-01
μ=
453.470
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
60,381.680 42,706.547 -183,093.833 22,772.974 -33,921.107 -63,157.881 -23,422.709 54,160.173 167,940.448 -21,048.365 -61,419.965 61,273.608 72,683.317 -45,631.487 11,300.174 17,672.522 -62,720.855 17,537.914 151,150.040 -13,542.816 -28,147.632 -15,982.410 -7,218.521 -115,709.538 -184,620.483 -40,262.778 34,519.138 -25,315.208 37,709.669 -22,091.485 2,620.080 40,923.129
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-76,891.107 6,156.387 4,311.025 -53,870.038 -4,392.103 -161,513.620 -131,228.046 102,627.555 178,617.063 142,563.784 25,906.446 -73,467.135 -5,126.379 35,432.803 119,426.471 101,374.216 20,616.138 -14,856.085 10,359.223 245,795.168 -173,919.796 8,447.875 -66,326.866 -7,705.819 -95,990.263 -52,931.636 47,954.382 -9,507.947 -13,062.867 31,597.815 -58,918.141 108,908.117
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-8,308.972 -5,182.143 -15,483.796 -153,041.747 177,669.251 20,912.788 163,245.616 9,754.563 122,141.093 54,809.740 43,693.707 -31,944.354 -53,722.455 48,161.463 -58,221.354 -441.157 -37,480.258 -210,471.101 86,940.459 -86,787.064 -120,240.230 -74,345.860 -51,188.365 -121,565.459 52,711.217 -91,561.161 87,459.110 -40,099.449 11,851.393 -43,209.110 -2,346.676 42,910.405
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-125,295.622 37,462.571 -36,932.687 -119,427.128 75,707.395 -55,431.773 78,718.715 122,367.260 9,851.300 6,301.887 -139,945.866 -48,563.809 -11,228.241 43,420.700 12,389.530 34,442.831 -56,985.289 40,202.739 147,649.992 -174,948.843 -67,879.615 146,930.214 -11,813.663 -64,212.046 -104,164.818 -37,100.186 91,150.309 356,702.509 -3,843.092 271,422.945 -71,706.779 -100,292.489
324
Table N.11: Summary of Reduced-Order (LH011 Model) Tmin Coefficients. θ1 =
6.332E-08
θ3 =
0.000E+00
θ5 =
3.723E-08
θ7 =
4.125E+00
θ2 =
0.000E+00
θ4 =
6.534E-11
θ6 =
7.140E-02
μ=
229.099
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-10,614.515 187,282.795 10,458.387 23,760.361 -20,687.617 -12,552.274 12,151.609 -410,338.366 -166,151.180 -12,177.196 -246,458.664 87,617.278 75,708.516 312,019.925 213,572.221 104,390.851 27,848.531 6,986.940 -204,340.390 -44,166.690 188,965.775 66,912.361 -297,627.708 -37,918.366 99,492.215 171,144.659 -99,390.394 20,149.849 -44,569.339 -89,616.994 -62,450.081 7,944.656
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
232,137.752 -250,807.712 80,612.305 66,041.572 124,847.636 8,730.468 235,685.874 -150,597.495 108,610.220 -277,735.731 76,663.436 79,537.203 -95,330.357 109,069.898 -129,810.140 -177,008.566 99,442.616 106,977.359 162,362.488 31,964.036 -37,705.461 62,920.300 16,700.856 -155,387.609 65,151.259 -30,349.791 -202,741.230 -95,144.498 -212,700.330 9,077.546 -166,252.578 221,539.840
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-176,132.995 -134,206.476 150,591.777 -21,407.658 -234,469.014 164,143.013 -150,278.457 43,308.688 49,993.581 15,596.188 57,298.791 137,587.811 -91,323.218 -201,440.454 32,833.642 -341,656.004 23,007.209 -232,973.599 -200,489.376 -39,879.308 -41,656.180 45,513.212 69,249.920 50,925.729 158,654.482 25,231.969 -272,341.452 305,301.635 154,539.073 -47,031.702 281,840.877 -240,662.400
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
7,414.636 23,619.473 242,116.547 126,818.607 10,562.728 24,452.215 -180,138.812 -56,387.646 183,426.288 73,965.471 112,838.066 21,784.798 -89,427.755 21,413.066 -79,748.754 54,952.694 -9,139.057 -101,215.028 172,237.777 -425,056.075 94,884.853 -133,079.221 136,000.762 63,823.436 417,463.217 7,807.667 -30,542.632 164,327.289 112,221.291 177,197.401 -117,507.713 -140,603.216
325
Table N.12: Summary of Reduced-Order (LH011 Model) Tmaxd Coefficients. θ1 =
3.144E-07
θ3 =
1.198E-07
θ5 =
3.949E-05
θ7 =
1.006E-02
θ2 =
1.863E+00
θ4 =
2.262E-07
θ6 =
1.143E+00
μ=
88.133
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
690.241 -79.844 126.248 671.840 415.621 -689.874 -719.591 555.123 -88.858 -973.243 30.366 -301.477 758.474 -918.826 -1,246.385 -639.840 -1,568.545 1,076.529 2,483.592 -273.114 156.026 161.224 -147.532 -149.767 -2,017.491 -119.732 -669.116 2,005.465 793.637 914.319 -327.887 -1,070.596
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-226.819 384.870 -37.068 1,399.829 -899.309 803.211 -120.575 1,400.629 478.126 294.882 36.355 -91.336 141.214 -1,076.146 523.332 707.150 1,033.153 -125.942 -71.150 87.566 -1,252.522 -316.815 756.028 -152.934 -827.618 -657.914 -85.614 199.119 -995.273 -423.520 -985.654 1,192.119
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
47.742 353.984 -795.777 -1,647.039 4,217.616 -1,227.344 -295.247 -744.245 244.541 1,989.876 -251.013 -2,329.813 419.626 171.030 -882.489 103.942 174.085 -1,872.755 817.941 -485.231 563.901 -783.801 -617.175 -62.110 -1,133.952 -1,123.571 61.012 679.405 -633.220 51.828 -938.075 335.099
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-625.602 1,333.846 -437.631 -211.801 65.807 29.609 324.331 -224.773 -1,455.952 -241.759 55.299 -56.425 1,399.049 782.275 247.054 280.264 -188.656 835.841 745.933 -722.465 891.467 8.542 461.532 -845.843 -344.254 131.452 463.446 2,598.154 -975.761 859.071 -496.905 977.723
326
Table N.13: Summary of Reduced-Order (LH100 Model) Tmax Coefficients. θ1 =
3.136E-08
θ3 =
1.368E-08
θ5 =
8.328E-07
θ7 =
1.316E+00
θ2 =
2.316E-01
θ4 =
1.575E-08
θ6 =
1.437E-01
μ=
978.645
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33,000.947 59,572.404 -309,706.229 178,466.050 -22,920.829 -82,448.553 144,645.674 254,829.895 -107,806.732 6,878.463 -25,392.180 -82,636.938 43,761.704 -43,618.924 -394,502.158 -190,968.822 -63,264.787 38,495.920 292,315.982 49,387.900 -139,102.856 1,063.997 37,399.628 -154,078.864 -236,189.123 -7,799.766 -115,719.747 -77,968.683 28,874.256 -77,705.127 143,752.467 -17,605.687
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-61,480.489 713.390 175,536.998 -47,065.979 -10,136.803 -178,892.897 -302,246.641 170,471.033 401,416.978 129,793.427 -231,880.372 121,677.290 -38,794.646 242,544.049 257,689.161 -24,765.014 130,537.317 142,860.745 30,985.958 248,412.574 -418,363.658 74,742.521 -75,220.080 -210,437.363 -70,674.581 -45,812.453 7,818.400 124,083.331 -69,558.238 83,009.212 121,483.293 231,698.686
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
90,408.224 126,267.674 161,023.791 -123,841.239 -88,319.741 93,513.848 -51,151.093 -39,833.054 97,175.949 -87,701.357 107,154.961 -134,283.887 12,851.780 190,089.297 -34,337.661 -224,486.141 18,172.381 -157,429.696 133,015.231 -159,952.351 -293,030.588 -246,561.644 -64,360.000 -285,480.235 55,019.982 140,423.392 16,829.625 -179,425.170 134,551.658 55,410.938 -30,234.066 75,413.662
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-178,409.379 152,926.349 -198,965.064 34,019.654 272,569.791 -23,333.605 82,124.305 8,360.670 -248,230.506 102,075.886 -87,140.134 -63,701.723 -338,316.936 252,318.601 -75,855.439 -40,182.776 26,980.606 27,410.515 115,474.016 -234,920.873 -125,337.198 90,653.529 64,674.031 112,136.509 111,687.049 50,692.624 76,600.969 717,304.299 -176,442.632 248,937.873 -113,862.613 -22,297.300
327
Table N.14: Summary of Reduced-Order (LH100 Model) Tmin Coefficients. θ1 =
2.352E-08
θ3 =
1.418E-12
θ5 =
1.445E-07
θ7 =
3.895E+00
θ2 =
2.589E-03
θ4 =
1.103E-07
θ6 =
6.703E-02
μ=
-21.895
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
107,544.040 71,855.461 182,006.824 -103,889.960 -200,241.666 180,666.823 -31,876.190 -77,778.944 207,496.929 -78,512.845 29,944.940 305,458.818 107,292.377 185,736.762 34,475.497 -15,472.927 511,251.952 242,660.166 -222,470.204 111,605.006 148,415.008 553.122 -103,468.068 36,190.793 326,171.486 298,529.971 -79,997.078 185,170.346 -115,639.980 -83,629.788 -189,119.883 132,365.419
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
40,705.344 -502,490.861 -40,608.399 74,554.153 -67,849.640 -175,221.048 -3,563.660 -224,591.983 142,759.439 -125,459.541 38,455.934 27,734.031 -45,525.527 -18,923.517 -65,332.484 -29,271.149 107,447.064 109,743.418 21,671.960 186,298.054 -208,978.794 129,044.946 -163,252.078 -72,422.786 -252,954.517 -101,013.104 -46,173.240 120,603.914 158,827.026 55,096.659 -36,730.621 42,060.197
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
53,968.990 -426,261.987 -113,417.005 -128,580.345 -139,826.116 168,818.534 84,389.772 -232,981.483 240,230.721 -454,369.530 501,674.881 -88,189.247 -335,623.661 -395,368.268 -203,664.318 -394,393.310 501,109.403 -207,715.658 -426,468.884 -171,353.154 -233,536.495 191,981.256 17,348.007 -578,831.934 168,418.294 123,817.433 -52,128.307 -136.296 -170,560.720 -158,107.072 527,699.163 -255,527.206
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
304,660.067 46,860.654 128,708.466 155,182.363 136,567.635 -288,930.968 64,747.017 -119,481.329 110,274.661 -91,221.049 -147,315.451 -157,435.654 -291,998.055 -265,370.402 -26,387.458 -40,896.115 -138,617.308 172,544.637 361,983.365 -333,634.945 128,967.383 115,448.239 204,768.364 15,493.768 121,216.477 -57,171.615 124.042 180,582.816 331,528.028 359,961.830 163,083.962 265,407.720
328
Table N.15: Summary of Reduced-Order (LH100 Model) Tmaxd Coefficients. θ1 =
2.169E-07
θ3 =
4.917E-08
θ5 =
1.160E-05
θ7 =
1.967E-01
θ2 =
9.662E-01
θ4 =
4.278E-07
θ6 =
4.686E-01
μ=
209.618
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
281.372 9,224.318 -18,122.341 15,218.879 670.995 -6,527.240 4,815.229 14,878.681 -457.289 4,852.147 631.152 5,419.116 -8,264.238 3,907.237 -14,630.777 -12,833.535 -9,749.415 6,764.884 19,499.484 -5,465.926 -11,265.388 -9,234.304 9,439.830 -808.579 -7,225.673 3,662.820 2,937.072 -747.138 -3,815.358 -5,613.279 7,959.088 -7,558.667
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
3,970.925 -956.996 -255.611 6,617.276 -4,439.763 3,698.101 4,451.290 8,476.399 4,860.824 6,058.258 222.052 7,516.437 -3,282.608 4,082.960 7,924.035 -3,061.037 7,703.307 -6,103.969 -2,846.611 -6,536.201 -11,396.892 5,240.847 1,623.493 -5,499.260 2,835.441 3,168.428 -7,010.065 5,085.005 4,576.474 2,896.771 1,769.352 15,353.018
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
4,192.633 -974.036 14,833.287 -5,950.479 12,211.268 -16,866.971 1,600.884 -17,442.836 -7,082.864 2,526.684 5,005.559 -10,479.873 2,375.469 -2,987.603 3,333.908 -1,492.119 -5,792.493 2,015.806 14,528.497 -15,468.204 -12,423.734 -18,041.807 1,723.486 -7,760.426 -1,239.213 1,954.287 -2,676.922 12,451.002 -9,333.691 -4,009.321 -7,641.998 -397.633
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-7,921.395 2,110.893 -7,030.361 5,988.317 -5,299.972 8,249.341 -2,795.705 -6,103.161 -1,385.387 2,051.061 5,108.388 -6,235.112 -103.362 16,829.174 12,614.807 1,540.028 -5,330.237 -8,711.384 3,848.418 -6,038.583 -842.133 -4,603.286 -1,044.035 1,882.011 5,977.227 -1,544.001 6,467.315 34,004.762 -9,770.141 4,360.860 834.480 -12,384.213
329
Table N.16: Summary of Reduced-Order (LH101 Model) Tmax Coefficients. θ1 =
4.953E-08
θ3 =
4.585E-11
θ5 =
1.590E-07
θ7 =
2.272E+00
θ2 =
6.002E-04
θ4 =
3.034E-08
θ6 =
4.466E-01
μ=
589.788
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-83,566.901 40,378.393 -42,699.464 31,459.775 -36,711.451 -24,123.193 38,052.930 104,547.968 36,217.076 -20,932.830 -28,278.833 -27,839.559 -8,680.016 -117,901.110 -35,959.157 -34,072.592 -41,693.062 -17,214.965 67,303.230 -11,602.440 -107,797.692 22,314.600 9,469.610 -4,775.703 3,793.632 -25,390.585 -23,542.196 -9,548.024 -4,118.576 -12,902.366 58,815.025 41,067.765
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-115,461.978 30,361.065 -11,952.125 11,618.082 15,471.558 -59,577.851 -52,033.367 35,844.190 82,516.599 15,949.754 -25,407.313 -61,998.846 26,330.850 43,248.179 -40,771.449 18,172.808 27,119.505 -1,334.330 -3,948.454 43,076.774 -38,337.293 -1,723.890 46,478.714 3,285.019 -5,586.894 -13,008.138 -2,591.165 -28,169.181 109,197.715 -1,195.736 -9,846.991 4,600.615
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
9,581.553 51,903.279 13,678.181 -40,456.749 6,519.212 41,629.213 27,425.920 -25,326.528 44,092.966 -11,147.813 75,627.794 125,778.950 20,260.645 46,638.208 -8,949.491 -17,783.296 -24,725.083 -20,298.534 52,316.495 180,646.287 -93,774.603 5,418.609 -7,201.803 -14,356.510 36,105.419 -27,000.635 25,281.747 9,603.224 13,173.977 17,312.438 -19,875.572 32,390.408
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-30,711.308 -39,516.645 -13,459.203 -4,978.433 20,820.564 -28,460.907 3,141.758 29,359.483 -28,261.650 -2,553.752 -41,267.306 50,755.349 -44,455.973 -8,711.566 -4,212.807 -3,404.274 5,182.758 -33,964.655 16,579.436 -59,964.354 -30,392.137 16,244.543 -12,467.442 19,152.407 21,713.388 -1,024.937 20,569.698 -13,260.566 -34,920.255 -6,324.125 -4,943.575 -55,147.134
330
Table N.17: Summary of Reduced-Order (LH101 Model) Tmin Coefficients. θ1 =
4.965E-08
θ3 =
2.682E-10
θ5 =
3.451E-08
θ7 =
2.700E+00
θ2 =
6.806E-04
θ4 =
2.421E-08
θ6 =
5.278E-02
μ=
68.316
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
8,221.059 97,946.140 -15,485.773 72,699.700 12,293.832 -10,931.806 -9,455.861 76,553.838 45,042.053 25,351.936 -6,444.545 79,799.929 -138,181.211 33,190.469 22,326.814 -59,427.861 -31,117.629 90,183.958 -135,576.341 26,189.277 9,715.475 59,492.700 -46,838.654 -897.759 25,948.557 32,841.955 53,255.827 -43,911.670 -21,295.916 14,120.797 -27,766.228 -502.806
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-16,182.451 149,301.366 89,534.928 20,194.585 115,416.461 -111,491.913 -64,068.247 -24,569.988 -14,499.107 -6,989.990 59,241.557 -38,468.702 -95,108.897 40,336.580 -40,294.353 -77,008.364 27,552.075 42,578.372 79,441.671 93,802.514 -63,912.213 -14,126.017 -73,173.946 29,787.238 -55,914.881 13,647.609 -111,774.191 35,715.757 -20,154.533 -14,300.860 -77,287.164 -101,104.188
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
10,103.154 58,423.111 -158.090 7,176.966 -57,603.518 84,427.597 41,743.875 16,190.799 -32,006.048 -76,137.666 88,863.687 9,832.587 -13,877.745 -129,043.183 -42,598.818 -78,083.990 60,269.023 -65,748.765 -6,219.426 8,565.007 -6,491.680 -33,744.295 64,850.202 11,525.308 107,299.142 76,533.006 -117.876 -61,397.576 -51,630.035 92,081.014 41,923.543 -137,002.700
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
1,008.147 55,908.615 116,610.460 -34,317.315 1,558.905 -101,898.241 -34,875.675 14,149.707 34,363.050 12,104.122 3,057.976 -9,328.489 69,255.544 -55,148.386 -62,866.516 16,289.975 25,752.957 -49,787.418 -29,102.843 -94,209.669 -61,092.659 36,393.245 51,261.179 -69,307.417 36,552.224 -149,225.447 29,167.521 181,741.574 33,555.601 107,417.470 64,815.558 -141,211.334
331
Table N.18: Summary of Reduced-Order (LH101 Model) Tmaxd Coefficients. θ1 =
3.647E-07
θ3 =
0.000E+00
θ5 =
6.159E-06
θ7 =
8.345E-02
θ2 =
3.903E-04
θ4 =
3.952E-07
θ6 =
9.641E-01
μ=
127.870
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-6,824.668 725.480 -824.957 797.932 1,983.448 398.630 645.837 5,116.047 -1,580.473 -842.413 -911.963 -3,238.096 5,109.163 -4,661.643 -2,427.910 1,007.120 -2,792.551 -2,286.120 10,434.145 -797.952 -3,225.741 -1,774.798 716.037 141.803 769.581 -3,225.583 -2,759.601 2,272.596 -825.400 -751.784 5,220.375 2,030.488
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-3,435.090 -9,150.703 -5,199.175 2,632.530 -6,541.494 -2,243.790 3,870.149 920.496 4,442.306 -127.603 -2,834.399 -4,199.562 6,440.764 1,162.109 -1,178.793 301.724 2,249.204 885.888 -2,442.011 -1,637.020 -1,110.510 -3,087.770 7,019.440 -612.998 1,181.159 1,104.381 626.757 -5,523.286 9,669.167 678.667 2,699.597 3,742.584
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-21.784 -1,349.127 2,922.307 -4,557.252 919.672 -1,951.116 -2,779.163 -1,829.090 2,759.844 889.015 1,577.252 2,175.303 -1,075.755 5,099.229 3,543.366 44.720 -2,559.368 5,637.270 2,101.951 12,626.030 -4,401.607 291.530 -3,360.169 -1,124.435 -1,016.320 -7,820.702 2,328.463 5,867.460 1,359.710 -4,073.839 -5,808.100 7,109.302
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-2,484.718 -2,142.578 -6,081.706 650.410 -2,282.019 1,599.257 4,030.494 2,494.101 -4,308.473 452.164 1,108.215 2,474.602 -4,238.394 1,437.232 5,435.398 -642.904 694.876 -1,281.528 1,237.508 -2,596.776 1,731.502 -1,642.155 -2,166.286 5,335.295 -1,144.939 7,111.743 -3,806.358 -6,634.421 -3,102.025 -5,503.472 -1,445.864 4,267.476
332
Table N.19: Summary of Reduced-Order (LH110 Model) Tmax Coefficients. θ1 =
2.978E-08
θ3 =
1.640E-08
θ5 =
1.154E-06
θ7 =
1.479E+00
θ2 =
1.630E-01
θ4 =
1.605E-08
θ6 =
1.543E-01
μ=
859.128
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33,266.681 16,219.499 -274,594.367 166,351.625 -20,138.567 -73,302.508 71,351.874 191,107.264 -21,894.450 -21,038.728 -56,087.693 -44,502.845 22,829.407 -1,376.902 -310,539.387 -79,139.878 -12,541.624 41,753.887 232,499.179 17,751.591 -117,266.223 70,507.952 -25,818.246 -130,920.275 -134,540.466 -83,523.909 -84,534.213 -47,237.836 53,328.499 -43,757.883 83,110.344 -4,988.809
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-16,511.039 47,016.902 115,341.105 40,054.222 25,914.232 -161,508.648 -263,977.751 193,048.820 296,126.442 52,348.740 -187,974.345 23,165.582 -14,226.411 146,605.742 132,690.674 -18,856.472 76,501.889 105,902.735 -21,257.957 193,535.684 -323,210.160 39,304.412 -62,931.022 -146,196.371 -42,598.177 -11,523.534 -119,254.110 62,006.660 -83,374.465 89,180.169 23,795.455 192,404.763
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
81,761.753 61,807.738 129,974.211 -104,430.031 -52,645.487 72,477.443 20,873.227 -68,920.715 166,574.930 -81,576.511 112,787.103 -137,865.362 23,903.153 103,702.539 -88,812.655 -161,469.954 58,867.267 -218,232.288 205,307.329 -81,674.450 -274,379.183 -209,771.856 -137,885.253 -167,343.621 69,295.923 92,221.439 39,016.692 -112,334.642 147,160.016 20,695.349 -33,184.339 78,647.087
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-161,134.352 142,627.916 -140,231.039 14,850.230 169,521.227 -48,944.342 114,942.463 81,510.850 -168,318.019 67,073.678 -50,274.350 -123,489.394 -177,313.133 174,277.299 13,920.430 -28,608.092 -20,600.751 4,287.128 116,068.047 -187,818.552 -120,505.601 92,500.028 32,137.177 36,975.993 132,795.380 60,353.179 62,657.378 633,959.121 -211,382.575 261,131.372 -90,901.175 -30,491.140
333
Table N.20: Summary of Reduced-Order (LH110 Model) Tmin Coefficients. θ1 =
2.525E-08
θ3 =
2.562E-12
θ5 =
2.490E-07
θ7 =
4.273E+00
θ2 =
3.493E-03
θ4 =
1.134E-07
θ6 =
1.088E-01
μ=
116.230
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-2,969.584 -18,477.031 40,903.359 30,555.567 -133,772.121 15,946.507 38,183.614 -3,262.221 104,291.900 -28,424.669 -10,645.396 124,198.328 -39,710.899 3,854.453 94,516.703 -35,442.491 235,929.024 112,250.372 -77,773.001 61,165.104 132,804.468 -4,019.944 -6,441.452 -60,865.712 95,513.754 221,210.482 12,828.508 59,186.015 -75,133.114 -16,782.271 -159,670.223 98,660.272
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
78,047.886 -180,042.335 -12,023.782 80,653.378 -715.320 -86,576.569 38,063.968 -84,930.205 39,046.717 -124,252.727 3,793.570 35,322.262 17,770.411 -38,486.515 131,205.646 -75,421.745 52,701.355 90,525.618 45,635.097 128,354.805 -58,147.263 -4,802.066 -21,062.350 -20,313.833 -91,111.121 4,212.230 -77,204.005 67,077.220 16,956.763 36,298.006 -57,037.839 51,497.498
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
24,751.348 -139,566.493 -24,029.164 -69,120.214 -48,266.821 100,981.830 25,811.364 -11,954.002 -11,313.861 -286,371.421 179,392.638 -5,926.301 -151,911.629 -142,628.653 -193,300.995 -267,506.613 142,162.390 -54,221.235 -192,673.940 -117,937.515 -40,363.217 74,470.751 37,587.290 -210,248.342 47,534.867 45,480.600 -31,751.727 -53,472.956 -32,436.100 -58,203.909 166,354.012 -95,687.583
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
135,925.676 10,993.247 137,121.727 49,594.331 -7,355.970 -114,170.012 9,801.879 -30,996.664 169,000.576 11,514.490 8,216.737 -45,173.352 -74,163.805 -78,939.808 -40,902.240 -33,007.943 -105,599.729 76,020.991 195,941.212 -152,301.412 -11,608.803 67,279.796 147,143.704 -74,735.048 62,481.955 -87,190.730 -27,455.007 -14,615.867 180,376.694 128,170.776 89,738.453 19,616.691
334
Table N.21: Summary of Reduced-Order (LH110 Model) Tmaxd Coefficients. θ1 =
1.681E-07
θ3 =
8.475E-08
θ5 =
3.643E-05
θ7 =
2.208E-01
θ2 =
1.071E+00
θ4 =
6.886E-07
θ6 =
1.020E+00
μ=
154.245
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-657.825 2,700.943 -3,969.250 4,275.654 921.180 -3,442.531 659.672 5,818.557 -3,572.156 362.360 546.730 2,343.725 -1,897.787 56.622 -3,838.654 -4,996.619 -2,256.083 641.888 4,507.100 -2,095.758 -2,961.220 -2,633.845 2,434.027 460.248 745.116 1,262.865 1,075.332 -211.007 -603.938 394.428 2,499.030 -3,139.505
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
887.016 2,649.813 -453.851 4,842.100 -1,841.744 -1,249.874 -446.325 5,212.391 4,575.263 2,082.108 -11.698 -876.426 -499.781 -1,229.221 -410.317 -1,237.336 2,807.223 -3,701.046 -2,656.828 -1,091.729 -6,284.637 1,379.286 1,581.172 -1,173.477 976.813 -1,091.960 -2,646.592 1,482.886 2,463.715 302.534 -324.646 4,842.938
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
3,948.757 -1,002.376 1,673.344 -1,429.831 3,464.184 -6,724.456 2,036.893 -6,118.179 1,591.352 3,501.389 2,327.847 -1,065.604 1,423.842 -3,478.454 1,315.923 -529.426 -1,521.946 -1,976.624 5,334.652 -3,831.089 -4,605.539 -6,193.595 -3,416.457 -1,324.358 302.603 2,375.364 1,104.368 6,108.799 -214.878 1,292.803 -3,728.777 989.934
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-4,008.747 851.744 -8,727.066 130.212 136.212 2,688.401 -125.403 -3,232.011 -5,160.203 -589.565 466.827 594.545 -763.699 1,817.398 3,611.056 2,367.994 -2,052.251 -5,469.427 3,659.795 -1,520.170 -536.400 -2,366.606 1,041.678 3,064.427 -160.556 503.279 1,028.822 17,166.992 -2,130.793 5,645.959 -1,200.553 -2,649.432
335
Table N.22: Summary of Reduced-Order (LH111 Model) Tmax Coefficients. θ1 =
3.697E-08
θ3 =
2.687E-11
θ5 =
7.193E-06
θ7 =
3.201E+00
θ2 =
3.862E-04
θ4 =
3.565E-08
θ6 =
6.278E-01
μ=
578.010
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-142,811.549 -44,963.964 -47,704.456 -41,670.695 -76,556.321 -77,952.772 58,848.092 -64,758.897 121,531.438 11,828.039 -85,912.168 2,803.057 68,581.371 -22,799.092 -35,602.955 -15,846.317 -119,077.508 -77,265.248 205,333.491 26,010.610 -55,990.276 -47,443.280 -57,480.969 54,077.442 -1,072.152 -47,316.408 -1,320.467 40,949.987 -54,963.081 -34,551.525 54,681.178 51,241.726
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-111,052.243 74,155.147 -27,782.393 -5,504.035 50,991.134 -115,511.497 -124,009.436 45,825.130 154,432.785 3,195.379 27,642.449 -29,361.693 67,897.754 186,737.905 -178,311.708 69,959.202 -32,420.204 15,825.723 -19,376.277 94,931.765 61,554.042 44,859.831 29,619.835 -66,522.990 -50,565.627 -20,554.624 12,024.506 -114,184.768 -67,570.498 14,680.135 -37,748.962 82,045.001
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
49,374.221 -27,268.301 37,266.355 -102,700.458 -84,648.113 54,806.840 15,109.234 -87,022.381 98,228.824 -44,237.560 146,330.420 193,198.438 -5,469.853 9,645.037 36,912.583 -64,108.510 98,281.471 -388.342 160,241.738 156,537.536 -199,921.540 -97,291.586 13,496.109 -62,185.787 140,286.814 79,573.136 12,679.400 70,366.800 73,298.565 -104,952.750 -89,919.140 -14,295.885
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-42,678.424 22,445.246 -63,092.608 61,194.742 -145,706.656 -77,226.408 8,566.295 -22,106.386 26,625.074 40,649.365 22,580.787 22,236.259 108,359.642 35,905.844 -12,147.097 19,961.937 -18,887.015 -36,854.983 27,023.695 -40,558.376 -12,182.450 -82,041.625 -67,199.970 16,440.377 92,187.526 101,351.059 -22,636.081 29,814.356 -71,584.216 19,492.848 -14,632.085 34,748.913
336
Table N.23: Summary of Reduced-Order (LH111 Model) Tmin Coefficients. θ1 =
3.096E-08
θ3 =
3.656E-11
θ5 =
4.353E-08
θ7 =
2.050E+00
θ2 =
1.494E-04
θ4 =
1.608E-08
θ6 =
6.571E-02
μ=
10.427
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
21,300.144 296,882.125 25,373.447 146,186.134 -22,621.945 -143,984.146 28,387.362 414,181.786 180,691.448 58,169.784 -30,712.118 112,711.524 -62,137.354 -62,559.120 154,446.445 -104,922.938 254,714.741 -15,249.096 -271,552.953 -202,299.134 190,159.551 34,073.913 87,014.135 -19,175.881 -31,464.179 86,183.635 70,816.936 -66,305.069 -86,170.080 -27,398.029 -112,859.790 8,467.672
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
32,799.992 87,787.084 -34,836.724 22,369.631 217,791.295 -334,928.338 183,772.456 -80,133.679 112,091.924 132,932.338 35,039.656 -201,821.435 -1,293.768 126,149.061 31,913.042 -178,130.341 24,382.454 98,695.463 45,470.852 244,732.767 -277,202.418 -170,348.453 41,262.668 70,566.759 -48,832.014 155,072.264 -50,690.443 88,306.950 -20,139.069 -133,542.144 -60,191.061 -14,695.213
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
130,256.469 -291,912.266 76,606.246 -57,103.412 -118,313.727 214,310.556 -85,692.714 316,164.866 17,376.236 -362,522.233 71,424.656 -26,934.313 -126,761.063 -22,790.178 -36,699.232 -234,705.621 61,264.282 -149,216.411 -422,551.472 -15,239.367 17,100.190 37,680.009 42,640.953 -251,638.520 -95,323.330 128,955.159 186,896.564 -100,589.427 -57,786.133 94,918.616 97,745.473 -158,120.281
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
-16,069.677 82,052.330 234,576.115 6,493.537 225,462.547 -84,909.623 8,362.022 -30,802.198 -100,769.527 32,871.574 -77,167.186 106,590.771 -45,824.059 -186,358.129 61,238.509 96,554.679 -83,949.382 55,497.353 116,574.316 -137,669.589 -242,427.386 16,128.803 -41,440.067 -168,566.401 -237,873.230 -233,923.883 -26,828.981 435,638.544 107,808.816 259,022.745 209,065.618 -273,500.042
337
Table N.24: Summary of Reduced-Order (LH111 Model) Tmaxd Coefficients. θ1 =
1.645E-07
θ3 =
1.553E-12
θ5 =
3.356E-05
θ7 =
6.923E-02
θ2 =
0.000E+00
θ4 =
4.597E-07
θ6 =
7.671E-01
μ=
154.683
i
Ci
i
Ci
i
Ci
i
Ci
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-2,172.687 -5,193.679 1,644.074 -2,643.593 6,660.361 3,805.783 5,716.220 -19,472.072 -7,030.473 1,777.194 11,100.219 -3,785.542 4,012.236 10,072.595 -8,522.071 -1,909.623 -18,775.658 -2,270.981 18,093.793 6,882.788 2,788.940 6,434.645 -591.846 -246.299 1,317.143 -3,202.480 -2,115.607 6,592.904 2,048.194 -566.259 -1,345.883 3,612.009
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
-11,568.155 -14,996.349 -181.735 5,518.900 -12,042.613 -2,684.724 -4,716.089 4,469.753 6,515.595 -4,818.948 3,792.351 10,539.699 11,033.455 -2,222.756 -10,476.545 -2,216.506 1,736.615 -7,494.389 -3,850.511 -1,231.471 3,206.154 -2,910.170 11,863.590 -508.886 -2,763.008 1,038.383 -1,628.645 -3,841.516 -2,263.265 -553.453 -7,144.898 -5,090.691
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
-426.581 -412.977 -3,226.772 -4,748.826 -6,291.289 -1,521.930 1,616.724 -13,844.846 12,431.067 12,731.117 8,086.437 5,969.622 2,117.380 -6,364.473 4,886.971 -2,894.552 -5,646.766 9,256.300 30,149.609 8,861.480 -11,313.582 -8,108.571 -3,984.823 10,228.385 14,621.468 9,811.966 -6,710.867 8,974.920 9,766.930 5,500.581 -11,973.233 2,209.261
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
601.709 -4,143.166 -7,744.707 -6,307.013 -1,999.270 -4,358.028 -5,258.037 1,725.065 4,960.184 -831.020 -183.045 -1,499.175 -9,294.279 289.499 -7,815.071 2,665.273 -3,233.269 -13,023.290 -1,008.768 -1,509.960 1,184.345 -9,977.896 6,220.831 11,934.491 10,860.860 14,539.579 1,137.106 -21,447.130 -1,986.712 -5,620.134 2,756.209 11,391.175
VITA
338
VITA
Mr. Derek William Hengeveld received his Bachelor of Science Degree in Mechanical Engineering with a minor in Computer Science from South Dakota State University (SDSU) in 1997. During this time, he joined the Industrial Assessment Center at SDSU, which provided energy, productivity, and waste assessments for manufacturing facilities throughout the Midwest. In 1998, he received his Master of Science Degree in Engineering from SDSU. The title of his thesis was “Comparison of the Performance of the H.M. Briggs Library’s Cooling System with Manufacturer’s Rated Performance and a Common Mathematical Model”. For the next six years, Mr. Hengeveld worked at SDSU as both a researcher and instructor. He was active in building energy efficiency projects and taught several undergraduate courses. In 2002-2003, he was awarded SDSU’s College of Engineering Teacher of the Year. Since 1998, Mr. Hengeveld has been a part owner of BTU Engineering. This company specializes in energy engineering for residential, commercial, and industrial clients. He currently serves as vice-president. Mr. Hengeveld is pursuing his Ph.D. degree at the School of Mechanical Engineering at Purdue University. He has been co-advised by Professors James E. Braun and Eckhard A. Groll. During his Ph.D. studies, he was awarded the American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) Grant-in-Aid Award (20082009); the American Institute of Aeronautics and Astronautics (AIAA) Foundation Open Topic Graduate Award (2008-2009); the Winkelman Davidson Fellowship (2006-2008); the Carrier ASHRAE Fellowship (2005-2006); and was selected as an Air Force Research Laboratory Space Scholar (2006-2009). Upon graduation, Mr. Hengeveld will join LoadPath, a company specializing in aerospace design, testing, and materials and will continue his role in BTU Engineering.
