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GENERALIZED A SIZING METHOD FOR REVOLUTIONARY CONCEPTS UNDER PROBABILISTIC DESIGN CONSTRAINTS
A Thesis Presented to The Academic Faculty by Taewoo Nam
In Partial Fulllment of the Requirements for the Degree Doctor of Philosophy in the School of Aerospace Engineering
Georgia Institute of Technology May 2007 Copyright 2007 by Taewoo Nam
A GENERALIZED SIZING METHOD FOR REVOLUTIONARY CONCEPTS UNDER PROBABILISTIC DESIGN CONSTRAINTS
Approved by: Professor Dimitri N. Mavris, Committee Chair School of Aerospace Engineering Georgia Institute of Technology Professor Daniel P. Schrage School of Aerospace Engineering Georgia Institute of Technology Dr. Danielle S. Soban School of Aerospace Engineering Georgia Institute of Technology Professor Vitali V. Volovoi School of Aerospace Engineering Georgia Institute of Technology Mr. Craig L. Nickol Langley Research Center National Aeronautics and Space Administration Date Approved: April 2, 2007
To my wife, Joung Yun for her patience, perseverance, sacrice, and support but most of all for her unending love
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ACKNOWLEDGEMENTS
I would rst like to thank my advisor, Professor Dimitri Mavris, for his support throughout my entire graduate studies. His technical expertise, unagging ardor, and persistent encouragement were the bedrock of this work. My gratitude extends to my entire committee for their guidance and valuable comments regarding my endeavor. This research would not have been possible of it were not for the nancial support provided by NASA and the Department of Defense under the University Research, Engineering and Technology Institutes (URETI) program. I am also grateful for the nancial support that I have received from the Handeul Scholarship Foundation. My fellow graduate students have been of great help in completing this document. Specically, I would like to thank Taeyun Choi for providing his fuel cell propulsion system analysis model and tireless proofreading. My appreciation extends to all other URETI team members, especially Davis Balaba, Eric Upton, Bhuan Agrawal, and Blake Mott for their suggestions and assistance. I thank to Kelly Griendling, Jean Charles Domercant, Dongwook Lim, Santiago Balestrini, and Henry Won for reviewing various versions of my thesis manuscripts. Special thanks are in order for Kyunghoon Lee for his magical assistance with all things LaTeX. I am also very appreciative of the valuable comments given to me by Mr. Mark Water, known as a great wizard of ASDL. The nal version of this document would not be what it is, had it not been for the professional proofreading done by Mrs. Osburg. I am also in debt to Dr. Chae, Dr. Lillie, Dr. Cheon, Mr. McKeon, and Dr. Ahn for their friendship and support. They have made my time in Atlanta more enjoyable. A very special thanks has to go to Mr. Hoschwelle, who mentored me while I
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was at KAI. To me, he shall remain as the one who most inuenced my career in aerospace engineering by inspiring me to love studying airplanes. The last but not the least, I would like to thank my family and loved ones. No word can suciently express how thankful I am of my parents and parents-in-law enough for their unending love and support. I have been a long way from home for a long time. Particularly, the last ve years have been the hardest they had to endure, without me being able to support them as much as they deserved. Finally, my wife, Joung Yun, my daughter, Jihye, and my son, Jiung; without your encouragement and love, I could never have achieved this. I am a lucky man and proud to be called your husband and your father. Thanks everyone, Atlanta, Georgia, Spring 2007
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TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv xi xii
LIST OF SYMBOLS OR ABBREVIATIONS . . . . . . . . . . . . . . . . . . xvii SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Promoters for Energy Alternatives . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 Mounting Concern for the Environment . . . . . . . . . . . Technological Advancement . . . . . . . . . . . . . . . . . . Economic Aspects . . . . . . . . . . . . . . . . . . . . . . . National Energy Security . . . . . . . . . . . . . . . . . . . Unconventional Missions . . . . . . . . . . . . . . . . . . . . 1 3 3 7 7 9 11 13 14 15 18 21 22 24 25 29 29 35 36
1.2 Energy Sources for Aviation . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 Conventional Aviation Fuels . . . . . . . . . . . . . . . . . . Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar Energy and Beam Power . . . . . . . . . . . . . . . . Nuclear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Alternatives . . . . . . . . . . . . . . . . . . . . . . .
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . II AIRCRAFT SIZING AND SYNTHESIS . . . . . . . . . . . . . . . . . . 2.1 Denition of Aircraft Sizing and Synthesis . . . . . . . . . . . . . . 2.2 Traditional Sizing Methods . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Overall Process of Aircraft Sizing . . . . . . . . . . . . . . .
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2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7
Inputs of the Aircraft Sizing Process . . . . . . . . . . . . . Thrust Balance . . . . . . . . . . . . . . . . . . . . . . . . . Fuel Balance . . . . . . . . . . . . . . . . . . . . . . . . . . Weight Estimation . . . . . . . . . . . . . . . . . . . . . . . Actual Value-Based Approach and Weight Specic ParameterBased Approach . . . . . . . . . . . . . . . . . . . . . . . . Iteration of the Aircraft Sizing Process . . . . . . . . . . . .
36 38 42 45 48 49 51 51 55 58 62 63 67 69 69 70 72 77 77 85
2.3 Aircraft Sizing under Uncertainty . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 2.3.3 Uncertainty Sources . . . . . . . . . . . . . . . . . . . . . . Traditional Approaches to Aircraft Sizing Under Uncertainty Recent Probabilistic Approaches to Aircraft Design . . . . .
2.4 Deciencies in Traditional Sizing Methods . . . . . . . . . . . . . . 2.4.1 2.4.2 III Inexibility toward Unconventional Concepts . . . . . . . . Inability to Account for Uncertainty . . . . . . . . . . . . .
RESEARCH OBJECTIVE, QUESTIONS, AND HYPOTHESES . . . . 3.1 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Substantiation of Hypotheses . . . . . . . . . . . . . . . . . . . . . 3.4.1 3.4.2 Mathematical Representation of Hypothesis 1 . . . . . . . . Mathematical Representation of Hypothesis 2 . . . . . . . .
IV
FORMULATION OF THE ARCHITECTURE-INDEPENDENT AIRCRAFT SIZING METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1 Generalized Constraint Analysis . . . . . . . . . . . . . . . . . . . . 4.1.1 4.1.2 Formulation of a Single Constraint Analysis . . . . . . . . . Constraint Analysis Matrix . . . . . . . . . . . . . . . . . . 87 87 90 94 94 96
4.2 Generalized Breguet Range Equations . . . . . . . . . . . . . . . . 4.2.1 4.2.2 Flight by Consumable Energy . . . . . . . . . . . . . . . . . Flight by Non-consumable Energy . . . . . . . . . . . . . .
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4.2.3
Example Application to Zero-emissions Aircraft . . . . . . .
96
4.3 Generalized Mission Analysis . . . . . . . . . . . . . . . . . . . . . 102 4.3.1 4.3.2 Consumable Energy Sizing . . . . . . . . . . . . . . . . . . 104
Non-consumable Energy Sizing . . . . . . . . . . . . . . . . 107
4.4 Generalized Weight Estimation . . . . . . . . . . . . . . . . . . . . 111 4.5 The Process of AIASM . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.6 Non-dimensional Aircraft Mass (NAM) Ratio . . . . . . . . . . . . 119 4.7 Extension to Solar-Powered Propulsion Architecture . . . . . . . . 131 4.7.1 4.7.2 V Application to Solar-Powered Aircraft . . . . . . . . . . . . 134 Application to Solar-Powered Regenerative Propulsion Aircraft137
FORMULATION OF THE PROBABILISTIC AIRCRAFT SIZING METHOD 143 5.1 Approaches to Probabilistically Constrained Optimization Problems 144 5.2 A Numerical Example of Optimization Under Uncertainty . . . . . 147 5.2.1 5.2.2 5.2.3 5.2.4 Deterministic Solution Sampling (DSS) Method . . . . . . . 148 Two-Stage Stochastic Programming Method . . . . . . . . . 149 CCP Method . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Lessons Learned . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Standard Form of the Probabilistic Aircraft Sizing Method . . . . . 155 5.3.1 5.3.2 5.3.3 Integration with AIASM . . . . . . . . . . . . . . . . . . . . 155 Decision Variables as Random Variables . . . . . . . . . . . 156 Standard Form of PASM . . . . . . . . . . . . . . . . . . . . 158
5.4 Solution Techniques of CCP and RBDO . . . . . . . . . . . . . . . 160 5.4.1 5.4.2 5.4.3 5.4.4 Deterministic Equivalent . . . . . . . . . . . . . . . . . . . . 160 Constraint Sampling Approach . . . . . . . . . . . . . . . . 161 Optimization with Reliability Analysis . . . . . . . . . . . . 162 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . 163
5.5 Extended Formulation of PASM . . . . . . . . . . . . . . . . . . . . 168 5.5.1 Joint Probabilistic Constraints . . . . . . . . . . . . . . . . 169
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5.5.2 5.5.3 5.5.4 5.5.5
Multidisciplinary Design Optimization . . . . . . . . . . . . 169 Probabilistic Objective Function . . . . . . . . . . . . . . . 170 Multi-Objective Function . . . . . . . . . . . . . . . . . . . 171 Robust Design . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.6.1 5.6.2 VI Sensitivity of Objective Function to Target Probability . . . 176 Sensitivity of Constraint Functions to Distributions of Random Parameters . . . . . . . . . . . . . . . . . . . . . . . . 177
METHOD IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . 181 6.1 Fuel Cell-Powered General Aviation . . . . . . . . . . . . . . . . . 181 6.1.1 6.1.2 6.1.3 Deterministic Solutions . . . . . . . . . . . . . . . . . . . . 186 Code Verication . . . . . . . . . . . . . . . . . . . . . . . . 188 Probabilistic Sizing . . . . . . . . . . . . . . . . . . . . . . . 189
6.2 Regenerative Solar-powered Aircraft . . . . . . . . . . . . . . . . . 193 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.2.7 Mission and Conguration . . . . . . . . . . . . . . . . . . . 193 Deterministic Sizing . . . . . . . . . . . . . . . . . . . . . . 196 Impact of Technology Advancement . . . . . . . . . . . . . 202 Conguration Optimization . . . . . . . . . . . . . . . . . . 204 Probabilistic Sizing . . . . . . . . . . . . . . . . . . . . . . . 206 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 213 Joint Probabilistic Constraints . . . . . . . . . . . . . . . . 221
6.3 Lessons Learned from Implementation Studies . . . . . . . . . . . . 224 VII CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . 227 7.1 Research Questions Answered . . . . . . . . . . . . . . . . . . . . . 227 7.1.1 7.1.2 Research Questions 1 . . . . . . . . . . . . . . . . . . . . . . 227 Research Questions 2 . . . . . . . . . . . . . . . . . . . . . . 229
7.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . 229 7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.3.1 Comprehensive Sizing Method . . . . . . . . . . . . . . . . . 231 ix
7.3.2 7.3.3
Uncertainty Modeling . . . . . . . . . . . . . . . . . . . . . 235 Multi-Stage RBDO with Recourse . . . . . . . . . . . . . . 236
7.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 246 APPENDIX A APPENDIX B CONSTRAINT EQUATIONS . . . . . . . . . . . . . . . . 247 WEIGHT FRACTION EQUATIONS . . . . . . . . . . . . 258
APPENDIX C WEIGHT-SPECIFIC PARAMETERS AS DECISION VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 APPENDIX D 263 APPENDIX E ELECTROLYZER MODEL AND ROUND TRIP EFFICIENCY SUPPLEMENTAL CHARTS . . . . . . . . . . . . . . . . . 268
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
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LIST OF TABLES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Characteristics of solar cells . . . . . . . . . . . . . . . . . . . . . . . Energy contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 79
FLOPS analysis results of zero-emissions aircraft . . . . . . . . . . . . 101 Input parameters and results of the generalized Range Equations of zero-emissions aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 102 The assumptions of architecture options . . . . . . . . . . . . . . . . 130 High altitude unmanned aircraft . . . . . . . . . . . . . . . . . . . . . 130 Transformation of the u-space to the -space . . . . . . . . . . . . . . 167 Deterministic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Probabilistic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Comparison of probability of failure . . . . . . . . . . . . . . . . . . . 192 The solution of joint probabilistic constraints . . . . . . . . . . . . . . 193 Baseline conguration sizing - solution D1 . . . . . . . . . . . . . . . 201 Baseline conguration sizing with advanced technologies - solution D2 204 Optimum conguration with advanced technologies - solution D3 . . 207 Assumed distributions of random parameters . . . . . . . . . . . . . . 208 Comparison of the results of probabilistic sizing by FORM and MCS 211
Lagrange multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Sensitivity of the objective function per target probability (%) . . . . 219 Comparison of the solutions of individual probabilistic constraints and joint probabilistic constraints . . . . . . . . . . . . . . . . . . . . . . 224 Comparison of the results from RBDO and MSRBDO . . . . . . . . . 243 Inputs of the constraint analyses of the GA study . . . . . . . . . . . 268 Mission analysis of the electric GA . . . . . . . . . . . . . . . . . . . 269
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LIST OF FIGURES
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 Examples of unconventionally powered and propelled aerospace systems Global land-ocean temperature anomaly (C) . . . . . . . . . . . . . . History of crude oil, rener acquisition cost . . . . . . . . . . . . . . . U.S. oil supply sources . . . . . . . . . . . . . . . . . . . . . . . . . . ARES platform for Mars exploration . . . . . . . . . . . . . . . . . . AeroVironments fuel cell-powered aircraft: the Hornet and Global Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of solar-powered aircraft . . . . . . . . . . . . . . . . . . . Examples of battery-powered aircraft . . . . . . . . . . . . . . . . . . A notional system of system analysis structure for assessing the impacts of alternative energy sources on all of society . . . . . . . . . . . . . . Thesis structure overview . . . . . . . . . . . . . . . . . . . . . . . . 2 5 8 10 13 18 20 23 26 27 32 37 40 42 45 49 50 53 55 59 60 62 63 73 78
Illustration of the fundamental concept of aircraft sizing . . . . . . . Mattinglys aircraft sizing process . . . . . . . . . . . . . . . . . . . . Forces on aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notional constraint analysis diagrams . . . . . . . . . . . . . . . . . . Discretized mission prole . . . . . . . . . . . . . . . . . . . . . . . . Iterative process of actual-value-based sizing approach . . . . . . . . . Iterative process of specic parameter-based sizing approach . . . . . Design evolution of the T-50 Golden Eagle . . . . . . . . . . . . . . . Impact of uncertainty on aircraft sizing . . . . . . . . . . . . . . . . . Overall process of Robust Design Simulation . . . . . . . . . . . . . . Bandtes JPDM process . . . . . . . . . . . . . . . . . . . . . . . . . Nams DSS method process . . . . . . . . . . . . . . . . . . . . . . . Simulation through deterministic sizing codes . . . . . . . . . . . . . Strategy for a generalized sizing method . . . . . . . . . . . . . . . . Generalized propulsion system model . . . . . . . . . . . . . . . . . .
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26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Comparison of specic energy and specic power for various power source technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple power-paths . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical k values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference Power in a power-path . . . . . . . . . . . . . . . . . . . . Design point selection of two power-paths . . . . . . . . . . . . . . .
82 83 84 89 91 92 93 97 98 99
Approach to a notional UCAV design with a combination of aerodynamic morphing and propulsion morphing . . . . . . . . . . . . . . . Notional constraint analysis setup for a morphing aircraft with hybrid power systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ratio of nal to initial vehicle weight vs. fuel fraction . . . . . . . Range vs. fuel fraction . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized range vs. fuel fraction . . . . . . . . . . . . . . . . . . . .
Range vs. WRF by a modied FLOPS and the generalized Breguet range (GBR) equations . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Discretized mission prole . . . . . . . . . . . . . . . . . . . . . . . . 104 Comparison of numerical errors in two dierent approaches to mission analysis with varying number of mission legs . . . . . . . . . . . . . . 110 AeroVironment WASP . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Weight estimation process . . . . . . . . . . . . . . . . . . . . . . . . 116 Overview of AIASM . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Mission Space Exploration . . . . . . . . . . . . . . . . . . . . . . . . 122 Dualistic relationship between energy weight fraction and propulsion system weight fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Comparison of an alternative energy-propulsion system architecture and a conventional energy-propulsion system architecture . . . . . . . 125 Mapping the energy weight fraction to the mission range . . . . . . . 126 Mapping the cruise velocity to the propulsion system weight fraction 127
Illustration of a notional NAM ratio diagram . . . . . . . . . . . . . . 128 The NAM ratio diagram of a high altitude unmanned aircraft . . . . 132 B747-400 vs P-51B/C Mustang . . . . . . . . . . . . . . . . . . . . . 134
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50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
Available power per unit wing area for several combinations of geographic locations and dates . . . . . . . . . . . . . . . . . . . . . . . 136 Constraint analysis of solar-powered aircraft . . . . . . . . . . . . . . 137 Illustration of a solar-powered regenerative propulsion system . . . . 138 Power prole of solar-powered aircraft with a regenerative propulsion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Constraint analysis of solar-powered aircraft with a regenerative propulsion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Sizing process of solar-powered aircraft with a regenerative propulsion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Design space of a deterministic optimization problem . . . . . . . . . 147 Optimum solutions from Monte Carlo Simulation . . . . . . . . . . . 149 CDF of optimum values of the objective function of the Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 The total cost function of the two-stage stochastic programming problem151 Optimum solutions per the value of the penalty coecient . . . . . . 152 Equivalent deterministic constraints . . . . . . . . . . . . . . . . . . . 154 Overview of PASM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 System optimizer integrated with a nested reliability analysis . . . . . 164 Approximation of the limit state function by FORM and SORM . . . 166 Simultaneous application of joint probability to the objective space and the constraint space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Illustration of the MPP-based sensitivity measure . . . . . . . . . . . 179 Mission prole of electric GA aircraft . . . . . . . . . . . . . . . . . . 182 Notional fuel cell propulsion system architecture . . . . . . . . . . . . 184 Integrated analysis environment . . . . . . . . . . . . . . . . . . . . . 184 Power vs. weight of the electric propulsion system . . . . . . . . . . . 185 The distributions of random variables considered in the electric GA study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Constraint analysis of the electric GA Aircraft, D1 . . . . . . . . . . 187 Thrust and fuel ow of the PEMFC propulsion system . . . . . . . . 190
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74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
Comparison of range performance . . . . . . . . . . . . . . . . . . . . 190 The mission prole of the SPHALE . . . . . . . . . . . . . . . . . . . 195 Baseline conguration of the SPHALE . . . . . . . . . . . . . . . . . 195 Comparison of drag polars . . . . . . . . . . . . . . . . . . . . . . . . 198 Convergence of weight and payload power-to-weight ratio . . . . . . . 198 Convergence of wing loading . . . . . . . . . . . . . . . . . . . . . . . 199 Power prole of the SPHALE . . . . . . . . . . . . . . . . . . . . . . 199 Constraint analysis at the converged solution . . . . . . . . . . . . . . 200 Weight breakdown of the design gross weight of D1 . . . . . . . . . . 201 Sensitivity of technology impact . . . . . . . . . . . . . . . . . . . . . 203 Weight reduction in weight groups by infusing advanced technologies 205
Weight breakdown of the design gross weight of D2 . . . . . . . . . . 205 Impact of wing aspect ratio on drag and airframe weight . . . . . . . 207 Dierences in probabilities meeting the constraints estimated by FORM to those estimated by Monte Carlo simulations . . . . . . . . . . . . . 210 Dierences of the objective function values (95th percentile of the objective function responses) estimated by FORM to those estimated by Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . 210 Monte Carlo simulations with dierent numbers of trials for P1 . . . 212 Dierence of probabilistic solutions, P1 and P2 to the deterministic solution, D3 in percentage . . . . . . . . . . . . . . . . . . . . . . . . 214 PDF of the objective function . . . . . . . . . . . . . . . . . . . . . . 214 PDFs of constraint functions . . . . . . . . . . . . . . . . . . . . . . 215 Objective function vs. target reliability . . . . . . . . . . . . . . . . . 216 The optimum values of sizing variables vs. target reliability . . . . . . 217 Sensitivity index obtained by the MPP-based sensitivity analysis . . . 220 Sensitivity index obtained by Crystal Ball . . . . . . . . . . . . . . 220
89 90 91 92 93 94 95 96 97 98
Reliability improvement of probabilistic constraints by variance reduction in random parameters . . . . . . . . . . . . . . . . . . . . . . . . 222 Illustration of samples of an MCS in the space of power balance energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
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99
Comprehensive sizing method . . . . . . . . . . . . . . . . . . . . . . 238
100 Distribution of knowledge, cost committed, and freedom in the design cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 101 Comparison of nondeterministic approaches . . . . . . . . . . . . . . 241 102 Distributions of z and f from the application of the MSRBDO . . . 244 103 The MSRBDO simulates a decision making process of a complex system design problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 104 Forces on aircraft in a sustained level banked turn . . . . . . . . . . . 249 105 Takeo terminology (hTR < hobs ) . . . . . . . . . . . . . . . . . . . . . 253 106 Takeo terminology (hTR > hobs ) . . . . . . . . . . . . . . . . . . . . . 253 107 Landing terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 108 Equations associated with generalized mission analysis . . . . . . . . 259 109 Design space of probabilistic aircraft sizing . . . . . . . . . . . . . . . 262 110 Illustration of a regenerative fuel cell system . . . . . . . . . . . . . . 264 111 Performance of the electrolyzer model . . . . . . . . . . . . . . . . . . 265 112 Hydrogen mass ow versus power . . . . . . . . . . . . . . . . . . . . 266 113 Illustration of samples of an MCS (P2) . . . . . . . . . . . . . . . . . 270 114 PDF of total power required at loiter (W) . . . . . . . . . . . . . . . 271 115 PDF of angle of attack at loiter . . . . . . . . . . . . . . . . . . . . . 271 116 PDF of loiter velocity (m/sec) . . . . . . . . . . . . . . . . . . . . . . 272 117 PDF of loiter eciency . . . . . . . . . . . . . . . . . . . . . . . . . . 272
xvi
SUMMARY
Internal combustion (IC) engines that consume hydrocarbon fuels have dominated the propulsion systems of air-vehicles for the rst century of aviation. In recent years, however, growing concern over rapid climate changes and national energy security has galvanized the aerospace community into delving into new alternatives that could challenge the dominance of the IC engine. A critical element required to pursue alternative energy ight is a quantitative assessment environment for aircraft sizing and synthesis that provides insight into the system-wide responses of these technologies and system architectures, thereby increasing the possibility of arriving at more informed decisions. Nevertheless, traditional aircraft sizing methods have signicant shortcomings for the design of such unconventionally-powered aircraft. First, traditional aircraft sizing methods are specialized for aircraft powered by IC engines, and thus are not exible enough to assess revolutionary propulsion concepts that produce propulsive thrust through a completely dierent energy conversion process. Another deciency associated with the traditional methods is that a user of these methods must rely heavily on experts experience and advice for determining appropriate design margins. Aircraft sizing achieves its goal by imposing two primary constraints: matching available power to required power and available energy to required energy. These constraints have always been assumed to be deterministic in the context of traditional sizing methods and even under the more recent probabilistic design paradigm. In reality, signicant uncertainty, including unsettled performance requirements and
xvii
environmental regulations, changes in drag and weight due to evolving airframe design, and lack of accuracy of subsystem performance prediction may permeate design parameters and contributing analyses associated with these constraints. The traditional approach to mitigate the risk associated with such uncertainty is to add design margin quantied by the empirical knowledge of experienced engineers. Such an approach, however, may result in either signicant risk or unnecessary increases in weight and cost. Furthermore, the introduction of revolutionary propulsion systems and energy sources is very likely to entail an unconventional aircraft conguration, which inexorably disqualies the conjecture of such connoisseurs as a means of risk management. Motivated by such deciencies, this dissertation aims at advancing two aspects of aircraft sizing: 1) to develop a generalized aircraft sizing formulation applicable to a wide range of unconventionally powered aircraft concepts and 2) to formulate a probabilistic optimization technique that is able to quantify appropriate design margins that are tailored towards the level of risk deemed acceptable to a decision maker. A more generalized aircraft sizing formulation, named the Architecture Independent Aircraft Sizing Method (AIASM), is achieved by modifying several assumptions of the traditional aircraft sizing method. First, fuel is generalized as a concept of on-board energy which can originate from a gamut of energy sources. Each source is categorized as consumable energy or non-consumable energy. In addition, the propulsion system is modeled as an integration of multiple power-paths, each of which is characterized by three parameters: the specic energy of the energy source, the specic power and eciency maps of power transfer devices. Lastly, generalized weight decomposition and weight dierential equations are employed to model the fuel consumption behavior of the revolutionary concepts that, unlike the IC engine, may sequester and retain specic by-products from being emitted during ight. In the course of the development of AIASM, a couple of expedients useful for the
xviii
rst-order estimation of aircraft performance are also developed: a set of generalized Breguet range equations and the Non-dimensional Aircraft Mass (NAM) Ratio Diagram. The generalized Breguet range equations can estimate the ferry range of an aircraft powered by alternative energy-propulsion architectures with little information pertinent to the aircraft. Utilizing the generalized Breguet range equations as well as the intrinsic duality between the mass of on-board energy sources and the mass of the propulsion systems, the NAM Ratio Diagram identies the performance frontiers in terms of range and velocity, which are achievable with a given energy-propulsion system architecture. Along with advances in deterministic aircraft sizing, a non-deterministic sizing technique, named the Probabilistic Aircraft Sizing Method (PASM), is developed. The method allows one to quantify adequate design margins to account for the various sources of uncertainty via the application of the chance-constrained programming (CCP) strategy to AIASM. The CCP aims to nd an optimum solution that minimizes the objective function within the probabilistically feasible space, in which each point satises a set of non-deterministic constraints in accordance with the target probabilities chosen by a decision maker. The PASM mathematically formulates an aircraft sizing problem into an optimization problem whose goal is to minimize the expectation of the objective function values subject to multivariate, nonlinear, individual or joint probabilistic constraints. In this context, the three design variables, power, wing area, and fuel (energy) quantity, are manipulated until all probabilistic constraints are satised with equal or higher probabilities than the target. In this way, PASM can also provide insights into a good compromise between cost and safety. The proposed methods are veried by applying them to two aircraft sizing studies for a fuel cell-powered general aviation (GA) aircraft and a Solar-Powered High Altitude Long Endurance (SPHALE) aircraft with a Regenerative Fuel Cell (RFC) system.
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CHAPTER I
INTRODUCTION
Since the Wright brothers rst ight, aerospace endeavors have achieved enormous advancement in technology, both commercial and military. The fabric-covered wooden frame and skid of the Wright Flyer has been replaced with metal or advanced composite material and retractable landing gear systems, respectively. Their descendants no longer lie down on the wing to control an airplane but sit in a glass cockpit supported by an fully integrated avionics system. Despite such prodigious development of aviation, one thing has not changed: machine-ights depend almost entirely on internal combustion engines that consume hydrocarbon fuels. Although the advent of the jet engine ushered in a new era of aviation, its thrust is produced by ultimately the same physics and sources: exothermic reaction of a fossil fuel derived from oilelds. Only a few momentous events, for instance the Solar Challengers ight across the English Channel solely by solar power in 1981, have reminded the aerospace community of the obvious but not always apparent fact that aircraft can y via dierent energy sources. About half a century ago, atomic energy and hydrogen were envisioned for aircraft propulsion, respectively, and attempts were made to experiment with replacing the hydrocarbon fuels with nuclear or liquid hydrogen for high-speed aircraft [1, 2]. For reasons that were valid at that time, these eorts did not lead to practical applications and were not actively pursued. Despite incessant subsequent eorts to fuel aircraft with alternative energy sources, the dominance of internal combustion (IC) engines had been deemed unbeatable. In recent years, however, alternate energy sources and revolutionary propulsion systems have been attracting renewed attention. An
1
Solar Cell
Beamed Power
Hydrogen
http://www.dfrc.nasa.gov/Newsroom/ X-Press/images/083101/helios.jpg
Nuclear Battery
Atmosphere (Methane in Titan)
Chemical Muscle
Human-Power
AA Batteries
Lithium-Polymer Battery
www.dfrc.nasa.gov/.../ Small/EC87-0014-8.jpg
http://www.aerovironment.com/news/newsarchive/local-images/wasp1.jpg6
Figure 1: Examples of unconventionally powered and propelled aerospace systems
ample anthology of literature [3, 4, 5] that foresees the future of aviation commonly advocates the transition to alternative energy sources to power the next generation aircraft, with fossil fuels becoming less reliable. AeroVironments Helios, funded by the National Aeronautics and Space Administration (NASA), accentuated the great potential of solar energy by establishing a new altitude record for non-rocket-powered aircraft of 96,863 feet in 2001 [6]. Particularly, hydrogen has also been rigorously re-examined for applicability to both jet engines and fuel cells [7].
2
1.1
Promoters for Energy Alternatives
In the sense that crude oil is a nite resource, the search for alternative energy sources has been predestined since the inception of the oil economy. It would be a very logical inference that the cost and availability of conventional aviation fuels will ultimately increase up to a point which is unattractive to aviation. However, the rigorous research of recent years has reached well beyond the level of a mundane response for the clich. In fact, several issues concerning the environment, technology, e economics, and national energy security have coalesced into an impetus, which is fostering the search for alternative energy ights. 1.1.1 Mounting Concern for the Environment
The rst issue is increased public concern over the environmental impact of engine emissions, which have adverse eects on global warming, ozone depletion, and local air quality. Global warming is referred to as the gradual increase in temperatures on the Earths surface. Scientists have amassed compelling evidence that recent warming coincides with rapid growth of anthropogenic greenhouse gases such as carbon dioxide, water vapor, methane, nitrous oxide, and ozone. According to a recent report by the National Research Council (NRC), the average surface temperature of the earth has risen 0.4 - 0.8C over the last century [8]. Moreover, climatologists at NASAs Goddard Institute for Space Studies (GISS) in New York City noted that the 2005 calendar year marks the highest global annual average surface temperature in more than a century and the past four years were identied as belonging to the top ve warmest years worldwide over past century [9]. This growing evidence indicates that such a strong underlying warming trend is continuing and even speeding up. The estimation about future global temperature trends depends heavily on assumptions concerning population and economic growth, land use, technological changes, energy availability, and fuel mix [10]. Estimates for this century range from 1 to 3.5C, any of
3
which is the greatest average rate of warming in the last 10,000 years [10]. Although this appears as a tiny perturbation, such a change in the global mean may result in a substantial variation in regional temperature and weather, which could have appalling ramications on the ecological system and human life1 . Aviation is believed to be less responsible for global warming than some other human activities2 . Nevertheless, the impact of aircraft emissions on the earths atmosphere and climate has caught the attention of transportation planners and policymakers for several reasons. First, jet aircraft are the primary source of greenhouse gases that are deposited directly into the upper atmosphere [10]. According to the Intergovernmental Panel on Climate Change (IPCC) and other experts, some of these jet emissions have a greater warming eect than they would have if they were released in equal amounts at the surface by, for example, automobiles [13]. Secondly, the impact of carbon dioxide emissions on the atmosphere can be amplied two- to vefold by other gases and particles3 emitted from jet engines. Lastly, aviation is one of the fastest growing contributors to climate change, with global air passenger travel projected to grow at a rate of 5 percent per annum through 2015 [13]. Such a solid growth in aviation, the IPCC predicted, will increase the radiative forcing from aircraft by up to a factor of 11 by 2050, compared with a predicted increase of roughly a factor of two in total anthropogenic forcing [10]. The IPCC recently concluded that the consequential increase in aviation emissions, concomitant to such an escalating demand for air travel, would not be fully oset by reductions in emissions achieved through technological improvements on conventional propulsion systems alone [13]. Another concern related to aircraft emissions is ozone depletion. The stratospheric
According to a recent analysis [11], global warming of more than 1C, relative to 2000, will constitute dangerous climate change as judged from likely eects on sea level and extermination of species. 2 Carbon dioxide and water emitted from aircraft engines account for only 3% of total anthropogenic green house gas emissions [12]. 3 They include water vapor, nitrogen oxide and nitrogen dioxide (collectively termed NOx ), sulfates, and soot.
1
4
Figure 2: global land-ocean temperature anomaly (C). Error bars are estimated with 2 (95% condence) uncertainty. [9]
ozone layer absorbs ultraviolet radiation that can cause skin cancer. Although ozonedepleting substances (ODS) such as chlorouorocarbons (CFCs) and halons are controlled under the Montreal Protocol, the Earths protective ozone layer is persistently being diluted. It has been reported that aircraft emissions can alter the chemistry of the stratosphere, resulting in changes in the concentration of ozone [14]. These changes, in turn, may indirectly aect the total amount of ozone and global climate through coupling with radiative and dynamic atmospheric processes [15]. NOx emissions by aircraft can aect atmospheric ozone in either positive or negative ways, depending on the altitude at which they are emitted [16]. Although there is considerable uncertainty in the estimates of the actual impact, it is generally agreed that emissions at altitudes above 50,000 to 55,000 feet will degrade the ozone layer to some degree [17]. Several analyses [18, 19, 20] indicate that water emissions from a eet of large commercial supersonic aircraft operating in the stratosphere would signicantly deplete ozone, even if NOx could be reduced to very low levels.
5
In addition to eects on the global environment, concerns over degradation in local air quality in the vicinity of airports are growing due to increasing air trac, increasing urbanization, increasing public, and regulatory attention [21]. Oxides of nitrogen (NOx ), carbon monoxide (CO), and unburned hydro carbon (HC) are the main concerns of aircraft engine emissions for local air quality. With growing concern about aviations eects on the global atmosphere and local air quality, relevant regulations are now more stringent, and new regulations are being prepared. Present aircraft emissions regulations apply only to the landing and take-o cycle, known as the LTO cycle, up to an altitude of 900 meters. In the United States, the rst regulation on vented fuel, smoke and exhaust (NOx , HC, CO) emissions was promulgated by the U.S. Environmental Protection Agency (EPA) in 1973, and was revised in 1976, 1980, 1982 and 1984 [22]. In 1997, the two-staged voluntary NOx standard and CO standard of the International Civil Aviation Organization (ICAO) was codied into the United States law to align with the international emission regulation [23]. In order to further reduce the NOx emission, the ICAO made a more stringent standard which aims for 16% average reduction for the engines certied after December 31, 2003, and EPA amended the existing standards for NOx emissions of new commercial aircraft engines 2005 to align with the new ICAO standard. [24]. Instead of developing the standard for CO2 , the ICAO is currently considering a market-base measurement, which includes voluntary measures, emissions-related levies, and emission trading to achieve environmental goals at a lower cost and in a more exible manner [17]. In response to this, from 1998 to 1999, a series of high-level studies exploring aircraft with unconventional propulsion systems, which focused on reducing aircraft emissions and noise was jointly performed by NASA Langley Research Center (LaRC) and NASA Glenn Research Center (GRC) [25]. Such eorts culminated in the Quiet
6
Green Transport (QGT) study, one of the initial studies undertaken by the Revolutionary Aerospace Systems Concepts (RASC) Program [25, 26]. 1.1.2 Technological Advancement
One thing that held back earlier eorts at seeking alternative energy sources is that the propulsion systems and energy storage systems required for the use of nonhydrocarbon fuel were not mature. A rapid leap in revolutionary propulsion system technology during the past decades, however, has reopened the possibility of operating aircraft on alternative energy sources. For example, the rst photovoltaic (PV) cell built by Charles Fritts in 1883 was only around 1% ecient [27], and the rst commercial solar cells developed by Homan Electronics-Semiconductor Division in 1955 barely exceeded 2% [28]. However, eciencies of state-of-the-art PV cells are higher than 20%. Although the current technologies are not immediately ready for commercial applications, continuing ample investment on developing alternative propulsion technology in aerospace as well as other sectors will collectively bring forward feasible and viable aircraft applications. For example, recent eorts in fuel cell propulsion systems are being driven mainly by automobile communities. Aviation sectors will also benet from such advances in fuel cell technology and electrical components, promoted by automotive and other transportation sectors. 1.1.3 Economic Aspects
A continuous decrease in alternative energy costs due to technological advances, in concert with a steep increase in oil prices in recent years, has been also pushing the advent of commercially viable alternative energy sources forward. As depicted in Figure 3, world oil prices have increased signicantly over the past two years relative to historical levels. Crude oil prices, which hovered in the $15-25 per barrel range from the mid-1980s until 2002, have been above $40 since February 2005. Future oil prices 7
80 70 60 50 US $ 40 30 20 10 0 1975 Refiner Acquisition Cost of Crude Oil, Domestic Refiner Acquisition Cost of Crude Oil, Imported
1980
1985
1990 Year
1995
2000
2005
Figure 3: History of crude oil, rener acquisition cost. The source data [29] were published by the U.S. Department of Energy
are not easy to predict, heavily depending on capricious factors such as geopolitical instability and weather as well as a fundamental driver: supply-demand. Considering rapid world-oil-demand growth driven by all areas of economic activity, many analysts agree that the recent soaring oil prices are not a passing phenomenon like previous oil crises. As far as supply is concerned, the ever-increasing production of crude oil is believed to be approaching its peak4 . The peak is dened as the apex of worldwide oil production, that occurs when half of the ultimately recoverable reserves have been extracted according to Hubbert Bell Curve [31]. When this resource-limited peak is reached, oil production will start a gradual, but relentless decline that will trigger a cataclysmic rise in oil prices [32]. The date of the global peak heavily depends on the
According to a DOE report published in 2004 [30], no major new eld discoveries have been made in decades. Presently, world oil reserves are being depleted three times as fast as they are being discovered.
4
8
size of Middle-East reserves, estimations of which are notoriously unreliable [33]. The most optimistic estimate is that the peak of production will be 20 to 25 years from now [34]. Most studies expect crude oil production to peak much earlier, sometime around the year 2015, or even earlier [33]. Whether the peak occurs sooner or later is a matter of relative urgency, but does not alter a central conclusion; the United States needs to establish a supply base for its future energy needs using its signicant oil shale, coal, and other energy resources [30]. In contrast, the cost of renewable energy has been signicantly lowered. For example, the cost of wind power in the U.S. was $.80 per KWh in 1980, but $0.04 per KWh today [35]. The prices of solar cells have even more drastically decreased over a half century. The rst commercial solar cells developed by Homan ElectronicsSemiconductor Division in 1955 cost $1,785 per watt (in 1955 dollars) [28]. The costs dropped down to $100 a watt in 1976, and they sell for less than $3 a watt today. The costs are expected to continue declining 5 percent annually, even if there are no technology breakthroughs. [36] In addition to decreases in price tags on alternative energy, increases in oil prices will further narrow the gap, after which alternative energy sources will become increasingly competitive. 1.1.4 National Energy Security
Growing concerns about national energy security have made the search for alternative energy source even more apropos. Dependence upon foreign oil may undermine the national economy and energy security. Today, the U.S. imports 60% of its crude oil supply, and U.S. dependence on foreign sources of oil will increase as domestic resources are exploited [37]. In particular, the U.S. transportation network relies heavily on petroleum from overseas as an energy source [12]. Air travel uses around 10% of total energy consumed by the U.S. transportation network [38], and in 2002, the U.S. imported 62% of its petroleum from overseas. Demand for the transportation
9
20 18 16 14 12 10 8 6 4 2 0 2003
Oil Supply (million barrels per day)
net imported domestic
2008
2013
2018
2023
2028
year
Figure 4: U.S. oil supply sources [38]
sector is expected to continue to increase at a rate between two and three percent per annum until at least 2010 whilst domestic production is forecast to remain relatively static until 2030 [38]. The ramication is that import dependence is only likely to get worse (forecast to be at 68% by 2030 [38]), as shown in Figure 4. Most of the worlds oil reserves are concentrated in the Middle East, and over two-thirds are controlled by the members of OPEC whose self-appointed mission is to manipulate prices by turning the production spigots up and down [39]. A recent analysis [40] indicates that oil price shocks and price manipulation by OPEC have cost the U.S. economy dearly, about $7 trillion from 1973 to 2000, which is as large as the sum total of payments on the national debt over the same period. Furthermore, each of the oil market upheavals was followed by a recession. With growing U.S. imports and increasing world dependence on OPEC oil, the U.S. economy would be vulnerable to future price shocks [41]. In addition to increased economic vulnerability, dependence on foreign energy sources( particularly on oil imported from the Middle East, a very volatile region of the world) increases the political burden on the U.S.
10
government. Senator Joseph R. Biden says: that dependence means we pay a huge price militarily for access to a resource that we cannot do without. One estimate suggests we pay as much as $825 billion a year in security expenditures to project our inuence and secure access to oil [42]. To address these challenges, in 2001 President George W. Bush announced the National Energy Policy [43] envisioning a comprehensive long-term strategy that includes promoting domestic energy resources such as natural gas, coal, nuclear, and renewable energy sources. In addition, the Energy Policy Act of 2005 [44], and the U.S. Department of Energy (DOE) Strategic Plan [45] also call for developing a diverse portfolio of domestic energy supplies as well as improving energy eciencies. Furthermore, the Presidents Advanced Energy Initiative announced in 2006 provides for a 22% increase in research by the U.S. Department of Energy to nd clean alternatives to oil [46]. Eorts to alleviate the national economys addiction to oil are also being made across Europe. Such approaches include ourishing windmills in Denmark and a renewed debate in Britain about reversing promises to cut back on nuclear energy supplies [47]. Most impressive, the Swedish government announced in February 2006 national plans to be worlds rst oil-free economy by 2020 without building a new generation of nuclear power stations [48]. In addition, governments of many countries are espousing cutting-edge research to develop sustainable energy sources to reduce their dependence on imported energy sources. 1.1.5 Unconventional Missions
The development of revolutionary aerospace concepts such as high altitude, extremely long endurance aircraft and vehicles designed for planetary exploration that are not feasible with conventional propulsion systems has provided a need for revolutionary propulsion technologies. High Altitude Long Endurance (HALE) platforms such as
11
Helios and HeliPlat [49] have been envisioned as a possible alternative to communication satellites. They could also monitor weather, track hurricanes, and make substantial contributions in disaster management via more precisely directing emergency resources [6]. In addition, their remote sensing capability, continuously available over extended time periods of weeks and months, inspires several commercial applications such as precision agricultural management and wildre monitoring that have a need for near real-time high-spatial resolution imagery [50]. In comparison to orbital satellites, atmospheric satellites would oer better observational resolution, local persistence, and the capability of reuse [51]. However, such great potential is stymied by a conspicuous technological barrier: such a long endurance capability cannot be achieved with a conventional propulsion architecture and fuel. The U.S. Air Force tanker KC-10 Extender would be the apotheosis of a conventional propulsion aircraft designed for carrying as much fuel as permitted. The aerial refueling tanker, derived from the civilian DC-10-30 airliner, carrying 365,000 lbs. fuel [52], oers an unrefueled range of 11,500 nautical miles, holding the record for the worlds longest-ranged aircraft. However, even this aircraft cannot stay airborne more than two days without refueling. Therefore, alternative propulsion system architectures and energy sources suitable for such an unconventional mission must be pursued. Burgeoning interest in the exploration of terrestrial planets is also promoting the research of alternative propulsion aircraft. The idea of performing scientic observations on Mars or Titan using a winged-platform has persisted because of the clear advantages associated with an airborne platform. Just like on Earth, airborne observations would complement ground-based and space-based observations, permitting higher resolution than possible with space-based platforms and greater coverage than possible with ground-based platforms [53]. Airplanes oer an additional advantage over other airborne platforms such as balloons in that it can be maneuvered to specic locations of interest [53]. This advantage has brought winged-platforms onto the
12
Figure 5: ARES platform for Mars exploration [57]
table with other alternative concepts for atmospheric explorations of Mars and Titan. For Mars exploration, a concept of winged-platform, shown in Figure 5, was embodied for the Aerial Regional-scale Environmental Survey (ARES) of Mars mission [54]. This Mars Scout mission was proposed to provide high-value scientic measurements in the areas of atmospheric chemistry, surface geology and mineralogy, and crustal magnetism [53]. The feasibility of atmospheric ight by an airplane on Titan was also investigated for a post-Cassini mission [55]. Nevertheless, there are several technical challenges associated with a terrestrial airplane, many of which arise from signicantly dierent ight environments. Particularly their atmospheric properties such as density, temperature, and composition invoke a great challenge. For instance, the thin, carbon dioxide Martian atmosphere does not allow conventional air-breathing propulsion systems, which forces NASA researchers to investigate alternative propulsion systems [56].
1.2
Energy Sources for Aviation
Although the recent issues described in the previous section address the need of the search for new aviation fuels, alternative-energy ights have been pursued for a long
13
time in aviation history. This section provides a compendium of the previous eorts and the current research status for alternative energy sources focusing on hydrogen, solar, and nuclear power as well as the evolution of conventional fuels. 1.2.1 Conventional Aviation Fuels
Conventional aviation fuels are classied into two general categories: aviation gas and aviation turbine fuels, both of which are hydrocarbon liquids obtained from crude oil. Aviation gas, often called avgas, is a gasoline-based fuel. Since the edging aviation engines rst used to power ight were built based on the automotive gasoline piston engines of the day, it is a natural outcome that they were fueled with automotive gasoline [58]. Such a gasoline-based fuel has been improved for antiknock properties by increasing the octane rating over the years. Currently, these types of fuel are used mainly for reciprocating piston engine aircraft and light helicopters5 . The advent of jet turbine engines introduced another type of aviation fuel. Aviation turbine fuels, often called jet fuel are kerosene-based. A major problem with gasoline is its higher volatility, characterized by a low ash point, the temperature at which it produces fumes that can be ignited by an open ame [59]. Gasoline has a ash point of around -45 degrees Celsius [60], which makes the aircraft vulnerable to catching re in the event of an accident or combat. Jet fuel has much higher ash point than avgas (e.g. minimum 38C for Jet A [61]), which results in less risk of re during handling on the ground and higher survivability in crashes. Compared with reciprocating engines that prefer a low ash point to improve their ignition characteristics, continuous combustion turbine engines are less sensitive to a ash point of fuel, and work properly on kerosene. In addition to its higher safety, jet fuels yield fewer losses due to evaporation at high altitudes. These types of fuel are used for
Currently the two major grades in use internationally are Avgas 100 and Avgas 100LL, low lead version of the former [58].
5
14
powering jet and turboprop aircraft for both commercial and military applications6 . These conventional aviation fuels produced from crude oil have powered virtually all machined ight for over a century. The ascendancy is not a result of fortuitous circumstances but recognitions of their inherent advantages, principally their high (both volumetrically and gravitationally) energy content as well as competitive prices, over other energy sources. Nevertheless, the advantages, as discussed in 1.1, have been gradually eclipsed by their adverse impacts on the environment and national energy security. 1.2.2 Hydrogen
Hydrogen as a substitute for conventional hydrocarbon fuel has attracted strong interest from the aviation community for more than half a century for several reasons. First, hydrogen has three times higher energy content per unit mass than conventional hydrocarbon fuels (120MJ/Kg vs. 40MJ/Kg, in lower heating value) [32]. Secondly, hydrogen can be extracted from fossil fuels such as coal, oil or natural gas, or can be obtained via electrolyzing water [12], which may drastically palliate national energy security concerns. Strictly speaking, hydrogen is not an energy source in itself but rather an energy carrier. That is to say that it must, itself, be made from a primary energy source, which is not necessarily imported. The ability to source such a great deal more primary energy domestically or to diversify energy supply sources has great potential to help reduce the U.S. energy security challenges [12]. Lastly, hydrogen has the potential of drastically alleviating emissions problems [32]. Should hydrogen be sourced from fossil fuels coupled with carbon sequestration or generated from renewable energy sources such as wind, solar, and geothermal energy, the net carbon emissions, in the sense of well to wing, could be reduced to near zero [12].
6 Two main grades of turbine fuel in use in civil commercial aviation: Jet A-1 and Jet A, both are kerosene type fuels. There is another grade of jet fuel, Jet B, which is a wide cut kerosene (a blend of gasoline and kerosene) but it is rarely used except in very cold climates. Three grades of turbine fuel in use in military aviation include JP-4, JP-5, and JP-8.
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Despite their noteworthy advantages, several drawbacks of the use of hydrogen as an aviation fuel exists. First, its low volumetric energy density (about three times lower than that of conventional aviation fuels) necessitates larger and heavier fuel tanks, which are likely to oset signicantly the advantages of its lower energy mass density. In addition, development of technologies to ensure sucient safety as well as the assurance of the public is a big challenge. The construction of infrastructures related to hydrogen production, distribution, and storage also may bring on numerous technical, economic, and political problems. Hydrogen can power an airplane via two dierent energy conversion processes: thermodynamic conversion through combustor or electrochemical conversion through fuel cell systems. The aboriginal research of hydrogen as an aviation jet fuel is found in a report [62] by Silverstein and Hall of the NACA-Lewis Flight Propulsion Laboratory published in 1955, which identied the potential of the use hydrogen as an aviation fuel. Their research gave a birth to an experimental program7 that was intended to demonstrate the feasibility of burning hydrogen in a turbojet engine at a high altitude in the next year. In the same year, Lockheeds Advanced Development Projects organization, better known as Kelly Johnsons Skunk Works, was awarded a contract to build two prototype reconnaissance aircraft designated as CL-400 that would be capable of cruising at Mach 2.5 at an altitude of 100,000 feet [2]. In spite of the success in developing practical solutions to the problems encountered in handling cryogenic liquid fuel, the program was terminated with no aircraft built. In the light of technological advancements in the past several decades, however, in recent years hydrogen fueled jet aircraft have met with a renewed interest in the United States. Europe also commenced an industry-wide investigation entitled Cryoplane - Liquid
A U.S. Air Force B-57 twin-engine medium bomber was modied to carry liquid hydrogen in a tank located under the left wing tip and made the rst ight by hydrogen in 1956. The converted aircraft climbed up to altitude and speed specied for the test using conventional JP fuel for both engines. Upon reaching test conditions, the convertible J-65 turbojet engine on the left-hand side switched over to hydrogen fuel. [2]
7
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Hydrogen Fueled Aircraft System Analysis in 2000 [7]. In addition to its use as a jet fuel, hydrogen could be used as the fuel source for fuel cells. Fuel cells are electrochemical devices that convert the chemical energy of a reaction directly into electrical energy [63]. The electrochemical process, as opposed to a combustion process, is not limited by Carnot cycle eciency, and fuel cell systems have the potential to achieve signicantly higher eciencies than IC engines do. There are many varieties of fuel cells including proton exchange membrane fuel cell (PEMFC) and solid oxide fuel cell (SOFC), but they are all related by a single common chemistry; generating electrical energy by the electrochemical oxidation of fuel [63]. While most fuel cells operate on pure hydrogen, some cells can operate on hydrogen-rich fuels such as methanol. Hydrogen fuel cells can also be operated on other hydrocarbon fuels if a reformer, which acts as a mini-renery to separate the hydrogen from the other elements in the fuel, is used along with the fuel cell [64]. The use of a reformer or direct methanol fuel cell could eliminate concerns over hydrogen production and storage, but would result in higher in-use emissions, compared to those of a hydrogen fuel cell [64]. Fuel cells have already been used to power electrical systems of spacecraft on every manned space ight of NASA [65]. Capitalizing on enormous advancement in specic power and power density in recent years, fuel cells began to be considered for aeronautical applications [5]. Several pioneering attempts by industry and academia have already been made to y an airplane on fuel cell-based electric propulsion systems as demonstrated with AeroVironments Hornet and Global Observer8 shown in Figure 6, as well as a fuel cell-powered UAV designed and constructed by Georgia Institute of Technology [67]. Currently fuel cells are mostly envisioned for low speed and long endurance applications, in which propellers driven by electric motors produce thrust.
After three years development, AeroVironment commenced ight testing of a subscale prototype of the liquid hydrogen-powered UAV having eight electric motors mounted along a 50 ft wing in 2005 [66].
8
17
Figure 6: AeroVironments fuel cell-powered aircraft: the Hornet (left) and Global Observer (right)
Fuel cells are also proposed as a power plant for magnetoplasma jet engine (called magjet) that is capable of air-breathing ight in the supersonic and hypersonic regime [68]. 1.2.3 Solar Energy and Beam Power
Solar-powered aircraft have also attracted the attention of several agencies over the past several decades because of their promising potential in military and civilian applications. This type of aircraft utilizes electric energy transformed from solar rays via PV cells. The most appealing feature of solar energy is that it can be obtained continuously during ight, and thus could yield a nearly fuel-less, emissions-free ight. Furthermore, if PV cells produce and store sucient extra energy during the daytime for ight at night, solar-powered aircraft could possibly y for a virtually unlimited duration. Another noteworthy advantage of a solar powered propulsion system over air-breathing engines is that available power is nearly insensitive to the variation of air density. Since the solar ray is attenuated as it travels through the atmosphere, its strength at high altitude is considerably higher than that at sea level. These two primary advantages of solar power are greatly valued for high altitude long endurance missions.
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Table 1 compares characteristics of various PV cells commercially available. Sunpower single crystal silicon PV cells are widely used for airplane applications such as Pathnder and Helios because of their superior specic power and eciency. Table 1: Characteristics of solar cells [69] Cell Type Si Si Si GaAs-Dual GaAs-Triple Manufacturer SunPower Mitsubishi Sanyo SpectroLabs SpectroLabs Weight/Area (Kg/cm2 ) 0.000081 0.00122627 0.000051 0.084 0.084 Pmax 0.01984 0.013054 0.00748 0.0295 0.0383 Eciency (%) 21.50 13.10 7.48 19.62 25.47 Cost/Watt ($) 10.32 12.90 6.03 Unknown Unknown 12.24
A number of solar-powered aircraft have been developed over several decades in the United States as shown in Figure 7. The rst solar aircraft was Sunrise I built by Astro Flight and own during the winter of 1974-75. It weighed 27.5 pounds, had a 32-foot wing span, and was powered by 450 watts provided from the solar cells [70]. By the fall of 1975, Astro Flight constructed an improved version called Sunrise II. This eort was followed by the development of the rst manned solar-powered aircraft, Gossamer Penguin, which used the solar panels from Sunrise II. Following Gossamer Penguin, Dr. Paul MacCreadys Solar Challenger, the solar cells of which could deliver over 4000 W at altitude and 2500 W at sea level, crossed the English Channel on July 7, 1981. In the same year, a classied program looked into the feasibility of long-duration, solar-electric ight above 65,000 feet, giving a birth to HALSOL (HighAltitude Solar Energy) built by AeroVironment. HALSOL proved the aerodynamics and structures of the approach, but subsystem technologies, principally for energy storage, were inadequate for the intended mission [71]. HALSOL was mothballed for ten years but later evolved into Pathnder [71] in NASAs Environmental Research Aircraft and Sensor Technology (ERAST) Program [72], which also spawned the 19
35
Power Delivered by Installed Solar Cells (KW)
Helios
30 25 20 15
Sunrise II Pathfinder Plus
http://www.dfrc.nasa.gov/Newsroom/ X-Press/images/083101/helios.jpg
10 5
Sunrise I Gossamer Penguin Solar Challenger
Pathfinder
0
1970
1980
1990
2000
Figure 7: Evolution of solar-powered aircraft
development of the Centelios and Helios vehicles. Despite the loss of the Helios in 2003, NASA has been pursuing a variety of options to continue the further development of solar and energy storage system technology for airborne applications [6]. These eorts are expected to lead to even more revolutionary HALE UAV aircraft capable of ying routinely as reliable atmospheric satellites on critical scientic and civil missions by 2010-2015 [6]. The European Space Agency is also promoting the development of a solar-powered airplane, the Solar Impulse, for which design and assembly are planned in 2007, followed by a rst ight attempt in 2008 [73]. In addition, solar power remains a promising energy source for the application of atmospheric ights on terrestrial planets where solar intensity is suciently high. For instance, a group of NASA researchers [74, 75] proposed a solar-powered aircraft designed to explore the atmospheric environment of Venus.
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1.2.4
Nuclear
Despite the appalling aftermath of its failure, nuclear energy has long been attractive to aerospace propulsion engineers. The most fundamental and compelling reason for the utilization of nuclear power in aerospace applications is that the nuclear explosive is the most compact of all known energy sources so far [76]. The attempt to operate aircraft on atomic power antedated eorts made to use hydrogen and solar power. Development of a nuclear-turbojet powered airplane was initiated by the Nuclear Energy for the Propulsion of Aircraft (NEPA) Project, launched by the United States Air Force (USAF) in 1946, and the follow-up Aircraft Nuclear Propulsion (ANP) Program, controlled by the joint Atomic Energy Commission (AEC) and USAF [77]. By the time President Kennedy delivered a statement canceling the program on March 28, 1961, about $1 billion had already been devoted to the development of a nuclear-powered aircraft for nearly 15 years [78]. In light of the advent of large commercial aircraft such as the Boeing 747, thanks to considerable advances in the airframe and engine, the USAF sponsored a conceptual study [1] for assessing whether an aircraft of one million pounds, with logical extensions of the technology, would allow for a useful payload capability with a fully shielded nuclear reactor installed. Annexed to the research, NASA also continued a low-level eort [79, 80] to determine and solve the major obstacles to practical, safe, and economical nuclear aircraft. However, to date, no aircraft has ever own with atomic energy. In contrast to their abortive eorts, nuclear power has been successfully implemented in the application of the U.S. Navys combatant eet. Currently, all submarines and nine aircraft carriers of the Navys aircraft carriers are nuclearpowered [81]. In addition, nuclear power has been widely used in space missions9 .
Since 1961, the United States has successfully own 41 radioisotope thermoelectric generators (RTGs) and one reactor to provide power for 24 space systems [82]. The former Soviet Union has reportedly own at least 35 nuclear reactors and at least two RTGs to power 37 space systems [82].
9
21
In recent years, nuclear power has been reinvestigated for extremely long endurance reconnaissance ights [83] as well as space exploration [84, 85]. The use of nuclear technology in the aerospace sector faces formidable barriers of public acceptance, however, especially if employed in airplanes. Therefore, the development of nuclear powered aircraft is contingent upon resolving a paramount safety issue and acquiring public acceptance. 1.2.5 Other Alternatives
In addition to solar energy, hydrogen, and atomic energy, several energy sources have been envisioned for alternative aviation fuels. Bob Saynor et al. [86] examined, in their research named The Potential for Renewable Energy Sources in Aviation (PRESAV) Project, the feasibility of several energy sources and identied Synthetic Fischer-Tropsch kerosene and biodiesel as those that warrant further detailed studies. Liquid methane has also been considered one of the options for aircraft engines. However, the use of liquid methane would result in a minimal reduction in pollutants. Moreover, the methane production peak is expected to occur a few years after the crude oil production peak [32]. Considering the expected price escalation, the complexity and cost of changing the infrastructure, methane is not considered to be a viable alternative. A cohort of researchers also introduced an interesting non-chemical propulsion concept of beaming power, which is based on wireless transmission of electrical energy, thus ending up with a nearly fuel-less, emissions-free ight with extremely long endurance capability. Compared with solar-powered aircraft, the operation of which is heavily limited by geographic location, especially latitude, due to the availability of solar light, the operation of beaming powered aircraft is more likely to be exible. Motivated by such potential advantages, researchers in several countries ew a variety of model aircraft using beamed microwave energy twenty years ago [87]. However, the
22
intrinsic nature of the microwave beam causes it to dissipate with distance, resulting in a commensurate decrease in power delivered to the target. With only lasers left as a potentially feasible option for practical power beaming, a group of researchers proposed the concept of a laser-powered transportation system [88]. Beamed power has been rigorously reinvestigated for many potential aerospace applications, including high-altitude airships, extra-terrestrial robotic rovers and aircraft, and small or swarming unmanned aircraft [84, 87]. Batteries and ultra capacitors are also being increasingly considered as preferable energy storage systems for a certain class of aircraft. Despite their remarkable power draw capability, the poor energy density of batteries and ultra capacitors have stymied them from appearing as a promising energy carrier for airplanes and even automobiles. Capitalizing on remarkable technological advancement in the past decade, however, batteries, especially lithium-polymer batteries, have shown their great potential as demonstrated by small radio-controlled (RC) aircraft and Micro Aerial Vehicles (MAVs). Batteries are expected to expand their usage such as booster power to subsidize primary power source as envisioned for the Boeings Fuel Cell Demonstrator Airplane [89] and a rechargeable energy source to sustain overnight ights for an extremely long endurance solar-powered aircraft as demonstrated in AC Propulsions solar powered unmanned aerial vehicle, the SoLong [90], shown in Figure 8.
Figure 8: Examples of battery-powered aircraft: Boeings Fuel Cell Demonstrator [89] (left) and AC Propulsions SoLong [91] (right)
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1.3
Motivation
When consent to transition to alternative energy sources is given, the following questions immediately arise: when will the transition happen?; what alternative energy sources should take the place of hydrocarbon?; and how can the transition be achieved? [2] Answering the questions is relevant, but not the scope of this research. In fact, those three questions are casting numerous research doubts on the whole society, because the transition is expected to bring tremendous impact on all aspects of the entire world. Fortunately, rigorous research during the past decades has led to a large consensus that hydrogen is very likely to be the answer to the second question. A transition from hydrocarbon fuels to hydrogen, or any other alternative, is expected to be a very complex, enormously expensive and arduous process even if all the necessary technologies were already available [32]. It must also be noted that the transition involves not only the development of new vehicles but also the construction of infrastructures related to the production and distribution of alternative energy, thereby inexorably aecting various aspects of our society and ecosystem. For example, zero-emissions aircraft, which store virtually all by-products on-board, are often conceived of as being nearly innoxious to our environment. Although the air-vehicle itself can approach the idea of an emissions-free energy source well, emissions are still produced during the production of the energy source and in the manufacture of the energy production plant and vehicle [31]. Therefore, in order to develop technically feasible, economically viable, environmentally acceptable new energy sources and aircraft systems, the government, academia, and private industry beyond the aerospace sector must interact as a whole and vigorously implement a holistic strategy addressing these challenges through increased levels of inter-agency collaboration [92]. To facilitate such large scale collaboration, a quantitative assessment environment that provides insight into the system-wide responses of alternative technology and policy evolution scenarios, as illustrated in Figure 9, would be valuable, thereby 24
increasing the possibility of arriving at more informed decisions. One critical aspect of such a quantitative assessment environment is aircraft sizing, which determines the overall size of aircraft and installed engines. Providing such a fundamental basis for most design and analysis activities, including internal layout, cost analysis, signature analysis, and system eectiveness analysis, it is considered a prerequisite task during the conceptual design phase. For instance, one of the results of aircraft sizing, the initial estimation of thrust or power required, is a primary input in the preliminary investigation of the engine company, especially if a new propulsion system is jointly developed. Therefore, aircraft sizing is the gateway that bridges technologies of the alternative propulsion systems and alternative energy sources to the assessment of impacts on our environment and economy. As discussed in 1.1, an alternative aviation energy source is being pursued to address not only global issues entailing a worldwide transition of the primary energy source, but also emerging scientic or military needs seeking a specic solution for an unique mission. Nevertheless, aircraft sizing is still regarded as a critical capability that is required to select the most appropriate energy source and system architecture in performing the given mission. Through a century of aircraft development, the traditional method for aircraft sizing has fully matured, and it is well understood. Nevertheless, it is doubtful that traditional aircraft sizing methods are immediately applicable to the sizing of revolutionary aircraft powered by a wide range of alternative energy sources, which is the aboriginal motivation that initiated this research.
1.4
Dissertation Overview
This introductory chapter has discussed the impetus toward the transition to alternative aviation energy sources, promoted by a combination of environmental, technological, economic, and national security issues. It has also reviewed that such rigorous
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Aircraft Sizing and Synthesis Vehicle System Level Subsystem Level
Global System Level Transportation System Level
Ecological Impact Economics National Security
Figure 9: A notional system of system analysis structure for assessing the impacts of alternative energy sources on all of society
eorts of implementing alternative energy (fuel) sources to aviation may necessitate improving aircraft sizing methods, which originally motivated this research. The remainder of this dissertation is organized as described in Figure 10. Chapter II surveys and summarizes the literature regarding the aircraft sizing process as it exists today and the eorts of many researchers to improve the aircraft sizing process to apply it to aircraft operating on alternative energy sources. The literature survey also identies the deciencies of current practice in the designs of alternative propulsion aircraft, which warrant the need for an advanced aircraft sizing method, as conceived in the previous chapter. Identifying what capabilities the emerging method must possess to ll the gap between the traditional sizing method and the emerging unconventional aircraft concepts, Chapter III establishes the research objective of this dissertation. The next logical step is to identify research questions to be answered in order to achieve the research objective. This eort crystallizes the research questions into how to develop two capabilities: 1) aircraft sizing methodology independent of the architectures of
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Chapter I : Introduction
Chapter 2: Aircraft Sizing and Synthesis
2.1 Definition of Aircraft Sizing 2.2~3 Classical Methods
2.4 Deficiencies
Chapter 3: Research Objectives, Questions, and Hypothesis
3.1 Research Objectives
Appendix A and B
3.3 Hypothesis Hypothesis 1 Hypothesis 2
3.2 Research Questions Question 1 Question 2
3.4 Substantiation of Hypothesis Hypothesis 1 Hypothesis 2
Chapter 5: Formulation of Probabilistic Aircraft Sizing Method Appendix C
Chapter 4: Formulation of Architecture Independent Method
Chapter 6: Implementation to Aircraft Sizing
Appendix D and E
Chapter 7: Conclusions and Future Work
Figure 10: Thesis structure overview
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energy and propulsion systems; and 2) intelligent allocations of opposite design margins against uncertainty. Subsequently, the hypotheses for the corresponding research questions are surmised and translated into appropriate mathematical representations. Built upon the substantiation of the hypotheses, Chapter IV and Chapter V present the formulation of the Architecture-Independent Aircraft Sizing Method (AIASM) and the Probabilistic Aircraft Sizing Method (PASM), respectively, as the solutions to the research questions. In the course of the development of AIASM, Chapter IV also introduces two expedients useful for rst-order estimations of aircraft performances: generalized Breguet range equations and the Non-dimensional Aircraft Mass (NAM) ratio diagram. In addition to PASM, Chapter V also discusses several solution techniques suitable for solving the formulated problem and several extended topics such as the application to multidisciplinary design optimization problems and probabilistic sensitivity analysis techniques to enhance the proposed method. In order to demonstrate the usefulness of these methods, Chapter VI presents the implementation of the proposed method into the sizing of fuel cell-powered electric general aviation aircraft and solar-powered electric HALE aircraft with regenerative fuel cell propulsion. Finally, Chapter VII provides the conclusion of this research and discusses future work envisioned to reinforce the proposed methods.
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CHAPTER II
AIRCRAFT SIZING AND SYNTHESIS
Discussion of the previous chapter has directed our focus to aircraft sizing. The primary purpose of this chapter is to review classical aircraft sizing methods as well as recent eorts to improve them and to identify the deciencies in their applications to alternative-energy ights. 2.1 contains a compendium of denitions and underlying physics of aircraft sizing. Subsequently, 2.2 reviews traditional aircraft sizing methods. 2.3 extends the discussion to how the traditional methods account for the implication of uncertainty inherent to the aircraft sizing process. Upon all the discussions, the last section summarizes the shortcomings of traditional sizing methods.
2.1
Denition of Aircraft Sizing and Synthesis
One of the imperative tasks performed during the conceptual design phase, aircraft sizing is well known to the aerospace community. Therefore, reminding readers of the meaning and concept of aircraft sizing tends to be commonplace. Nevertheless, the clarication of its denition is crucial in elucidating the remainder of this dissertation. According to Raymer [93], aircraft sizing is the process of determining the takeo gross weight and fuel weight required for an aircraft concept to perform its design mission. However, this denition does not contain sucient anity for the word, sizing. If determining aircraft weight is a main concern, it should rather be called weight estimation. In addition, his denition unevenly emphasizes mission fuel estimation, which is important, but not predominant in the process. DeLaurentis [94] describes aircraft sizing as a mathematical algorithm that determines the size and
29
weight of an aircraft based on a specied mission and contributing disciplinary analyses. This denition states the process is to determine the aircraft size as well as its weight, which inspires another question: what is the aircraft size? This question may be answered by reecting on how an airplane comes onto the design board from the creative imagination of designers. At the beginning of conceptual design, designers create and evaluate various congurations and select one or several baseline congurations for detailed evaluations. Each of the conceptual designs is created at rst in the form of a simple sketch that conveys general ideas for the airplane, leaving the following signicant questions: 1) how big the aircraft should be; 2) how powerful the engine should be; and 3) how heavy the aircraft will be. Three quantities: aircraft size, thrust and weight, given as the answers to the above questions, are essentially important in evaluating the goodness of a conguration, because they strongly aect the cost, including acquisition cost and operation cost. In addition, these three quantities are prerequisite information to follow-on design activities. For instance, without knowing the size of the aircraft, control surface sizing and internal layout are not possible. Another important aspect is that the three questions should be answered simultaneously since they are interdependent. If the aircraft size needs to be changed, the aircraft weight changes, and thus, the required thrust must change because drag and weight increase, which in turn increases aircraft weight due to bigger engines. The chain of this impact propagation will keep moving on until the quantities converge toward a solution, which is the objective of the aircraft sizing process. The next logical question would be the following: what drives these three quantities to a converged solution? Or, what is the criteria of the convergence of the solution? The aircraft must achieve three criteria: power (or thrust) matching, energy matching, and volume matching. Power matching is referred to as a balancing
30
act between the required power and the available power. The required power is dictated by the point performance requirements that specify the ability of performing certain maneuvering motions such as take-o, climb, sustained turn, instantaneous turn, acceleration, cruise, approach, and landing. The required energy is dictated by the mission performance requirements that specify the ability of performing a series of motions. In other words, the point performance requirements establish the demand for producing force or power, while the mission performance requirements establish the demand for containing energy. Volume matching is referred to as a balance between the required volume and the available volume. Traditionally, volume balance is veried through more detailed studies of the internal arrangement after aircraft geometry is initially established through the aircraft sizing process. In the case of traditional aircraft design, however, the volume balance is implicitly secured to a certain degree via the application of familiar (in the sense of evolutionary, not revolutionary) congurations and empirical weight estimations without the direct assessment of volume balance, simply because all existing aircraft, whose weight data are used to construct the regressed equations, contain all subsystems, structures, and fuel. In addition, it is not too far-fetched to regard the aircraft being designed as a small perturbation from the historical trend. For this putative reason, this section focuses on thrust balance and fuel balance. In this process, aircraft size and thrust are what the designer can control, while aircraft weight is what is to be computed accordingly, which means aircraft size and thrust are design variables of the aircraft sizing process. In this context, changing the size is referred to as photographically scaling up or down the notional conguration, which results in a geometry of the nal conguration that is not necessarily congruent to but similar to the initial conguration. In such a sense, aircraft size can be determined by setting wing area. Similarly, changing thrust is referred to as scaling rubberized engines. Therefore, aircraft sizing is to determine two scales of the given
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Notional Concept
Aerodynamics Propulsion Systems Empirical Weight Equation
Point Performance Requirements
Constraints Analysis
Mission Analysis
Balancing Thrust
TOFL Climb Acceleration Cruise Sustained Turn Service Ceiling etc.
Balancing Fuel
Fuel Fraction
Mission Performance Requirements
Mission Range Endurance Payload etc.
Thrust to Weight Ratio Wing Loading
Weight Estimation
Synthesis
Components Weight
Sized Configuration
Wing Area Thrust Weight
Figure 11: Illustration of the fundamental concept of aircraft sizing
concept: a geometric scale that is dictated by the wing area and a propulsive scale that is dictated by the amount of engine thrust. Aircraft sizing is often misconceived as aircraft conguration optimization, which seeks an optimum aircraft shape yielding the minimum aircraft weight and cost. Aircraft sizing is to determine the scale of the given conguration and not to attempt to modify the given conguration. Instead, aircraft sizing is a crucial element of conguration optimization. Most multidisciplinary design optimization (MDO) studies nd optimum aircraft design by wrapping around an aircraft sizing code with an optimization tool. Nevertheless, an optimization technique may be involved with the aircraft sizing process. Because a myriad of combinations of geometric scales and propulsive scales may satisfy design constraints, an optimization process is required to nd the
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best combination of the two scales. However, it must be noted that the optimization process tries to modify neither aircraft shape nor propulsion system characteristics, but two scaling parameters. In summation, aircraft sizing is dened herein as: an analytical process that determines the best combination of two scales of a baseline conguration, a geometric scale that is dictated by the wing area and a propulsive scale that is dictated by the amount of engine thrust so that the resultant aircraft should satisfy the three criteria: matching power, energy, and volume. Thus, the aircraft sizing problem can be formulated as a deterministic constrained optimization problem, in which the objective function, denoted as f , is minimized by varying design variables, denoted as x, subsect to a set of design constraints as follows: min f
x
s.t. T |available T |required WF |available WF |required
(1)
where T and WF denote the amount of thrust and fuel, respectively. The design variables denoted as x typically include available thrust (T |available ), wing area (S), and available fuel (WF |available ). In general, take-o gross weight or design gross weight is selected as the objective function of an aircraft sizing problem. The objective of aircraft sizing may dier depending on the mission objective of the aircraft and the scope of the design study. The ultimate gure of merit (FoM) for military aircraft would be the mission eectiveness or the exchange ratio of the aircraft. The FoM for commercial transport is often chosen among monetary metrics such as Return of Investment (ROI) and the average yield per Revenue Passenger seat Mile ($/RPM). However, the selection of such metrics as the objective function entails the integration of relevant analyses with aircraft sizing processes. The diculty in using economic cost as a design FoM 33
in aerospace vehicle design is that cost arises in a myriad of forms and can be quite complicated to estimate in its totality. This makes it unwieldy for use as a day-to-day design FoM in an engineering environment unless only the top few factors contributing to cost are tracked while the remaining many factors are ignored. These top few factors contributing to cost are well known in the aerospace industry and are strongly correlated to vehicle gross and empty weight. Therefore, aircraft weight is a good index on cost, and by extension, a good FoM for vehicle design [95]. The rst constraint in Eq. (1) represents the criterion of power balance, and the second one represents the criterion of energy balance. The available power in the power balance criterion is dictated by the characteristics of the propulsion system, and the required power is dictated by the performance requirement and airframe design. The available energy in the energy balance criterion is dictated by the amount of onboard fuel, and the required energy is dictated by the mission requirements, airframe design, and the eciency of the propulsion system. The mathematical formulation of the aircraft sizing process is based on an application of Newtons Second Law to the motion of an airplane, of which three-dimensional geometric characteristics and the properties of internal systems and structures are abstracted as a handful of parameters such as drag polars and mass properties. In reality, however, an aircraft is a complex system whose subsystems must meet physical and functional compatibility requirements for each other so that the whole system can work in harmony. The recomposition of these elements into an integrated whole is known as synthesis [96]. In such a sense, implementing proper synthesis increases the realism of aircraft sizing by accounting for the interconnections between the designs of elements. Synthesis comes along with aircraft sizing in many dierent ways. Jenkinson et al. [97] views synthesis as the designers eort to make an initial conguration for the aircraft sizing process balanced and workable by incorporating knowledge from all pertinent disciplines. Experienced design engineers are likely to come up with a less
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controversial, initial conguration of a notional aircraft that avoids any foreseeable technical issues by their engineering intuitions and previous experience. For example, they try to put the wing at a right location in consideration of many aspects such as stability and control, landing gear arrangement, and structural integrity. They also might consider various aspects regarding cockpit design such as vision envelopes and signatures as well as aerodynamic characteristics, although they do not precisely assess the design. Another important aspect of synthesis with respect to aircraft sizing is to provide guidance for a more realistic estimation of components weight. For instance, the weight of the landing system can be more accurately calculated along with determining its geometric size to provide sucient ground clearance. The degree of synthesis is limited by available information and design maturity. The scarcity of information on the aircraft system in the conceptual design phase makes only very low level of synthesis practical. However, the interconnections between major components must be captured and implemented into the aircraft sizing process. For example, the amount of power extraction from the main engine for subsystems may be signicant for some aircraft equipped with a high electric powerdemanding payload. In such a case, power extraction must be properly taken into account for sizing the propulsion system and the amount of required fuel. Otherwise, the results of aircraft sizing will not be realistic.
2.2
Traditional Sizing Methods
A copious anthology of literature presents dierent aircraft sizing methods. Some authors [98] discuss the aircraft sizing process in the course of explaining the development of conguration such as the aircraft shape, notably wing geometry. If only the parts regarding aircraft sizing from the textbooks are compared, however, a remarkable conuence in their logic and approach is found, despite some dierences
35
in details such as terminology and the sequence of the process. Therefore, review of the traditional methods herein is focused on one representative method by adding notable dierences from other methods. Mattinglys method is selected as just such a paragon method. 2.2.1 Overall Process of Aircraft Sizing
The overall process of Mattinglys method is pictorially described in Figure 12. The process is initiated by developing a notional concept. Determining the two scales, represented by wing area and thrust, of the given concept is the goal of the process. With its specic dimensions left undetermined, the initial concept encapsulates three major attributes related to aerodynamics, propulsion, and empty weight that are the basic inputs of the process. Therefore, those data must be prepared by preceding analyses before the process gets started. The process is comprised of three major parts: constraint analysis, mission analysis, and weight estimation. Constraint analysis establishes thrust balance by selecting thrust-to-weight ratio and wing load in the feasible solution region that avoids the violation of performance requirements. Mission analysis estimates the required fuel amount normalized by the take-o gross weight, which leads to fuel balance. These analyses, however, do not directly determine the size of the aircraft but secure thrust balance and fuel balance, because the analyses provide the information of wing area and thrust in normalized-by-weight forms, and aircraft weight is not yet determined at this point. Weight estimation combined by the information of the thrust-to-weight ratio and wing loading ultimately determined the three quantities: thrust, wing area, and aircraft weight. 2.2.2 Inputs of the Aircraft Sizing Process
As mentioned previously, the aircraft sizing process is performed with a specic airvehicle concept in mind. The initial concept does not have to be contrived, but must be specic enough to address all attributes of an imaginary aircraft required for the
36
Aerodynamics
CL CL
Point Performance Requirements
CD
Constraints Analysis
TST/WTO
Propulsion
Thrust Lapse
Gross Thrust
Alt.
Delta Drag
T SL W TO
WTO/S
W TO S
Alt.
TSL
Mach
Mach Fuel Flow
Fuel Flow
Fuel Flow
Alt. Mach
Weight Fraction
Notional Concept
Mission Analysis
Net Thrust
S WTO
Mach
T = f (m) WEng = g (m)
Scaling Law
Alt
WTO
WE , WF , WPL
TSL WTO
S WF
Thrust (sea level static) Take-off Gross Weight Wing Reference Area Fuel Weight Empty Weight Payload Weight
Weight Historical Data Empirical Eqs. W TO=(WE)
Figure 12: Mattinglys aircraft sizing process
sizing process. Most of such attributes are contained within the aircrafts external conguration and propulsion system characteristics. The former is developed based largely on engineering intuition and prior knowledge, and establishes basic aerodynamic characteristics, such as drag polar data and lift curves. The latter includes thrust and specic fuel consumption (SFC) variation with altitude, Mach number, and power settings. Most aircraft design projects begin with a rubberized engine that can be scaled up or down so that it matches the thrust required by the mission. Therefore, the thrust lapse behavior through ight conditions and power settings, other than the gure of the available thrust itself, is precisely what is required. In addition, empirical weight equations that are able to estimate empty weight are also required. Such equations were generally developed from regression analysis of the data extracted from large numbers of existing aircraft. The large diversity in the aerodynamic conguration, structural layout, materials, including subsystems of todays aircraft, does not allow universal weight equations, hence the weight equations were developed from a certain type of aircraft. Therefore, it is crucial to select correct 37
Radius
Mission Performance Requirements
WE WPL
weight equations that are properly customized to the class to which the aircraft belongs. Data sets of aerodynamics, propulsion, and weight equations described above are the input data of the aircraft sizing process and their delity ultimately determines the level of delity of aircraft sizing. 2.2.3 Thrust Balance
Ibrahim [99] proposed a method that determines wing load and thrust-to-weight ratio using semi-empirical relationships for performance measures, such as take-o distance, maximum rate of climb at sea level, maximum level speed, and landing distance, developed from historical data of military jet trainers and light attack airplanes. His method starts with developing an assumption regarding the response of a performance measure with respect to thrust to weight ratio and wing loading, each denoted as TSL /WTO and WTO /S, respectively. For example, the author propositioned the following relationship between take-o ground run denoted as STD and design parameters. STD 1.8 (T (WTO /S) SL /WTO )CLmax.T O (2)
where CLmax.T O and denote the maximum take-o lift coecient and ambient air density ratio, respectively. For a designated level of technology, it is assumed that the order of magnitude of the maximum lift coecient in the take-o conguration for a category of aircraft remains the same. Accordingly, STD (WTO /S) 1.8 (T /W ) SL TO (3)
Finally, based upon this relationship, a regression analysis from a historical database of military jet trainers and light attack airplanes results in a linear equation describing the take-o distance in terms of TSL /WTO and WTO /S as follows: STD = 167 + 1.029 38 (WTO /S) 1.8 (T /W ) SL TO (4)
The author developed semi-empirical equations for other performance constraints in a similar way. This method provides a comparatively accurate estimation of required thrust and wing loading only when a large amount of sample data for existing aircraft built with similar level of technologies is available. Thus, this method is not immediately applicable to the design of aircraft built with an unconventional conguration and/or a dierent level of technologies, for which sucient historical data do not exist. Mattingly [100] proposes a more systematic approach known as the constraint analysis for thrust balance. The equations of constraint analysis can be derived directly from the consideration of dynamic equilibrium of aircraft motion as shown in Figure 13. The velocity of free stream air has an angle of attack (AoA) to the wing chord line (WCL). Lift (L) and drag (D + R) forces are normal and parallel to this velocity, respectively. Thrust (T ) is at an angle () to the WCL. The application of Newtons Second Law to the aircraft motion yields the two following equations that describe the relationship of the net forces versus acceleration perpendicular and parallel to the velocity vector, V , respectively. T cos(AoA + ) W sin (D + R) = W dV W a = go go dt W a go (5) (6)
L + T sin(AoA + ) W cos =
where D is basic conguration drag; R is additional drag to D due to the changes of the conguration such as deecting control surfaces, extracting landing gear, and installing ordnance such as weapons and external fuel tanks; h is the ight altitude; V is the free-stream velocity; and go is the gravity constant. Multiplying Eq. (5) by V leads to the following equation: {T cos(AoA + ) (D + R)}V = W {V sin + d dt V2 } 2go (7)
Note that for most ight conditions the thrust vector is very nearly aligned with the direction of ight, so that the angle (AoA + ) is very small, which allows one 39
L WCL V AOA T
D+R WCL
W
Figure 13: Forces on aircraft [100]
to assume cos(AoA + ) 1. In addition, V sin is simply the time rate of change in altitude (h), known as the rate of climb (for positive values) or the sink rate (for negative values), given as V sin = dh dt (8)
Combining Eq. (7) and Eq. (8) yields the following equation. (T (D + R)) V = W dh W d + dt go dt V2 2 (9)
This equation describes the dynamic equilibrium between power input and the rate of change in the potential energy of the aircraft. The left-hand side of the equation represents the rate of net mechanical energy input to the aircraft, which is often called excess power. The right-hand side of the equation represents the time rate of change in the sum of kinetic energy and potential energy of the aircraft. Dividing Eq. (9) by aircraft weight, W , gives another equivalent equation: d (T (D + R)) V = W dt h+ V2 2go (10)
The left-hand side of the equation is called the specic excess power. This metric represents the ability of an aircraft to change its energy state, thus playing as one of
40
the important factors to determine the short range air combat capability of ghters. The right-hand side of the equation is called energy height, which is the sum of kinetic energy and potential energy normalized by the aircraft weight. Therefore, the equation states that the specic excess power of the aircraft equals the time rate of change in the energy height of the aircraft. Starting with Eq. (10), Mattingly derived a general form of constraint equations, named the Master Equation, that provides the desired relationships between the thrust-to-weight ratio and wing loading as follows:
2
TSL = WTO
qS K1 WTO
n WTO q S
+ K2
n WTO q S
+ CDo +
R 1 d + qS V dt
h+
V2 2go
(11)
where is the weight fraction; is the thrust lapse ratio; q is the dynamic pressure; K1 is the drag polar coecient for the 2nd order term; K2 is the drag polar coecient for the 1st order term; CDo is the zero lift drag coecient; and n is the load factor. With the master equation, Mattingly developed a number of constraint equations, each of which is customized to a specic performance constraint, such as take-o eld length, climb, acceleration, and sustained turn. The mathematical expression of the tailored constraints is listed in Appendix A. The beauty of this method is that it allows the designer to consider all performance constraints in one unied graphical environment. Most aerodynamic point performance measures, which include take-o eld length, climb, acceleration, sustained turn, and approach speed, can be expressed as functions of thrust loading (T/W) at sea level, and wing loading (W/S) for a given aircraft geometry, thrust lapse behavior and set of ight conditions. Thus, the feasible solution region is circumscribed by a set of constraint curves that represent each performance requirement as depicted in Figure 14. The design point is generally selected at the location of the lowest thrustto-weight ratio consistent with the greatest wing loading as illustrated in Figure 14(a), because lower thrust-to-weight ratios lead to smaller propulsion systems and greater 41
Thrust-to-Weight Ratio
Lowest Thrust-to-Weight Ratio And Highest Wing Loading
Thrust-to-Weight Ratio
Lowest Thrust-to-Weight Ratio
Highest Wing Loading
Wing Loading (a)
Wing Loading (b)
Figure 14: Notional constraint analysis diagrams
wing loading values lead to smaller wings [100], both of which always reduce aircraft weight and cost [93]. However, in some cases as illustrated in Figure 14(b), the lowest thrust-to-weight ratio may not be consistent with the greatest wing loading and a minimum point is not clearly identied. Such diculty can be resolved when the whole sizing process is mathematically optimized [101]. Another important aspect is to consider the proper design margins for the design constraints. However, a plethora of redundant thrust will most likely increase the propulsion system weight, and thus the take-o gross weight as a consequence without the proportional benet of reducing the risk. Therefore, the margins must be carefully determined so as to strike a good compromise between the risk and the impact on aircraft weight and cost. 2.2.4 Fuel Balance
Energy balance means the amount of on-board stored energy is no less than the amount of required energy to complete the mission. The equation that describes the amount of required energy can be established by integrating Eq. (9), as follows: P dt = (D + R)V dt + 42 W dze (12)
This equation states that the amount of energy added to the system by propulsive thrust work ( P dt) equals the sum of the amount of energy lost by the system due to work done by the system on its surroundings ( (D + R)V dt) and the increase in the conservative energy of the system ( W dze ) which is another representation of the energy conservation law that states: energy cannot be created or destroyed. Though containing the canonical idea of energy balance, this equation is not fairly practical to estimate the amount of required fuel as it is. First of all, since D, R, W , and V vary during the mission, it is not possible to obtain the simple closed form of Eq. (12). Furthermore, Eq. (12) describes the energy balance in terms of the mechanical energy of the aircraft as a point-mass with the exclusion of the energy conversion process taken inside of the aircraft. Therefore, even if we can estimate the amount of total energy by propulsive thrust work, P dt, from Eq. (12), in general we cannot directly
estimate from the information the amount of required fuel to complete the mission. This discussion reveals the need for another analysis that connects the time rate of fuel consumption (WF ) to the time rate of input mechanical energy (P )1 . The relationship is ultimately governed by thermodynamic laws, but can be described in simple form as follows: WF = ct T (13)
where ct , and T stands for thrust-specic fuel consumption and thrust, respectively. By assuming that the rate at which aircraft weight diminishes (W ) equals the fuel ow (WF ), Eq. (13) can be rewritten as T dW = ct dt W W For level ight, L = W and T = D. Thus, ct D dW = ds W V L
1
(14)
(15)
() notation represents dierentiation with respect to time.
43
Making an assumption that ct , V , and L/D are constant with their representative values during ight, Eq. (13) can be considered a rst-order ordinary dierential equation, which is solved for the boundary conditions of W (s = 0) = Winitial and W (s = R) = Wf inal with R= V L Winitial ln ct D Wf inal (16)
which is known as the Breguet range equation. For propeller aircraft, the Breguet range equation is given as R= pr L Winitial ln cp D Wf inal (17)
where pr and cp denotes propeller eciency and power-specic fuel consumption, respectively. A simple estimation of the required fuel, presented by several authors including Wood [98], is just to account for the fuel amount consumed for the cruise mission using the modied Breguet range equation as follows: WF Wf inal =1 = 1 exp WTO Winitial Rcp D pr L (18)
Since the greatest amount of fuel is consumed generally during cruise, this method is acceptable as a rst order crude guess. However, in the design of aircraft that must perform a complicated mission, notably, combat aircraft, this approach may yield signicant errors. A more accurate approach is to compute the required fuel fraction by calculating the product of all the weight fractions of discretized mission segments as shown in Figure 15, which is widely used in most sizing methods including Mattinglys method. The weight fraction of each mission interval can be obtained by one of a set of integrating dierential equations listed in Appendix B, which are developed from Eq. (14). Wf inal WF =1 =1 WTO Winitial where Wf inal
s (s)
Winitial
(s)
(19)
is dened as a product operator that multiplies all the values associated
with the variable in parentheses. 44
sth Segment Cruise s-1 Climb 1st Segment 1 Warm up Take-off and Taxi 0 Loiter n Landing s Descend
Figure 15: Discretized mission prole
2.2.5
Weight Estimation
The results from the two analyses are fed into the weight estimation module. Once the take-o gross weight as well as the T/W and W/S values are available, the amount of thrust and the wing area of the notional conguration can be established. The method of computing aircraft weight diers depending on the maturity of the design and the availability of the information. During early aircraft design, including most of the conceptual phase, simple empirical and parametric methods are widely used. Depending on the selected method, a dierent set of parameters may be required. However, the parameters used in such a method are limited to what can be predicted without detailed layout studies. The simplest method is to estimate the take-o gross weight simply by multiplying payload weight by a constant, in general, four [98]. More detailed methods break down the take-o gross weight into several weight groups; for example, empty weight, fuel weight and payload weight. Then, each group weight is calculated by appropriate methods such as numerical analysis or statistical estimation. Payload weight is usually available from the requirements; fuel weight is calculated by a mission analysis; and empty weight is estimated from a historical database. If most components and parts are modeled, then the element to element method may be used. This method 45
computes the aircraft weight by summing up the weight of all elements of an aircraft. The rst method is oversimplied. In contrast, the last method is not practical, since the aircraft components and parts are not designed at the time of aircraft sizing. Thus, the second method is widely used for aircraft sizing. The fundamental idea of this approach is that aircraft weight can be broken down into two parts: xed weight (Wf ix ) independent of the take-o gross weight such as the weight of the crew and payload, and variable weight (Wvar ) dependent on the take-o gross weight such as the weight of structures and subsystems. The former is known from the requirements and the latter can be predicted from a statistical database and/or relevant analyses. Therefore, a weight decomposition equation can be stated in generic form as follows [102]: WTO = Wf ix + Wvar Then, Eq. (20) can be reformed as follows: WTO = Wf ix 1 Wvar WTO (21) (20)
The take-o gross weight can be computed by Eq. (21), when the ratio of Wf ix to WTO can be estimated by utilizing statistical data. Components that Wf ix and Wvar account for may dier depending on the type of the aircraft. In general, Wf ix includes the weight of the payload and the crew, and Wvar includes empty weight and fuel weight. Then, Eq. (21) can be reformed as follows [103]: WTO = WPL + Wcrew WE WF 1 WTO WTO (22)
Even though Eq. (22) is one of the most common expressions of the take-o gross weight calculation, Eq. (21) must not be overlooked because it more eectively shows the bedrock idea of using a statistical database for the take-o gross weight in generic form. Multiple ways of estimating empty weight are available, depending on the availability of historical data. The simplest one is using a xed ratio of empty weight to 46
the take-o gross weight. For example, Anderson [103] determined the empty weight fraction as 0.62 for the design of ve-passenger, propeller-drive, business transport aircraft from an observation that the empty weight fractions of 19 airplanes covering the time period from 1930 to the present show a remarkable consistency of clustering around 0.62. However, this oversimplied method works only when the estimated weight is very close to the weight of the reference airplanes, because the empty weight fraction is aected by the aircraft weight. Thus, this method must be limited to an initial guess for the empty weight fraction. The accuracy of empty weight estimation may be improved by estimating the empty weight as a function of the take-o gross weight. WE = (WTO ) WTO (23)
However, this method is still too oversimplied to capture the impact of changes in wing area or engine thrust to aircraft weight. Therefore, a more advanced method is to compute empty weight as a function of engine thrust, wing area, and some other design parameters (p) as well as the take-o gross weight as follows: WE = f (WTO , TSL , S, p) (24)
A more complex method is to estimate the weight of components or subgroups separately. WE = Wcompi (25)
Selection of a method among various methods strongly depends on the availability of historical data. In general, a more detailed decomposition is preferred. However, it must be noted that employing a more detailed analysis does not necessarily improve the accuracy of weight prediction [104]. A more complex analysis requires more detailed information on the airplane. If accurate information is not properly provided, the results may be biased. Similarly, higher delity tools do not necessarily yield more accurate results, unless the aircraft design is matured enough to provide suciently 47
accurate information required for the tools2 . Therefore, the delity of analyses must keep pace with the maturity of designs, and the decision on a weight estimation method must be carefully made though prudent consideration of various factors. 2.2.6 Actual Value-Based Approach and Weight Specic Parameter-Based Approach
The aircraft sizing process generally includes three parts: point performance analysis, mission performance analysis, and aircraft weight estimation. There are two distinct ways of implementing these parts: the actual value-based approach and the weight specic parameter-based approach. The former uses actual values for thrust (power), wing area, and fuel quantity to establish thrust (power) balance and fuel balance, while the latter (Mattinglys method also belongs to this category) uses the parameters normalized or inversely normalized by aircraft weight, which are thrust-to-weight ratio, wing loading, and fuel fraction. In the actual value-based approach, one must estimate the take-o gross weight for computing point performances and mission performances. For example, the required thrust for level ight is given as lift-to-drag ratio times aircraft weight. Therefore, the process must iterate the three variables through constraint analysis and mission analysis as well as weight analysis until the solution is reached at the thrust and power balance as shown in Figure 16. In contrast, the weight specic parameterbased approach establishes thrust (power) and fuel balance in terms of thrust required per unit aircraft weight and the amount of fuel required per unit aircraft weight. Therefore, thrust balance and fuel balance can be established independently of the computation of aircraft weight, which means that the constraint analysis and the mission analysis do not need to be repeated due to updates of aircraft weight. Even
One example is that an initial weight estimation using an element-to-element method often yields larger errors than a parametric method based on historical data in actual aircraft development programs, even though the former is done by investing a substantial man-hours in engineering work including state-of-the-art computer-aided design and intensive structure analysis.
2
48
Constraints Analysis
~ T SL
S
~ S
TST
Weight Estimation
~ ~ (0 ~ ~ WE = f (WTO ) , TSL , S ,...)
~ (0 WTO )
Weight Fraction
~ (1 ~ ~ WTO) = WE + W f + WP
Mission Analysis
Alt
~ Wf
Radius
~ (1 WTO)
Yes
TSL
~ ( 0 ~ (1 WTO ) WTO) <
No
S WTO
Figure 16: Iterative process of actual-value-based sizing approach
if an iteration process is required to estimate aircraft weight, it is restricted to the inside of the weight estimation module as shown in Figure 17. This benet may not be noticeable for a one-time run, but it would be substantial for such applications that require a large number of simulations. If one wants to employ an optimization process to nd a solution to thrust, wing area, and fuel quantity that minimizes aircraft weight, however, the benet of the weight specic parameter-based approach over the actual value-based approach is no longer valid because the optimizer iterates the variables in anyway. 2.2.7 Iteration of the Aircraft Sizing Process
In addition to iterative processes described above, overall iteration is required for several reasons. First, the designer would want to improve aircraft design by changing the aircraft conguration and/or propulsion system design if the solution obtained from the process was not satisfactory. Alternatively, the designer would create an automated optimization environment by integrating the aircraft sizing process with disciplinary analyses to nd optimum aircraft conguration and subsystem design
49
Constraints Analysis
TST/WTO
TSL W TO
WTO/S
W TO S
~ (0 WTO )
Weight Fraction
Mission Analysis
Alt
T ~ ~ (0 T SL = SL W TO ) W TO ~ ~ (0 W S = W TO ) / TO S
Wf W TO
Radius
Weight Estimation
~ ~ (0 ~ ~ WE = f (WTO ) , TSL , S ,...)
~ (1 ~ ~ WTO) = WE + W f + WP
~ ( 0 ~ (1 WTO ) WTO) <
Yes
TSL
No
S WTO
Figure 17: Iterative process of specic parameter-based sizing approach
parameters. Another cogent reason for the overall iteration is that aerodynamic properties and propulsion system characteristics may change due to the scaling of aircraft geometry and engine thrust. Generally, aerodynamic characteristics stay constant in the process of scaling aircraft geometry up and down with the following the assumptions: 1. Aircraft is scaled photographically 2. Reynolds number eect is negligible within the range of scaling However, these assumptions may not hold in certain situations. In the design of a subsonic commercial airline, fuselage geometry is usually determined by an internal layout subject to strict volume requirements for cabin design. If such fuselage sizing is already reected on the notional conguration, changing only wing area makes more sense than photographically scaling the whole aircraft. Taking the former approach requires an update of aerodynamic properties at every change in the wing area of the 50
aircraft sizing process, since the aircraft geometry is not similar to the conguration of the previous iteration anymore [105]. Similarly, an assumption that the engine characteristics are only dependent on engine cycle and are not aected by scaling eects, is not always reliable. For instance, when more thrust is required than the level of a baseline turbofan engine, thereby enlarging engine geometry, fan tip speed will increase. As the tip speed of the fan approaches the divergence Mach number of its airfoil, aerodynamic losses will be substantial, and the real operating condition will diverge from what engine cycle analysis is based on. Therefore, once the aircraft sizing is done, one must assess the impact of the scaling eect on aerodynamics properties and engine characteristics and, if necessary, reiterate the whole aircraft sizing process after an appropriate update of aerodynamics properties and engine data.
2.3
Aircraft Sizing under Uncertainty
All design parameters involved in the aircraft sizing process can be neither accurate nor certain, particularly because aircraft sizing is performed during the pre-conceptual or conceptual design phase, when the least knowledge about the system is available. Therefore, how to account for the associated uncertainty sources is a crucial element in avoiding signicant rework during the subsequent design processes. 2.3.1 Uncertainty Sources
A multitude of sources of uncertainty are involved in the aircraft sizing process. One signicant source of uncertainty is the lack of model delity and the imprecise knowledge of the system. This type of uncertainty is unavoidable in the sense that it is impossible to develop a perfect mathematical representation of the behaviors of complex systems such as aircraft. Moreover, the desirable degree of delity is often compromised by the lack of time, resources, and the degree of design maturity, even when high delity models are available. Consequently, high delity models have
51
seldom been used during the early phases of aerospace systems design. Recent advancements in meta modeling have paved the way for accelerating the utilization of high delity models in conceptual design phases. Notwithstanding, the very act of approximation by meta models introduces yet another source of model uncertainty or errors. In addition, incomplete knowledge of the details of the system may also introduce a considerable amount of uncertainty. For instance, the precise amount of miscellaneous drag that accounts for additional parasite drag incurred by surface roughness, cavities, environment control systems, control surface gaps, and protuberances cannot be accurately calculated until a detailed description of their geometries, usually not available when aircraft sizing is performed, is established. Another source of uncertainty is airframe design evolution. Modications in the external conguration, often referred to as the Outer Mold Lines (OML), directly aect the aerodynamics of an aircraft. On the other hand, changes in the internal conguration may aect the weight of the aircraft. Such changes may collectively result in an increase or a decrease in the required power and energy for ight. The design changes may be attributed to various factors, most of which are motivated by the desire to resolve currently-known design issues. Performed in the embryonic stage of such a long evolution process, aircraft sizing inevitably involves uncertainty associated with design changes for the rest of the development period. While the majority of such changes occur during the conceptual design phase, the aircraft conguration continues to evolve after the aircraft is sized by eorts in the conceptual design phase, as shown in Figure 18, which summarizes the evolutionary process of the T-50 Golden Eagle developed for the Republic of Korea Air Force. As the design matures, the degrees of freedom to alter the design decrease, and the design is less likely to change. Nevertheless, both external and internal conguration of the aircraft will be subject to change through the rst ight, even until production. In addition to the airframe, the design requirements also evolve with time, which
52
Trade-Off Study
Wind Tunnel Test (6000 hrs) Air-Load Analysis Store Separation Analysis
Diffuser Analysis
Antenna Analysis
Pre-conceptual Design
Conceptual Design
Preliminary Design
Detail Design
12 Major Configuration Updates
101
Figure 18: Design evolution of the T-50 Golden Eagle [106]
53
201 301
104
406
506 Rev.C
Internal Load Analysis (FEM)
Flutter Analysis
Flight Simulation
Structural Test
Bird Strike Test
Thermal Analysis
introduces another signicant source of uncertainty in aircraft sizing. It is a popular misconception that design requirements are static. On the contrary, in almost all of the actual aircraft development programs, the requirements are seldom established rmly before design activities are initiated. They continue to evolve after being passed on to the design team. Particularly during the conceptual design phase, designers often explore the requirement space and the concept space while varying the requirements and design options in order to nd a correct combination of capabilities and cost. In the light of such interactive trade-o studies, customers or decision makers may revise the initial set of design requirements. However, the evolution of design requirements continues even after the conceptual design phase. Since knowledge of the system increases with design maturity, valuable compromises between capabilities and cost, not previously recognized, may become critical. In addition, notable changes in the market or the operating environment may force the management to update their requirements. Regulatory uncertainty refers to the risks inherent in obtaining any necessary licenses to construct or operate a project from the appropriate regulatory authority. Regulatory requirements may include safety regulations, environmental regulations, and maintenance regulations. Among those regulations, environmental regulations, in particular, emissions and noise regulations, have created signicant uncertainty, causing consternation for airframe and engine designers. Historically, environmental regulations have become increasingly stringent, and this trend should continue in the future due to the necessity for environmental protection and quality of life. In addition, most airplanes and engines are very likely to have an extended period of service life. During this time, present environmental regulations could be re-examined in light of technological improvements, and new regulations could be enacted, thereby forcing early product retirement unless these potential scenarios are accounted for adequately [107].
54
Propulsion System Performance Available Power Power Extraction Performance Requirement Drag Weight Available Energy Propulsion System Efficiency Usable Fuel Volume Required Energy Required Power
Figure 19: Impact of uncertainty on aircraft sizing
Implementing new technology also introduces uncertainty into the sizing process. For complex systems, the search for feasible and viable solutions often requires the application of multiple new technologies,[108] even though infusing technology may incur penalties in other disciplines as a price for the benets. Generally, the impact of a technology, the benet and the price cannot be precisely predicted in the conceptual design phase, especially if the technology is ranked low in terms of its technology readiness level (TRL), and if the propagation of technological impact is obscure. Revolutionary propulsion aircraft would be the case. Such advanced concepts will require years to develop, and the expected performance and system attributes could vary dramatically throughout their development. In addition to the uncertainty inherent in the propulsion system, the impact on other disciplines becomes more signicant and less predictable. 2.3.2 Traditional Approaches to Aircraft Sizing Under Uncertainty
As outlined in Chapter I, various sources of uncertainty are involved with aircraft sizing. What does it mean to size an aircraft in the presence of such uncertainty? How can the full impact of uncertainty be correctly accounted for? In order to answer
55
these questions, one needs to return to the denition of aircraft sizing. The goal of aircraft sizing is to nd the two scales, the geometric scale and the propulsive scale, that minimize the objective function, which is usually the take-o gross weight, while satisfying all requirements [109]. As illustrated in Figure 19, the sources of uncertainty permeate through both the available and required quantities, which indicates that both quantities must be treated as random variables, rather than deterministic design variables, for aircraft sizing. Therefore, the deterministic expression of the aircraft sizing problem given as Eq. (1) is rewritten as a probabilistic expression as follows: min f
x
s.t. Pavailable Prequired Eavailable Erequired
(26)
where ( ) indicates that the variable under the symbol is a random variable or a function of random variable(s). This problem is seeking an optimal solution of decision variables that minimize a probabilistic objective function under probabilistic constraints. Even though the aerospace community has been cognizant of the signicance of the probabilistic nature of design constraints and objective functions, a deterministic approach to aircraft sizing has been the dominant norm. The traditional approach to mitigate the risk associated with uncertainty is to add design margins as a hedge against foreseeable changes in the involved assumptions [107]. In the constraint analysis, designers choose the design point in the proximity of the active constraint curves, so that the movement of any of the constraint curves caused by revised estimates of aircraft and/or engine performance and requirements does not violate any of the design constraints. Then, the distance between the design point and a constraint curve represents the size of a design margin for the corresponding performance requirement(s). The margin for the mission performance requirements is usually not 56
considered in the aircraft sizing process in which the weight of aircraft is calculated with the assumption that the amount of available fuel equals that of required fuel. Such a margin, however, can be included by adding an allowance to the amount of required fuel. In fact, a military specication known as MIL-DTL-7700G reads that 5 percent penalty in fuel mileage must be accounted for in estimating the amount of required fuel. Nevertheless, this surplus fuel amount is intended to account for engines with poorer-than-normal fuel consumptions due to manufacturing tolerances on the production line or fuel mileage deterioration. Therefore, the allowance cannot be regarded as a design margin for uncertainty surrounding aircraft design activities such as the lack of analysis delity or changes in drag and weight due to design evolution. The margin for mission performance is accounted for by maintaining the extra available fuel in the following design process. As stated earlier, the amount of available fuel, the same as that of required fuel, is assumed to be obtainable in the conceptual aircraft that is embodied by preliminary internal layout studies, a design task following aircraft sizing. Then, the amount of available fuel is veried by calculating the available fuel volume. If the conrmed amount of available fuel is less than the amount of required fuel, designers must take several measures to rectify this issue. The rst is to rearrange the internal layout to acquire sucient fuel volume. If these eorts end up failing, the designers may need to reshape the external conguration to accommodate more fuel volume or to resize the aircraft to satisfy the criterion of volume balance. On the other hand, if an extra amount of available fuel is identied, the designers may want to make the conguration slimmer to improve the aerodynamics and/or reduce the aircrafts weight. However, the designers usually try to maintain sucient, but not extraneous, surplus fuel volume as a margin during the conceptual design phase. These activities continue throughout the preliminary design phase, when the external conguration and internal layouts continue to evolve,
57
and the amount of required fuel varies at every update of aerodynamics and weight databases, until a detailed internal arrangement is completed. In summation, the traditional approach has been converting the probabilistic problem given as Eq. (26) to the following deterministic problem by adding predetermined design margin: min f s.t. Pavailable = Prequired + P Eavailable = Erequired + E 2.3.3 Recent Probabilistic Approaches to Aircraft Design (27)
In the past decade, the aerospace community has made signicant progress in nondeterministic approaches to aerospace systems design. In particular, the Aerospace System Design Laboratory (ASDL) at the Georgia Institute of Technology has advanced various probabilistic design methodologies. Mavris, Bandte, and DeLaurentis introduced a probabilistic approach called Robust Design Simulation (RDS) to aerospace systems design [110]. The method was originally developed for determining a robust design solution that is the most insensitive to noise (random) variables. Since its conception, the method has been implemented in many applications, including the conceptual design studies of a high-speed commercial transport (HSCT). Even so, the RDS method is not entirely suitable for aircraft sizing under probabilistic constraints for several reasons. First, the uncertainty sources that the method can implement are limited to economic variables such as load factor, fuel cost, and learning curve. The propagated impact on system level metrics due to the variability of economic (noise) variables is still computed in a deterministically-sized aircraft. Secondly, the method seeks a solution that maximizes the probability of meeting a single, static, overall evaluation criterion. However, the constraints involved with aircraft sizing are multiple, interdependent, and probabilistic in nature. 58
Screening
Design/Control Variables
Baseline Configuration
Economic Noise Variables
Design of Experiments for all Variables
Synthesis Code (FLOPS/ALCCA)
Response Surface Equation For Objective Function ( /RPM) and Constraints (Fuel Requirements, Approach Speed, Landing, and Take-off Field Length)
Design of Experiments For Control Variables
Monte-Carlo Analysis
Variability Distributions for Noise Variables
Response Surface Equation For Probability of Achieving Values Below the Target
Maximize Probability of Achieving Values Below the Target
Figure 20: Overall process of Robust Design Simulation [110]
59
Alternatives Criteria (Z) Objectives/Requirements
Baseline
Control Variables x=(0.5,4,0.07,) Certain Criterion Values (zmin, zmax)
Uncontrollable/ Noise Variables Y Uncertain
Weights (w)
Requirement Trade-off Change Preference
Simulation Z=f(x,Y)
Optimization Iteration
Joint Probabilistic Decision Making Product Selection
Criterion 2 Criterion 2
Joint Probability Distribution (Five Methods)
Optimization
Five Methods
p O
Criterion 1
ize tim
MCS EDF
MCS JPM
RSE MCS EDF
RSE MCS JPM
AMV JPM
Criterion 1
Figure 21: Bandtes JPDM process [112]
Bandte proposed the Joint Probabilistic Decision Making (JPDM) method, which is a probabilistic, multi-criteria optimization approach for aerospace system design [111]. Unlike RDS, JPDM utilizes the joint probability that the system can simultaneously satisfy multiple design criteria to nd the design solution. The method was applied to various aerospace problems, such as the evaluation of the feasibility of design spaces, design optimization, and product selections. Still, this method is not immediately applicable to optimization problems with probabilistic constraints, such as probabilistic aircraft sizing. One of the primary reasons is the dierences in types of the criteria according to which the probability of success is evaluated. The criteria used in JPDM are determined at xed values, while the constraints used in the aircraft sizing problem are given as probabilistic multivariate functions of the decision variables. Nam et al. [22] proposed a probabilistic method to identify the best propulsion system architecture under environmental and regulatory uncertainty associated with
60
NOx , CO2 , and noise emissions. Such regulatory metrics are associated with design constraints for the design of an aircraft as well as a propulsion system. In the design constraints, the regulatory metrics appear as target values that associated performance measures as responses of design variables must satisfy. As discussed in 2.3.1, however, the regulatory metrics as target values of the associated constraints are subject to uncertainty such as future regulation changes. Under inherently probabilistic design constraints, their method aims to nd the best propulsion system architecture, rather than an optimum solution for one architecture. The overall process of the method is illustrated in Figure 22. The performance measures including NOx , CO2 , and noise emissions, which are functions of a set of design variables are approximated with a set of response surface equations. An optimization tool, wrapped with a Monte-Carlo simulation (MCS), solves a series of optimization problems, each of which is set up with randomly selected values of the regulatory constraints. A Cumulative Distribution Function (CDF) can be created from a sampling of the optimum deterministic solutions giving two pieces of information: the probability of success and expected optimum OEC range, which guides designers to select the best architecture among the options under consideration. Although useful for the selection of a system architecture, this method is not appropriate for quantifying design margins for a given system architecture. A compelling reason is that the method simulates uncertainty and evaluates the propagated impacts on system design through a deterministic sizing tool, as depicted in Figure 23. Therefore, all the events that produce the CDFs in the gure represent a possible and desired design from a wait and see approach, in each of which designers wait until the uncertainty is clear, and then they deterministically size the aircraft. The consequence of the variation in the random variables in this approach is the variation in the size of the aircraft. If designers need to determine the size of aircraft without knowing the nal settlement of the random variables in the future, they must take a
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Probability of Success
Alternative A
Alternative B
TOGW (or OEC)
...
Alternative Engine #n
...
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Figure 22: Nams DSS method process [22]
here and now approach, as opposed to wait and see approach [113]. The consequence of the variation in random variables in this approach appears as a variation in the performance measures of the production aircraft, by which the feasibility of the design can be estimated. Despite remarkable progress made in non-deterministic approaches for aerospace systems design in the past decade, no method provides a consolidated treatment of the probabilistic constraints associated with aircraft sizing.
2.4
Deciencies in Traditional Sizing Methods
Traditional sizing methods have been successfully applied to thousands of aircraft designs and developments. However, the method confronts some serious challenges, as aerospace endeavors are pushing the envelope by pursuing unconventional concepts and implementing revolutionary technology.
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Alternative Engine #2
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4 % &$
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" " %5 %5 # # # #
Baseline A/C Configuration
# # (% % ) ) (
Input Variables
P
Responses
x1 x2
Deterministic Sizing Code
P
y1 y2
Figure 23: Simulation through deterministic sizing codes
2.4.1
P
xn
ym
Inexibility toward Unconventional Concepts
One of the challenges is due to the inexibility of traditional sizing methods for incorporating unconventionally-powered revolutionary concepts such as zero-emissions aircraft and regenerative propulsion aircraft. Traditional aircraft sizing methods are signicantly specialized for aircraft powered by internal combustion engines. A propulsion system is a device that produces propulsive thrust through a series of energy conversions. This process is typically aected by the design parameters of the system and operating conditions, such as Mach number, altitude, and ambient temperature. Because the technology behind traditional air-breathing combustion engines has matured, thrust lapses and SFC behaviors of traditional air-breathing combustion engines are well understood. Thus, aircraft design engineers need not track every detail of the internal energy conversion process inside the engine. Instead, they can obtain the thrust lapse, SFC behavior, and the scaling laws of a notional engine from well-established historical data by selecting the most suitable engine type and cycle, depending on the mission requirements. In addition, the propulsion system data can
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also be created from physics-based legacy codes in which substantial knowledge obtained from several decades of engine development experiences was implemented. In contrast, emerging revolutionary propulsion systems have not yet been fully developed. They are continuously evolving, and thus, their thrust and fuel consumption behavior and scaling laws are not well established. Another challenge, closely related to the above, is inexibility of traditional analysis codes for revolutionary aircraft concepts. Most legacy codes that are used for conceptual aircraft design extensively refer to empirical or semi-empirical data, which may not work for this revolutionary propulsion aircraft. In addition, most aircraft sizing synthesis codes are specically adjusted for conventionally powered aircraft, which leads to inexibility for revolutionary propulsion systems such as zero-emissions aircraft and regenerative propulsion aircraft. Therefore, designers must reexamine the assumptions that the existing analysis models hold and determine whether the models are appropriate for the revolutionary propulsion system. Thereby, if necessary, new physics-based analysis environments must be developed. Strictly speaking, however, the scarcity of historical database or immaturity of propulsion system analysis environments, themselves are not a deciency in the traditional method, but must be referred to as limitation of implementing the method. The exact point where the traditional method fails is where emerging energy sources introduce coupling between propulsion component sizing and aircraft sizing. The successful integration of the propulsion system is one of critical aspects of aircraft development. Therefore, both physical and functional interfaces between the engine and airframe call for intensive engineering work including engine mounting; the connection of bleed air and mechanical power extraction; and the design of in-takes and diusers. As far as aircraft sizing is concerned, however, airframe design generally does not have to be bothered by details of propulsion system design. Since both engine deck and engine scaling law is of interest to the propulsion system, the propulsion
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system can be considered a sort of black-box. Nevertheless, certain emerging electric aircraft propulsion concepts introduce more ambiguous boundaries between the airframe, propulsion system, and energy storage. An example is an ambient energyharvesting aircraft whose such energy-collecting components cannot be sized without considering the airframes geometry. Another example is a fuel cell wing [26], which produces aerodynamic lift as well as propulsive thrust. These problems can be solved by combining a partial or entire propulsion system sizing into aircraft sizing process, which can be achieved by constructing a more generalized formulation for modeling the propulsion system and integrating that into aircraft sizing formulation. Another deciency of the traditional sizing methods is their inexibility in dealing with aircraft propelled by a set of dierent propulsion systems or hybrid propulsion systems. Except for a few aircraft such as KB-50J, a six-engine aerial refueling tanker equipped with four prop engines and two turbojet engines [114], most existing aircraft are equipped with a single engine or multiple identical engines. Therefore, the thrust available and fuel consumption for most conventional aircraft can be established by one engine deck. However, introducing new energy sources is very likely to call for hybrid propulsion systems for various reasons. First, economics appears as one of the most important factors. The cost associated with infrastructure changes and the sustained use of legacy aviation systems logically demands a transition period as new energy sources are introduced [5]. Thus, an appropriate use of hybrid propulsion systems may facilitate timely introduction of edgling new power systems. A hybrid system may also be preferred when some of the requirements are conicting with each other, thereby any single propulsion system results in infeasible solutions. Especially electric propulsion system architectures may take advantage of integrating a combination of dierent types of power devices into a single propulsion system thanks to the versatility of electric power. For example, the combination of high specic-power devices such as lithium-polymer batteries and ultra-capacitors and high specic-energy
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devices such as fuel cells may provide an optimum solution for an electric aircraft whose power prole has high peaks for short periods. If dierent types of propulsion systems and energy sources are equipped, and the power contribution rate of each propulsion system varies with ight conditions, then the traditional sizing formulation cannot handle this situation properly. The last deciency of the traditional sizing methods is related to the assumptions behind mission analysis. The most widely used mission analysis technique for conventional aircraft is based on the assumption that the time rate of change in aircraft weight equals the fuel ow, which leads to a historical equation known as the Breguet range equation. However, aircraft such as the Helios, which is equipped with regenerative power systems, maintains the same weight throughout the entire mission. Furthermore, more stringent emissions regulations in the future may force aerospace engineers to innovate propulsion systems so that the system can separate specic by-product components from engine emissions and store them on-board during ight. Furthermore, zero-emissions aircraft, which sequester and retain by-products on board during their operation and discharge them on the ground or at least a lower altitude, will get heavier as fuel burns. This behavior cannot be analyzed by the traditional mission analysis based on the Breguet range equation. Most recent system-level studies [32, 115, 116, 117, 118] evaluate unconventional propulsion systems by retrotting an existing aircraft and assessing the impact on aircraft performance without sizing the aircraft. The QGT study [26, 119] presented unconventionally powered aircraft sizing practices for two specic congurations: a conventional wing-tail combination with over-wing, hydrogen-fueled engines, and a blended wing-body concept powered by hydrogen fuel cells. The analyses were achieved with signicant modications to a traditional sizing code, the Flight Optimization System (FLOPS) code.
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Smith and his colleagues [120] developed an electric aircraft sizing method suitable for battery powered electric aircraft. Harmats and Weihs [121] proposed a cohesive mathematical formulation for sizing of a high-altitude, long-endurance remotely piloted vehicle powered by a hybrid propulsion system combining solar power and internal-combustion engine. However, such methods were limited to certain types of propulsion systems and missions. A comprehensive, structured, generalized aircraft sizing method that is applicable to a wide range of unconventionally-powered aircraft has not yet been fully developed. 2.4.2 Inability to Account for Uncertainty
Another challenge is due to the inability of the traditional methods to account for uncertainty. Generally, uncertainty is greatest in the conceptual design phases because of the scarcity of information about the new product being designed [122]. The traditional approach to mitigate the risk associated with uncertainty is to add design margins, which generally cause direct and adverse impact on aircraft performance and cost. A question that immediately arises is how to quantify proper design margins. If extraneous margins are included, the solution may be to increase the weight and/or the cost without a proportional decrease in the associated risk. Therefore, determining the proper amount of design margins is crucial in achieving aordable risk with a minimal impact on aircraft cost. In a traditional aircraft design environment, the appropriate margins are usually determined by experts based on their prior experience. Such a conjecture of grey beards, however, may result in either a signicant risk or unnecessary increases in weight and cost. Furthermore, this current practice with revolutionary concept design, whose solution is more likely to deviate from historical precedents, may conjure even more risk. Therefore, emerging revolutionary propulsion aircraft warrant the need for a novel, structured method that allows one to optimize the design with
67
the correct design margins, which tailor the risk to a level deemed acceptable by the decision maker. Nevertheless literature review has also shown that the various probabilistic approaches developed in the area of aerospace system design during the past decade are not able to fulll such a need.
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CHAPTER III
RESEARCH OBJECTIVE, QUESTIONS, AND HYPOTHESES
The literature review presented in Chapter I has identied an emerging need for a method that is capable of sizing aircraft powered by alternative energy sources. However, the continued review of traditional sizing methods as well as state-of-the-art design methods, presented in Chapter II, has revealed that existing methods are not able to fulll the need. These observations resulted in the objective of this dissertation, which is presented in 3.1. Subsequently, 3.2 describes the research questions that must be answered to achieve the research objective, which is followed by a discussion of the hypotheses. The last section presents the mathematical representations of the hypotheses.
3.1
Research Objective
The objective of this dissertation is to develop a comprehensive, generalized, aircraft sizing formulation that is capable of: 1. Sizing revolutionary aircraft concepts that are powered by a wide range of unconventional energy sources 2. Mitigating the risk associated with uncertainty to the level deemed acceptable by a decision maker The rst part of the research objective encapsulates the original research motivation of this thesis: introduction of alternative energy sources to aviation. Therefore, the emerging method must possess a sucient degree of generality to cope with various
69
types of propulsion system architecture utilizing dierent alternative energy sources. The second part is a logical adjunct to the rst part, because the incorporation of alternative energy sources necessarily introduces uncertainty sources that traditional methods cannot properly handle as discussed in 2.4.2. While no legacy code is publicly available for the analysis of revolutionary propulsion systems powered by alternative energy sources, a range of applicable analysis tools with varying levels of delity can be found in the literature. Therefore, the emerging sizing method must be made adaptable in order to to extend its applicability. Although not explicitly stated in the research objective, such a exibility regarding analysis delity is pursued throughout the development of the new sizing method.
3.2
Research Questions
The aforementioned research objective can be fullled by developing sound solutions to the following two research questions.
Question 1: How can a generalized aircraft sizing method, independent of the
architecture of energy storage and power generation, be formulated? This question is directly addressed by the eorts of pursuing the rst capability of the emerging method, capable of sizing aircraft powered by alternative energy sources. The use of alternative energy sources, however, is very likely to accompany revolutionary, unconventional propulsion systems, energy storage systems, and energy reception systems (if applicable). Understanding the implication of the dierence in the way of storing energy and producing power into aircraft sizing is as important as the dierence in the properties of fuels (energy sources). Therefore, this research question is not just about addressing the issue of alternative energy sources, but is ultimately seeking an architecture-independent aircraft sizing formulation.
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According to the SAE Dictionary of Aerospace Engineering [123], an architecture is dened as the structure, functional, and performance characteristics of a system, specied in an implementation-independent way. Thus, implementationindependency implies consistency throughout individual product designs. For example, turbofan engines share a common structure regardless of the dierences in the product-to-product details, compared with other type of engines, such as a turbojet, scramjet, and so forth. Similarly, alternative energy sources are also likely to have multiple options for incorporation into aircraft propulsion system architectures. For instance, hydrogen can fuel turbine engines as well as fuel cell power plants, each of which is associated with totally dierent architecture from another. Therefore, a new formulation must possess a common structure that is applicable to any architecture and is able to capture the impact on aircraft sizing due to the choice of an architecture. In meeting the second part of the research objective, the following research question must be addressed.
Question 2: How can adequate design margins, required for mitigating the risk
associated with uncertainty having minimal impacts on the design objective(s), be quantied in an aircraft sizing problem? As mentioned previously, the best way to mitigate the risk associated with uncertainty in a system design is to add proper design margins. Although the concept of robust design is intended to produce a design that is insensitive to randomness, robustness itself does not ensure the design will meet the design constraints with sucient probabilities. Therefore, the emerging method must be able to intelligently allocate design margins that ensure the target probability of meeting the design constraints without committing too much cost on the design objective(s).
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3.3
Hypotheses
The hypotheses presented herein are an attempt to answer the research questions given in the previous section. The hypotheses themselves serve as the cornerstones upon which the emerging method is constructed.
Hypothesis 1: A generalized aircraft sizing formulation that is independent of
propulsion system architectures and energy sources can be formulated based on the traditional energy-based sizing approach by making the following modications: The propulsion system architecture is modeled as an integration of powerpath(s), each of which is characterized by three parameters: the specic energy of the energy source, specic power, and eciencies of power transfer devices. Fuel is generalized as the source of energy on board a vehicle and is categorized into based on its nature of conversion: consumable and nonconsumable. Aircraft weight is decomposed into more generalized weight groups, which leads to a more general weight dierential equation. The fundamental idea of aircraft sizing, which is represented by a power balance and an energy balance, is still valid for the sizing of revolutionary aircraft powered by alternative energy sources. In addition, the overall process of the traditional aircraft sizing method briey outlined in Chapter II appears to be applicable to revolutionary aerospace vehicle concepts that operate on the consumption of alternative energy sources. Therefore, it is conjectured that the rst research question can be answered by modifying the traditional methods as much as necessary rather than starting completely anew. Upon this recognition, Mattinglys method is selected as the bedrock
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Technical Issues
Coupling between propulsion system sizing and aircraft sizing
Resolution plans
Generalized propulsion system modeling
Limited flexibility to vary power mix of hybrid propulsion systems
Concept of multiple power paths
Applicable to only limited types of fuel
Multiple energy sources Consumable energy Non-consumable energy Generalized weight decomposition and weight differential equation
The time rate of change in aircraft weight does NOT necessarily equal fuel flow
Figure 24: Strategy for a generalized sizing method
for a couple of reasons. Above all, the method is based on fundamental physics, Newtons Second Law, to which the emerging method also aspires. In addition, Mattinglys original formulation provides a structure that allows designers to employ a wide range of analysis tools of variable delity, unlike others that heavily depend on regressed equations developed from empirical data, which may not be valid for alternative energy-propulsion system architectures. Such an extensive applicability, however, outstandingly requires a considerable amount of modications to the traditional formulation. The modications can be addressed by seeking proper remedies to the deciencies of traditional sizing methods identied in Chapter II, and summarized here as Figure 24. The rst modication is to generalize the concept of a propulsion system. Should the use of alternative energy sources be accompanied by the revolutionary, unconventional propulsion system architectures, the emerging method must have an architectureindependent structure that properly incorporates the sizing of propulsion systems into
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the aircraft sizing process. Traditional aircraft sizing methods are applicable to internal combustion engine architectures consuming hydrocarbon fuels. With some modications, their basic structures also seem applicable to certain propulsion systems using alternative fuels, whose architectures are similar to conventional propulsion system architectures. For example, a hydrogen-fueled jet engine has a signicant proximity to its conventional counterpart. Nevertheless, the existing methods are mostly limited to what has been done in the past. Therefore, an architecture-independent formulation must be constructed based on the abstracted properties commonly found in all propulsion system architectures. Such abstracted properties must formally represent the following three attributes - fuel consumption, available power, and propulsion systems weight - that are the primary gures of merit of a propulsion system in view of aircraft sizing. This hypothesis is developed based on the belief that the abstraction of propulsion system architectures by specic energy of the energy source, the specic power, and the eciencies of power transfer devices can properly capture the above three attributes. The second modication is the generalization of the concept of fuel as an energy carrier. In addition to the propulsion system architectures, an architectureindependent method must have a proper means of capturing the attributes of various energy sources and their implications to aircraft sizing. As discussed in Chapter II, traditional aircraft sizing methods are, for the most part, specic to the use of conventional fuels, the weight of which gets reduced during ight due to consumption. However, this may not always be the case for certain alternative energy sources, whose weight remains constant in the process of power generation. Therefore, it is desirable to classify aircraft energy sources into two groups: consumables and nonconsumables. A separate treatment of each group would allow the designer to properly capture the variation of energy source weight with ight. Henceforth, consumable energy is dened as a form of energy that is derived from a source whose weight is
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reduced during power generation, such as traditional hydrocarbon fuels. Alternatively, non-consumable energy is dened as a form of energy derived from a source whose weight stays constant or changes negligibly during power generation, such as an electric battery, a nuclear battery, or human power. The last modication is the incorporation of more generalized weight decomposition and weight dierential equations. A revolutionary propulsion system with alternative energy source may debunk the primary assumption of traditional approaches to fuel balance: the time rate of change in aircraft weight equals fuel ow. This issue ensues from the use of a non-consumable energy source. The assumption may no longer be valid for certain types of aircraft that y with only non-consumable energy sources, such as the Helios, which was conceived to use re-circulating hydrogen to fuel the regenerative fuel cell propulsion system, whose weight would not change throughout the entirety of its operation. An improper consideration of the aircrafts weight variation due to fuel consumption may result in a considerable error in the estimation of its mission range or the amount of energy sources, and thus consequential errors in its sizing. Since drawing non-consumable energy, by denition, does not result in any change in aircraft weight, the relationship between the time rate of change in consumable energy weight and aircraft weight is of interest. In addition, the relationship depends on how much of the by-products from the energy conversion process of a consumable source are emitted from the aircraft. Unless such by-products are fully and instantaneously expelled from the aircraft, as is the case of conventional propulsion systems, the time rate of change in aircraft weight would not necessarily equal the time rate of change in consumable energy weight. The weight of the by-products from most energy conversion processes would be reasonably in proportion to the weight of consumed fuel, the ratio of which can be determined by symbolic representation of a chemical reaction associated with the energy conversion process. Incorporating a generalized formulation that establishes the relationship between the time rate of
75
change in aircraft weight and the time rate of change in consumable energy weight would thusly allow the sizing of any kind of revolutionary aircraft concept, regardless of its energy-conversion mechanism.
Hypothesis 2: It is possible to determine adequate design margins that result
in a solution satisfying all probabilistic constraints under consideration with a target probability, while searching for a design optimum. Although the initial problem statement of the probabilistic approach for aircraft sizing, given per Eq. (27) is a realistic representation of the real-world, in which one needs to make a decision for an optimal solution under uncertainty, it is not yet clearly dened. The most ambiguous of them all is how to interpret and implement the probabilistic constraints, and more specically, how to determine the feasibility of the probabilistic constraints. One way of accounting for the probabilistic constraints is to nd the design solution that, with 100% probability, satises all constraints on any realization of random parameters. However, such an extremely conservative solution is not likely to exist in reality. Furthermore, even if such a solution were to be found mathematically, the resulting impact on the objective function is expected to be prohibitive. The underlying idea of this hypothesis is that the adequateness of design margins can be measured by two gures of merit: the probability of meeting non-deterministic constraints and their impacts on the design objective function. It is believed that a practical way to address the impact of uncertainty in the aircraft sizing process is to set design margins that tailor the probability of meeting the design constraints to the level deemed acceptable by decision makers. At the same time, it is desired that the impacts of such design margins on the objective function are minimized.
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3.4
Substantiation of Hypotheses
The hypotheses verbally introduced in 3.3 are substantiated with mathematical interpretations that allow the intrinsic ideas to be assimilated into the emerging formulation. This section provides such a mathematical representation for each hypothesis. 3.4.1 Mathematical Representation of Hypothesis 1
The fundamental idea behind the rst hypothesis is that the basis of the traditional energy-based sizing method is also valid for the sizing of unconventional energypropulsion system architectures. The two axes of the traditional energy based sizing method - thrust balance (or power balance) and fuel balance (or energy balance) are based on the fundamental physics laws, Newtons Second Law and the Second Law of Thermodynamic Law, respectively. These two fundamental principles do not assume any specic energy source nor any specic propulsion system architecture, and thus are valid for describing the motion of any aircraft, regardless of its energy carrier and propulsion system architecture. Founded on the same cornerstones, however, the architecture-independent sizing formulation can be constructed by signicantly modifying the old approaches. The three key modications addressed by Hypothesis 1 are elaborated as follows. 3.4.1.1 Generalized Propulsion System A generalized modeling of propulsion system architectures begins with the decomposition of the system itself. A propulsion system consists of a series of power generation or conversion devices, as depicted in Figure 25. Such devices can be categorized into a power generation device (PGD), a power transformation device (PTD), and a power output device (POD) - the last power transformation device. All parameters which describe a given propulsion system in view of aircraft sizing - output power, fuel consumption, and system weight - can be represented by the following component-level parameters: specic energy of energy sources, specic power 77
Pref / k
0 k
Pref / k
1 k
Pref / k
2 k
Pref / k
Pref
k +1 n 2
Pref
k +1 n 1
Pref
k +1 n
Pref
Energy (Fuel)
Power PO
PGD
0
PD
PTD1
0
1
PD
...
10 PO
0 k 1
PTDk
k
1
PD
...
0 k
PTDn-1
n-1
k
PD
POD
n
n 1
PD
n
0 PO
( ) PO
( ) PO
0n 2
( ) PO
0 n 1
( ) PO
0n
( ) PO
Figure 25: Generalized propulsion system model
of each power device, and the eciency of each power device. Specic energy, denoted as E , is the energy per unit weight of an energy carrier. The values of specic energy1 for various energy sources are listed in Table 2. The specic power of a power device, PD , is the amount of output power produced by the device per unit weight. The eciency, , of the device is the ratio of the amount of output energy to input energy of each device. Then, the nal propulsive power, P , is related to the rst power input from the energy source, Po as follows: P = n n1 1 0 Po = Po (28)
where denotes the overall eciency of the propulsion system. This tank-to-wing eciency is computed by multiplying the eciencies of all energy conversion associated with a propulsive power generation process, that is = (). The overall
eciency of an internal combustion engine is given as a multiple of thermal eciency (th ), the ratio of net output from thermal cycle to thermal energy input, and propulsive eciency (p ), the ratio of propulsive work to net output from thermal cycle as follows [126]: = th p = V ct LHV (29)
where ct stands for thrust specic fuel consumption (TSFC) and LHV stands for lower
The specic energy of chemical energy sources are often dened in terms of HHV (higher heating value) or LHV (lower heating value). The former is determined by bringing all the products of combustion back to the original pre-combustion temperature. The latter, also known as net caloric value, is determined by subtracting the heat of vaporization of the water in the by-product from the higher heating value results. The lower heating value is what is typically used for IC vehicle engine analysis.
1
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Table 2: Energy contents Fuel Energy contents (MJ/Kg) 1) Reference 2) Coal 33.3 [124] CO 10.9 [124] Methane 50.1 [124] 3) Natural Gas 38.1 [124] Propane 45.8 [124] Gasoline 4) 42.5 [124] 5) 43.0 [124] Diesel 42.8 MIL-DTL-83133E JP-8 Hydrogen 120.1 [124] Natural uranium 500,000 [125] 1) Lower heating value for chemical energy sources 2) Anthracite, average 3) Groningen (The Netherlands) 4) Average gas station fuels 5) Average gas station fuels
heating value. Fuel consumption, WF , is given as P WF = E (30)
Finally, the weight of a propulsion system (WPS ) is given as the sum of all power devices weight (WPDi ), each of which can be computed by dividing its sizing power by specic power (PS i ) as follow:
nPD nPD
WPS =
i=1
WPDi =
i=1
PPDi PDi
(31)
Therefore, if the specic energy of the energy source and the specic power and the eciency of each power device are established, then the system characteristics and behavior of the propulsion system can be fully described for the purpose of aircraft sizing. Most aircraft propulsion systems also provide power for other subsystems such as cooling systems, hydraulic systems, and electric systems by means of engine bleed air 79
and mechanical power extractions. In order to capture the loss due to such power extraction, a small modication to the previous equation is needed. If Pext is taken out at the k th power device, then the modied eciency, k , of the power device can be expressed as follows: k = (k ) (32)
where is the ratio of the amount of power extraction to the amount of input power of the power device. 3.4.1.2 Consideration of Multiple Energy Sources In order to produce a generalized formulation, the aircraft is assumed to have multiple power-paths that have either consumable or non-consumable energy sources. Then, the total energy stored on-board, E, is the sum of these two types of energy as follows:
nCE
E=
i=1
i ECE
nNE
+
j=1
ENE
j
(33)
where nCE is the number of power-paths of consumable energy, and nNE is the number of power-paths of non-consumable energy. Therefore, the total weight of the on-board energy carriers, Wenergy is
nCE
Wenergy = WCE + WNE =
i=1
i WCE
nNE
+
j=1
WNE
j
(34)
It must be noted that this formulation calculates the energy weights in terms of power-paths and not energy types. For instance, if JP-8 is used for both a conventional jet engine and a fuel cell system that powers electric motors and propellers, then this system has two power-paths, and the required fuel weight for each of the two powerpaths is estimated separately. As new energy sources start to be utilized, hybrid propulsion systems and/or a combination of dierent types of propulsion systems that utilize multiple energy sources, may power future aircraft. Each of the power devices and their associated energy sources have dierent characteristics in terms of specic power and specic 80
energy. To reduce the weight of an aircraft, higher specic power and specic energy are always desired, but nature does not allow both in one source, as depicted in Figure 26. In general, higher specic energy may be favorable for range and endurance capability and higher specic power may be favorable for maneuvering capability. When multiple choices are available, designers may want to optimize the propulsion system architecture by properly mixing dierent types of power devices and energy sources. For instance, in the case of sizing an aircraft that is required to perform highpower demanding maneuvers for short periods during its mission, combining a high energy ecient primary power system with a booster would be a better approach than sizing the entire propulsion system with one combination of a power device and an energy source as demonstrated in previous research [83, 117]2 . However, the decision must be made based on a balanced consideration of the impact on producibility, reliability, and maintainability due to increased system complexity as well as cost. The consideration of heterogeneous propulsion systems fueled by single or multiple energy source(s) entails a further modication of the generalized propulsion system model presented in 3.4.1.1. A mathematical representation for multiple power or energy sources can be developed by introducing the concept of power-path. A powerpath is dened as a set of power devices along which a series of energy conversion processes take place. If a vehicle is powered by nT dierent power-paths, each consisting of multiple energy sources and energy transformation devices (ETD) as depicted in Figure 27, the power available to the aircraft is given as the sum of the power from all individual power-paths. P =
i=1
2
nT
P
i
(35)
A similar concept for a hybrid propulsion system has been also premeditated for Boeings Fuel Cell Demonstrator Airplane. According to Boeing [127], the aircraft will be powered by a hybrid fuel cell/battery propulsion system that combines lithium-ion batteries as a secondary power system with a PEMFC as a primary power source. The PEMFC system is sized to provide sucient electrical power for level ight. For takeo and climb, the batteries cut in to supply an additional boost.
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Favorable to Range and Endurance
Favorable to Maneuvers
Figure 26: Comparison of specic energy and specic power for various power source technologies [128]
This expression can be modied by introducing a power fraction factor ( ) as follows: P where,
i
= iP
(36)
nT
i=1
i
= 1, and
i
>0
i
(37)
The rst power input from the original energy source, Po , is eventually converted to the nal propulsive power through multiple energy transformation processes. The nal output power is given as the product of the eciencies and Po to account for the losses associated with each transformation/conversion. P
i i i
= i Po i
(38)
where represents the overall eciency of the ith power-path that is computed by the product of all eciencies associated with the energy conversions. 82
Power Path 1
Energy 1
PGD
PTD
POD1
Po
1
P
1
= P
1
Power Path 2
Energy 2
PGD
. . . PTD E.T.D
POD2
Po
2
P
2
= P
2
P
Power Path nT Energy n
PGD
PTD
PODnT
Po
nT
P
nT
=
nT
P
Figure 27: Multiple power-paths
3.4.1.3 Generalized Mission Analysis In addition to the generalized conceptualization of fuel, a more generalized weight decomposition and weight dierential equations are required to account for the impact of using of alternative energy sources. A more generalized weight decomposition has been developed as follows: W = WE + WPL + WCE + WR (39)
where WR is the weight of the retained by-products from energy conversion. The nonconsumable energy weight is included in the empty weight. Taking the derivative of Eq. (39) with respect to time produces the time rate of change in aircraft weight as W = WCE + WR (40)
By assuming that the weight of the retained products is proportional to fuel consumption, WR = WCE (41)
83
Conventional Aircraft
Regenerative Aircraft
Zero Emission Aircraft
Figure 28: Typical k values, from left Boeing 777 [129], Helios [6], and Emissionless aircraft [119]
where is the retained-product-to-fuel ratio. Substituting Eq. (41) into Eq. (40) yields W = (1 )WCE (42)
By introducing a constant k, the relationship between the time rate of change in the aircraft weight and the time rate of change in the consumable energy weight is given as W = k WCE where k =1 (44) (43)
The value of k is determined by the characteristics of the given energy conversion process. Several typical k values are illustrated in Figure 28. In the case of traditional IC engines, k numerically equal one. If an aircraft is equipped with a regenerative fuel cell system as envisioned for the Helios3 , the aircrafts weight does not vary with fuel consumption, resulting in k = 0. In the case of a zero-emissions aircraft that
The fuel cell systems designed for the Helios were of two types: regenerative and nonregenerative. However, the Helios never ew on a fuel cell propulsion system. The value of k would be zero if the former were equipped.
3
84
breathes external oxygen to burn hydrogen and store water on-board [26], the aircraft weight will grow as fuel burns, and thus, k is a negative number. If the aircraft is powered by multiple power-paths, k is given as
nCE
k=
i=1
1i
nCE ii=1
ii i
CE
ii
i
i
(45)
CE
ii
3.4.2
Mathematical Representation of Hypothesis 2
The key idea behind the second hypothesis is that the feasibility of the probabilistic constraints can be determined by the probability of meeting the constraints. For example, consider a probabilistic constraint, g(x, ) > 0, where x is a vector of decision variables and is a vector of random parameters. Then, the design represented by x is considered to be located in a feasible region if and only if P[g(x, ) > 0] , where is the target probability of meeting the constraints that is determined by management decisions. This implementation allows a means to determine the feasibility of the probabilistic constraints. Therefore, the aboriginal statement for the probabilistic sizing problem, given as Eq. (1), can be rewritten as follows: min f
x
s.t. P Pavailable Prequired P Eavailable Erequired
(46)
This equation mathematically formulates an aircraft sizing problem into an optimization problem whose goal is to minimize the objective function values subject to multivariate, nonlinear, individual or joint probabilistic constraints. In this context, the three design variables, power, wing area, and fuel (energy) quantity, are manipulated until all probabilistic constraints are satised with equal or higher probabilities than the target.
85
CHAPTER IV
FORMULATION OF THE ARCHITECTURE-INDEPENDENT AIRCRAFT SIZING METHOD
Research Question I described in Chapter III is seeking a generalized aircraft sizing formulation that is independent of the energy-propulsion system architecture of an aircraft. A edging attempt to answer this question produced Hypothesis I, which includes the conceptual idea of the formulation. Building upon the substantiation of Hypothesis I presented in 3.4, this chapter presents the formulation of the Architecture-Independent Aircraft Sizing Method (AIASM) proposed as a solution to Research Question I. The AIASM includes three parts: generalized constraint analysis, generalized mission analysis, and generalized weight estimation. The rst section of this chapter presents the formulation of generalized constraint analysis. 4.2 introduces a set of generalized Breguet range equations. The equations are not only useful by themselves in quick estimations of range capability, but also provide a basis for facilitating the development of the more comprehensive generalized mission analysis presented in the following section. 4.4 discusses a more generalized weight estimation method. Based upon the mathematical formulations of the major three components, 4.5 describes the overall implementation process of AIASM. The rest of this chapter discusses a couple of extended topics. 4.6 presents a graphical tool that is able to compare the mission capability of multiple energy-propulsion system architectures in a unied visual environment. The last section presents an additional formulation required for the sizing of solar powered aircraft.
86
4.1
Generalized Constraint Analysis
Sharing the crux of Mattinglys constraint analysis method presented in 2.2.3, the generalized constraint analysis method is formulated upon the concept of generalized propulsion system modeling presented in 3.4.1. The most notable dierence is that the new method establishes the static or dynamic equilibrium of motion for an aircraft in terms of power in lieu of thrust. The compelling reason for using power is that it is a more universal metric to represent the size of most emerging propulsion systems using alternative energy sources. In addition, power is a more convenient means by which hybrid propulsion system architectures are properly handled. Formulation starts with the most simple case characterized by a single powerpath propulsion system and a xed external conguration, for which performance constraints can be visualized in a single design space comprised of power-to-weight ratio1 and wing loading. Once the formulation for the simplest case is constructed, it is extended to more complicated cases with multiple power-paths and/or morphing congurations, leading to a multi-dimensional constraint analysis in a matrix form. 4.1.1 Formulation of a Single Constraint Analysis
Power balance states that the available power must be greater than or equal the required power, Pava Preq . The required power-to-weight ratio can be derived from Eq. (9) as follows:
Preq = WTO qS n WTO K1 WTO q S
2
+K2
R 1 d V2 n WTO +CDo+ + h+ q S qS V dt 2go
V
(47)
In general, the available power of a propulsion system varies depending on ight conditions (e.g. altitude, velocity, side slip) and power settings. Therefore, in order to size the propulsion system, we must have an invariant reference of the amount of power that represents the scale of a rubberized propulsion system. As such a reference,
1 This term is often considered interchangeable with power loading. However, having an opposite connotation from power-to-weight ratio, power loading means the weight of the aircraft divided by engine power.
87
traditional methods often use the amount of maximum thrust and shaft power at the sea level static condition, for jet airplanes and propeller airplanes, respectively. The choice of such a reference generally involves three decisions: a reference power device along the power-path; a reference operating condition; and a reference ight condition. The propulsive power used in Eq. (47) may not necessarily be the best choice for a metric that establishes power balance for most alternative propulsion systems. Particularly when the reference ight condition is set at a static condition (V = 0), the use of propulsive power as a reference immediately yields the conspicuous problem of the propulsive power always being zero, which makes the use of the propulsive power at a static condition impractical. Furthermore, the scale of the propulsion system may be better represented by the amount of power at a specic stage amid the energy conversion process rather than the output power at the nal stage (the propulsive power). Therefore, the reference power must be carefully selected considering various aspects including characteristics of the propulsion system architecture and the availability of data. If an aircraft sizing process is conducted with a dedicated propulsion system analysis tool integrated with an aircraft sizing code, the choice of the sizing reference power is very likely to be on-design point of the propulsion system analysis. If the k th power device in Figure 29 is selected as the reference power device, then the available propulsive power is given as Pava = + Pk , where + represents the product of eciencies associated with the energy conversions following the stage of reference power, that is,
k+1n ().
Note that Pk varies depending on ight
conditions, which can be accounted for by introducing the power lapse ratio () that relates Pk to a reference power, denoted as Pref , as follows: Pk = Pref (48)
The value of Pk equals that of Pref at an on-design point specied with engineoperating parameters, leading to = 1. For the aircraft propelled by a power-path, 88
Pref / k
0 k
Pref / k
1 k
Pref / k
2 k
Pref / k
Pext
k +1 n 2
Pref
k +1 n 2
Pref
k + 1 n
Pref
Energy (Fuel)
Power
PO
0
PGD
po
PTD1
1
p1
...
1 0 PO
0 k 1
k
PTDk
pk
0 k
...
PTDn-1
n-1
p n 1
POD
n
0 n
pn
0 PO
( ) PO
Figure 29: Reference Power in a power-path
the relationship between the reference power and the propulsive power is given as Pava = + Pref (49)
For example, in the case of a propeller aircraft, if one chooses Pref to be the maximum shaft power at the sea level static condition, then + would be eciency of the propeller, and accounts for the variation of shaft power with mach number, altitude, and engine power settings. By combining Eq. (47) and Eq. (49), the equation of power balance is written as
Pref WT O + qS n WTO K1 WT O q S
2
+K2
n WTO R 1 d V2 +CDo+ + h+ q S qS V dt 2go
This equation is a power-based master constraint equation, equivalent to Mattinglys master equation (Eq. (11) presented in 2.2). The equation denes the required reference power (Preq ) in terms of aerodynamic coecients, wing loading, rate of energy height, and other parameters. If the appropriate assumptions for each performance requirement are applied to this master equation, a corresponding constraint equation can be derived in a reduced form. A set of such tailored equations is included in Appendix A. The weight fraction values () for mission segments are not available at this moment. Therefore, these values must be assumed reasonably for this analysis and should be updated by the mission analysis, if necessary. Once the constraint equations are developed, the feasible solution area can be graphically identied as bound by associated constraint curves. 89
( ) PO
0 n2
( ) PO
0 n 1
( ) PO
( ) PO
V
(50)
4.1.2
Constraint Analysis Matrix
If the aircraft is powered by hybrid propulsion systems encapsulated by multiple power-paths, power balance must be considered for each power-path. The power constraint equation for multiple power-paths can be obtained by combining Eq. (50) with Eq. (36) as follows:
Pref WT O
i
=
i
i
+ i
n WTO qS K1 WT O q S
2
+K2
n WTO R 1 d V2 +CDo+ + h+ q S qS V dt 2go
V (51)
where
i
is a power fraction factor for ith power-path. Multiple power-paths create
the same number of constraint analysis domains, so the design point selection process becomes more complicated. An example of the constraint analysis for a notional aircraft propelled by two power-paths is illustrated in Figure 30. The propulsion system consists of the primary power-path that provides power throughout the entire mission and the secondary power-path that provides ancillary power on take-o, during climb, on missed approaches, and in emergency situations. The design point of the primary power-path is selected at the location of the lowest power-to-weight ratio. Then, the design point of the secondary power-path must be determined at a location along the vertical line, a-a, because wing loading of the aircraft must be maintained through all the constraint analysis domains. If a dierent wing loading on line b-b is selected, the power-to-weight ratio of the primary power-path increases, while the power-toweight ratio of the primary power-path decreases. Another possible trade-o is to change the power fractions (
1
,
2
) of the two power-paths. The constraint curves
1
of both the primary power-path and the secondary power-path are functions of and
2
. As the value of
2
increases, the primary power-path is allowed to have
a lower power-to-weight ratio. Therefore, the optimum values of power distributions and aircraft sizing parameters may be found by employing an optimization process. In addition, the method can be extended for the sizing of a morphing conguration. Morphing means herein reshaping aircraft external conguration with an aid of fully integrated embedded smart materials and actuators to achieve higher aerodynamic 90
250.0
Primary Power Path
200.0
<1 Pref >
b a
150.0
W TO
100.0
50.0
0.0 0 50.0 Take-off Roll Cruise Landing Roll Climb Approach Design Point 10 20
Secondary Power Path
45.0 40.0 35.0
<2> ref
P
30.0 25.0 20.0 15.0 10.0 5.0 0.0 0 10 20
W TO
b a
W TO S
Figure 30: Design point selection of two power-paths
eciency throughout most ight regions. An interesting aspect of morphing aircraft with regards to the constraint analysis is that multiple design values for wing loading may exist because the aircraft is able to vary wing area, as multiple design values for power-to-weight ratio should exist if an aircraft were powered by a hybrid propulsion system architecture. A constraint analysis as well as the comprehensive sizing practice of a morphing aircraft that recongure the wing to have two dierent values of wing area were demonstrated in Ref. [130]. Along with aerodynamic morphing, mission adjustability may benet from propulsion morphing, which is built of a hybrid propulsion comprised of a high power sector
91
Configuration for Low Speed
Configuration for High Speed
Fuel Cells (Primary Power)
Battery (Secondary Power)
Figure 31: Approach to a notional UCAV design with a combination of aerodynamic morphing and propulsion morphing, (Source of the gure of mission prole: Ref. [130])
and a high energy sector. Figure 32 illustrates a notional extended constraint analysis setting for aircraft having a recongurable external conguration and propulsion system. The aircraft has two dierent congurations, each of which is preferred at low speed and high speed regions, respectively, and the propulsion system has two power-paths: fuel cells as a primary power and battery or super-capacitor as secondary power. Thus, the constraint analysis for the sizing of this morphing aircraft includes two power-to-weight ratio variables ( Wref , Wref ) as well as two wing loading TO TO
TO variables ( WS 1 2
p
p
1
TO , WS
2
), which create four combinatorial design spaces constrained
by the following equation:
92
Cruise/Loiter
Landing
Dash
WT O
Take-off Sustained Turn
Pref
1
Fuel Cells Primary Power
Landing
Dash
WT O
Take-off Sustained Turn
Pref
2
Battery or Super Capacitor Secondary Power
WTO S
1
WTO S
2
Figure 32: Notional constraint analysis setup for a morphing aircraft with hybrid power systems
pref WTO
i
=
i
i
+ i
q S j n WTO K1 WTO q S
j
2
+K2
n WTO q S
j
+CDo+
R 1 d V + h+ V qS V dt 2go (52)
2
Therefore, the constraint analysis is given in the form of a two-by-two matrix as depicted in Figure 32. It must be noted that in such a matrix, the values of wing loading must be the same along the column, and power-to-weight ratios must be the same along the row. The constraint curves vary due to the change in the power fractions of the two power-paths as well as aerodynamic characteristics of two congurations. Thus, an extended version of the optimization problem in which the power fractions and conguration are included as additional design variables is of interest.
93
4.2
Generalized Breguet Range Equations
The Breguet range equation is useful for various design assessments, providing qualied estimates of the cruise performance of an aircraft with only a few pieces of information about the modeled aircraft. As mentioned in 2.4.1, the classical Breguet range equation, as it is, would not work on certain unconventionally-powered airvehicles. This section presents a set of generalized Breguet range equations that are also applicable to such aircraft. As useful as the classical Breguet range equation is for a quick estimation of the range capability, the development of the generalized Breguet range equations provides the basis of the generalized mission analysis presented in
4.3.
Compared with the original single Breguet range equation, multiple equations are formulated because the tendencies of the variation in aircraft weight due to fuel consumption could dier depending not only on the type of energy source but also how an alternative energy-propulsion system architecture treats the by-products from its power generation. The derivations of the generalized equations are made separately for consumable energy sources and also for non-consumable energy sources. 4.2.1 Flight by Consumable Energy
When an aircraft equipped with a consumable energy source ies, the amount of consumed energy for an innitesimal period of time, dt, is given as: dECE = Po dt = P dt (53)
The change in the amount of consumable energy is proportional to the change in the weight of the energy source(s), dECE = CE dWCE where CE is the specic energy of the fuel in the power-path. (54)
94
Combining Eq. (54) with Eq. (53) yields dWCE = P dt CE (55)
Eq. (55) can be restated with respect to the time rate of change in aircraft weight from Eq. (43) as follows: dW = k or dW k D = ds W CE L (57) P dt CE (56)
If k is not zero, assuming the lift-to-drag ratio (L/D) and overall eciency ( ) are constant during ight, the integration of Eq. (57) leads to the following equation: R= CE L ln k D Winitial Wf inal (58)
where Winitial and Wf inal denotes the aircrafts weight at the beginning and the end of the mission, respectively. The ratio of the initial weight to the nal weight in Eq. (58) is given as a function of k and
WCE WTO
as follows: (59)
Wf inal WCE =1k Winitial WTO Finally, combining Eq. (58) and Eq. (59), R= CE L ln k D 1 1 k WCE WTO
(60)
If the aircrafts weight does not change with ight, which is the case of k = 0, Eq. (57) yields a trivial solution, W = constant. To avoid this, a dierent approach is taken by modifying Eq. (55) as follows: 1 D dWCE = ds W CE L (61)
Note that W is constant in the above equation, thus the equation can be solved by direct integration, leading to the following equation: R = CE 95 L WCE D WTO (62)
This equation can also be derived from Eq. (60). As k approaches 0, ln asymptotically approximated as k WCE and Eq. (60) reduces to Eq. (62) WTO 4.2.2 Flight by Non-consumable Energy
1 W 1k WCE
TO
is
An aircraft powered by non-consumable energy would maintain the same weight throughout the mission unless it drops a payload. Therefore, the range equation can be developed by the same process applied to consumable energy with k = 0: R = NE 4.2.3 L WNE D WTO (63)
Example Application to Zero-emissions Aircraft
The range of a notional aircraft powered by hydrogen fuel is computed using the generalized range equations. In the order of L/D, CE , and , their numerical values are 15, 44.6 106 lbs ft/sec/lbs, and 0.3. Figure 33 illustrates the ratio of nal to initial weight computed by Eq. (59). The case of k = 0 maintains the same weight, so the ratio is 1 for all consumable energy weight fractions. In the case of a positive k, as the consumable energy weight gets reduced, the aircraft weight also reduces. As the value for k increases, the rate of change in aircraft weight also increases, which leads to a further reduction in nal aircraft weight. In contrast, in the case of a negative k, the aircraft weight increases with fuel burn, leading to a nal weight greater than what the vehicle started with in the beginning. Figure 34 plots the range versus consumable energy weight fraction for dierent values of k. It can be seen that as the consumable energy weight fraction increases, the range increases. In the case of k = 0, range is linearly related to energy weight fraction. As the numerical value of k grows, range increases more rapidly per consumable energy weight fraction, and exhibits a non-linear relationship. Figure 35 shows how range, normalized by values at k = 1, varies with consumable energy weight fraction for dierent k values. The penalty of accumulating the weight
96
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 k=2 k=1 k=0.5 k=0 k=-1 k=-4 k=-8
Figure 33: The ratio of nal to initial vehicle weight vs. fuel fraction
of by-products grows as the fuel fraction increases, which means that the penalty associated with the accrued reaction product weight becomes more severe in longerrange aircraft. The case where k = 8 is interesting because it is the case of a zero emissions aircraft that uses ambient air to oxidize hydrogen fuel and retains all by-products, essentially water, on-board. For an energy weight fraction of 0.1, the range at k = 8 is 70% of that of k = 1. Since the same aerodynamic eciency and propulsion system eciency are applied to this analysis, such a considerable dierence is caused by the dierence in the treatment of the by-products of the energy generating process. Therefore, in order to oset this penalty, signicant improvements in the aerodynamics, structures, and propulsion technologies must be made for a viable development
97
30000 k=2 k=1 25000 k=0.5 k=0 k=-1 k=-4 k=-8
Range (nautical miles)
20000
15000
10000
5000
0 0 0.1 0.2 0.3 0.4 0.5
Figure 34: Range vs. fuel fraction
of a truly zero emissions aircraft. For verication of the generalized Breguet range equations derived in this section, the equations are applied to the performance analysis of a 300-passenger, LH2 -fueled zero-emissions aircraft, whose mission range data are available in a study [119] performed by MSE Technology Applications, Inc. and NASA LaRC. The aircraft is propelled by electric ducted fan engines powered by a PSOFC (Planar Solid Oxide Fuel Cell) system. The fuel cells are operated on liquid hydrogen fuel, which is oxidized by ambient air induced by inlet ducts. The primary product of the electrochemical reaction is water vapor, which is stored in two spherical water tanks located in the front and back of the cabin. The original study investigated the feasibility of the zero-emissions aircraft via two
98
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
k=2 k=1 k=0.5 k=0 k=-1 k=-4 k=-8
Normalized Range
0.1
0.2
0.3
0.4
0.5
Figure 35: Normalized range vs. fuel fraction
sets of assumptions: a Near-Term (NT) scenario and a Long-Term (LT) scenario that describe the relevant assumptions regarding the aerodynamic parameters as well as the technologies related to the propulsion system, which include PSOFC electrochemical eciency, PSOFC power density, propulsion fan eciency, and percentage laminar ow. The NT scenario assumes electrochemical eciency at cruise, motor eciency, and fan eciency at cruise 50%, 99%, and 85%, respectively, which yields an overall eciency of 42.1%. In the LT scenario, the electrochemical eciency at cruise and fan eciency are raised to 60% and 87%, respectively. In addition, each scenario includes a technology adjustment parameter, named Weight Reduction Factor, (WRF) that simulates possible levels of airframe weight reduction by the application of advanced materials.
99
The study analyzes the range performance of the zero emissions aircraft with a customized version of FLOPS modied by NASA LaRC, to account for the gaining of aircraft weight due to the storage of water. Table 3 lists a portion of their analysis results relevant to this study. Based on the information in Table 3, the input parameters of the generalized Breguet range equations were prepared. Overall eciencies were computed by multiplying the electrochemical eciency at cruise, motor eciency, and fan eciency at cruise. Fuel fraction was computed by dividing the amount of used fuel by take-o gross weight. The values of L/D were given as an average of the lower limit and the upper limit of the L/D ranges listed in Table 3. The value of k is given as 7.94 per the original study. With these parameters, the mission range was computed by the Generalized Breguet range equations. Table 4 lists the parameters and the results of this analysis. The results of the generalized Breguet range equations are compared with the published results in Figure 36 that show the generalized Breguet range equations estimate the range of zero-emissions aircraft with fairly good accuracy. Except for a single case (the NT scenario with WRF=1), the range estimated by the generalized Breguet range equations is less than the results of FLOPS. The dierence grows as the range increases, hence the maximum dierence is found at 9.2% where the LT scenario is applied with WRF=3. These discrepancies are caused by several sources. The rst is that the generalized Breguet range equation computes the range without considering non-cruise segments such as take-o, climb, and descent. Another source for the dierence in the results is associated with an assumption of the generalized Breguet range equations: the overall eciency of the propulsion system, lift-to-drag ratio are constant throughout the whole mission. Lastly in Alexanders study, the aircraft is allowed to dump out the water from takeo up to 25,000 feet, because water emission would not be detrimental below that altitude. Such a procedure may either increase the range or reduce the aircraft weight. However, the generalized
100
Table 3: FLOPS analysis results of zero-emissions aircraft obtained by MSE and NASA LaRC [119] WRF 1 2.5 3 Wini(lbs.) 511,045 377,589 361,239 Wnal(lbs.) 609,680 592,759 590,444 Wfuel(lbs.) 22,718 38,088 39,997 Unconsumed 25.50 22.30 21.30 20.80 20.50 fuel(%) Used fuel 16,925 23,866 27,691 30,166 31,798 (lbs.) Electrochemical 50 50 50 50 50 eciency(%) Motor 60 60 60 60 60 eciency(%) Fan 99 99 99 99 99 eciency(%) L/D 24.3 - 24.7 23.3 - 24.5 22.8 - 24.4 22.4 - 24.5 21.9 - 24.5 Range(nmi) 2252 3599 4431 5019 5400 Near Term Scenario 1.5 2 441,320 402,588 601,222 596,185 30,715 35,186
WRF 1 2.5 3 Wini (lbs.) 447,979 308,993 292,678 Wnal (lbs.) 610,414 591,460 589,003 Wfuel (lbs.) 28,973 45,634 47,592 Unconsumed 19.70 18.30 17.90 17.70 17.60 fuel(%) Used fuel 23,265 31,122 35,082 37,557 39,216 (lbs.) Electrochemical 50 50 50 50 50 eciency(%) Motor 60 60 60 60 60 eciency(%) Fan 99 99 99 99 99 eciency(%) L/D 26.6 - 27.4 25.7 - 27.2 24.3 - 27.3 23.1 - 27.3 22.1 - 27.2 Range(nmi) 4834 7178 8547 9402 9975
Long Term Scenario 1.5 2 372,101 333,251 600,330 595,002 38,093 42,731
101
Table 4: Input parameters and results of the generalized Range Equations of zeroemissions aircraft Near Term Scenario WRF 1 1.5 2 2.5 3 Wce/Wto 0.033 0.054 0.069 0.080 0.088 overall eciency 0.421 0.421 0.421 0.421 0.421 Average L/D 24.5 23.9 23.6 23.45 23.2 Range 2362 3526 4247 4757 5077 % dierence 4.9 -2.0 -4.2 -5.2 -6.0 Long Term Scenario WRF 1 1.5 2 2.5 3 Wce/Wto 0.052 0.084 0.105 0.122 0.134 overall eciency 0.517 0.517 0.517 0.517 0.517 Average L/D 27.0 26.5 25.8 25.2 24.7 Range 4728 6832 7949 8634 9059 % dierence -2.2 -4.8 -7.0 -8.2 -9.2
Breguet range equations are not able to capture such a detail. This example study concludes that a makeshift useful for estimations of the mission range with only handful information about the aircraft, the generalized Breguet range equations may produce considerable errors due to the assumptions associated with their development. These shortcomings will be overcome by the generalized mission analysis formulation that will be presented in the following section.
4.3
Generalized Mission Analysis
Unlike the generalized Breguet range equations, the generalized mission analysis method accounts for a variation in the propulsion system eciency and aerodynamic parameters during the ight. In addition, the method provides proper equations for non-zero excess power conditions such as take-o, climb, acceleration, sustained turn, descent, and landing that had not been considered in the generalized Breguet range equations. Furthermore, the method is immediately applicable to hybrid propulsion
102
12000 Range (nautical miles) 10000 8000 6000 4000 2000 0 0.5 1.0 1.5 2.0 WRF 2.5 3.0 3.5
Short Term Technology
k=7.94
Long Term Technology
GBR FLOPS
Figure 36: Range vs. WRF by a modied FLOPS and the generalized Breguet range (GBR) equations
system architectures. The fundamental approach of the proposed method is similar to that of the traditional method. The mission prole is chopped into small legs, which allows us to assume that several parameters, such as the aerodynamic coecients and the overall eciencies of power-paths, are constant in each leg and simplify the associated equations a great deal compromising the analysis accuracy as little as possible. For each leg, the weight fraction or the amount of both consumable energy and non-consumable energy is calculated. By summing up the results, the amount of energy required to perform an entire mission can be estimated. For the purpose of generality, the propulsion system is assumed to comprise multiple power-paths that use both consumable energy sources and non-consumable energy sources. The value of k is allowed to vary across the legs. The weight of the aircraft may or may not stay the same for a certain leg, while other parameters are set to their representative values in each leg. Accounting for the variation of aircraft weight due
103
sth Segment Cruise s-1 Climb 1st Segment 0 1 Warm up Take-off and Taxi Payload drop Loiter n Landing s Descend
Figure 37: Discretized mission prole
to energy consumption in each leg, ensuing a considerable degree of complication in the mathematical formulation of the mission analysis, is found signicantly to reduce the numerical error, which is discussed in the last part of this section. 4.3.1 Consumable Energy Sizing
Consumed energy is power multiplied by time: dECE = poi dt =
i
iP
i
dt
(64)
The change in the amount of consumable energy is proportional to the change in the weight of energy sources. dECE = CE dWCE Combining Eq. (64) and Eq. (65) yields dWCE =
i i i i
(65)
iP CE
i i
dt
(66)
As mentioned before, two types of energy are being considered. The ways of calculating the amount of consumable energy dier, depending on how aircraft weight varies.
104
4.3.1.1 Variable Aircraft Weight (k = 0) If aircraft weight changes in the mission leg, the amount of consumable energy consumed in the mission leg can be calculated based on the generalized weight dierential equation presented in 3.4.1.3. Substituting Eq. (66) into Eq. (43) yields dW = W
nCE i=1
k
i
i i
CE
P W
dt
(67)
By solving Eq. (67) for the sth leg, the weight fraction of the leg obtained is W (s) = exp W (s1)
t(s) t(s1) nCE
k
i
i i i=1 CE
P W
dt
i
(68)
If the fragmented mission legs are small enough such that k i /CE i can be assumed to be constant, then the above equations can be approximated as follows: W (s) (s) = exp k (s) (s) CE (s1) W where (s) is the weight specic mechanical energy, which is given as
(s) (s) t(s)
(69)
=
t(s1)
P W
dt
(70)
and CE is the overall power specic fuel consumption (OPSFC), which is equivalent to the power-specic fuel consumption (PSFC) of conventional combustion engines and given as
(s) CE nCE
=
i=1
i
i i
CE
(71)
The ways of computing (s) dier, depending on the existence of excess power. In the case of positive excess power, d(h + V 2 /2go ) dze P dt = = W 1u 1u where D+R = nV u= T CD + CDR CL 105 /
i=1 nT + i
(72)
i Pref WTO
i
(73)
Therefore, Eq. (70) can be rewritten as
(s) ze
(s)
=
ze
(s1)
dze 1u
ze 1u
(s)
(74)
In the case of zero excess power, such that during cruise or a sustained turn, required power equals the total drag multiplied by free-stream velocity as follows: P dt = W or P dt = W Therefore, Eq. (70) can be rewritten as or (s) =
s(s) s(s1) (s) t(s)
D+R W
V dt
(75)
D+R W
ds
(76)
=
t(s1)
D+R W
V dt
CD + CDR CL
V t
(77)
D+R W
ds
CD + CDR CL
s
(78)
Then, the ratio of consumable energy weight used in the mission leg can be expressed as follows: WCE W (s1) W (s) (s1) = = WTO kWTO k where (s1) = When payload that weighs WP given as
(s)
1
W (s) W (s1)
(79)
W (s1) WTO
(80)
(s1)
is dropped at the beginning of the leg, (s1) is W (s1) WP = WTO
(s1)
(s1)
(81)
where W (s1) is the weight of the aircraft right before the payload drop.
106
4.3.1.2 Constant Aircraft Weight (k = 0) If aircraft weight does not change in the mission leg, and the propulsion system consumes a certain type of fuel, fuel consumption is in proportion to power consumption. By dividing Eq. (66) by the aircraft weight, dWCE = W
nCE
i
i i i=1 CE
P W
dt
(82)
Note that W is constant. By integrating Eq. (82) into the ight time or the ight distance of the sth mission leg, WCE = W or WCE (s) = (s1) (s) CE WTO
(s) (s) t(s) nCE t(s1)
i
i i i=1 CE
P W
dt
(83)
(84)
The weight of the total consumable energy normalized by the take-o gross weight, CE , can be obtained as the sum of the normalized weight of the consumable energy of the individual leg. CE where
CE
WCE = = (1 + WTO
m CE ) s=1
WCE WTO
(s)
(85)
is the consumable energy allowance ratio that accounts for a propulsion
system with poorer-than-normal energy consumption and the amount of unusable energy. In the traditional aircraft sizing method, the total aircraft fuel generally includes mission fuel as well as a 5% allowance for the reserve fuel which accounts for an engine with poorer-than-normal fuel consumption and an additional 1% allowance for trapped fuel [93]. In such a case,
CE
for the conventional aircraft is 0.06. However,
the proper allowance for unconventionally-powered aircraft will dier, depending on the characteristics of the propulsion system and energy storage systems. 4.3.2 Non-consumable Energy Sizing
If the aircraft is powered on solely non-consumable energy sources, then the value of k is zero. Therefore, energy weight fraction can be computed in the same way as 107
done for consumable energy sizing when k = 0 presented 4.3.1.2. If the propulsion system consists of multiple power-paths that include consumable and non-consumable energy sources, however, aircraft weight may vary in a mission leg. In this case the amount of non-consumable energy can be computed by using the information of the amount of consumable energy in the mission leg, which can be estimated by the process described in the previous section. The amount of non-consumable energy for the j th power-path is given as power multiplied by time: dENE = poj dt = Combining Eq. (65) and Eq. (66) yields CE dWCE = Combining Eq. (86) and Eq. (87) yields
j dENE i i j
j
P
j
dt
(86)
iP
i
dt
(87)
=
j i
i
j
CE dWCE
i
i
(88)
By integrating Eq. (88) for the mission leg and dividing by the take-o gross weight, the amount of non-consumable energy normalized by the take-o gross weight required for the mission leg can be computed as follows: ENE WTO
j (s1)
ENE WTO
j
(s)
=
j i
CE WCE j WTO
i
i
(s)
(89)
If the propulsion system consists of only non-consumable power-paths, this approach is not possible. However, in such a case, the aircraft weight stays unchanged during the mission segments, which leads to the following equation by modifying Eq. (86): j P dENE = j dt WTO W The amount of the j th energy consumed in the sth leg can be expressed as ENE WTO
j (s1) (s) j
(90)
ENE WTO
j
t(s)
=
t(s1)
(s1)
j
j
P dt W
(91)
108
which is an equivalent equation to Eq. (89) for the propulsion system that consists of non-consumable power-paths only. Total non-consumable energy can be calculated by summation of the consumed energy of individual segments that are computed by either Eq. (89) or Eq. (91).
j ENE (0)
WTO
j ENE
j ENE
(1)
WTO
(s1)
+
j ENE (s)
j ENE
(1)
WTO
j ENE
(2)
+ +
WTO
j ENE
(m1)
WTO
WTO
+ +
WTO ENE WTO
j (0)
j ENE
(m)
=
WTO ENE WTO
j (m)
=
ENE WTO
j
(92)
Therefore, ENE = (s) WTO and WNE =
j=1 nNE j
m
(s1)
nNE j
j
(93)
s=1
j WNE
=
j=1
ENE NE
j
j
(94)
The weight of the total consumable energy normalized by the take-o gross weight, CE , can be obtained as follows: NE Where
NE
WNE = = (1 + WTO
m NE ) s=1
(s1) (s) NE
(s)
(95)
is the non-consumable energy allowance ratio.
As aforementioned, the formulation presented in this section has been developed with a consistent consideration for the variation of aircraft weight in a mission leg due to energy consumption. Ignoring such a weight variation results in a simpler formulation. However, the simplication may cost a signicant error or require a larger number of mission segments for comparable accuracy. The following example is a simple analysis for a notional zero-emissions aircraft. The wing area and the
2 drag polar are 500 ft2 and 0.01489CL 0.0051CL + 0.0146. The aircraft cruises at
Mach 0.6 ight speed and a 60,000 ft altitude, where the air density and TAS (true
109
0
% Error of Aircraft Weight After Cruise
-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 0 50 100 150 200 Constant Weight Varying Weight
Number of Cruise Segments
Figure 38: Comparison of numerical errors in two dierent approaches to mission analysis with varying number of mission legs
air speed) are the 0.000224 slug/ft2 and 581 ft/sec, respectively. The aircraft weight after cruising 2,000 nmi is computed varying the number of segments in two ways: with and without consideration of the variation of aircraft weight. First, the cruise segment is divided into 200 small legs, and the aircraft weight at the end of the cruise segment in consideration of the variation of aircraft weight is computed at 35,736 lbs. Since two hundred is a suciently large number, this value is regarded as a true value for this analysis. Subsequently, the aircraft weight at the end of the cruise is computed both with and without consideration of the variation of aircraft weight lowering the number of legs from 100 to 1, and their errors with respect to the true value are depicted in Figure 38. It is found that discretizing the mission with the coarser legs leads to underestimation of increases in aircraft weight for both methods. However, considering the variation of aircraft weight in a mission leg signicantly reduces the error, which 110
means the method allows a lower number of mission legs for the same level of analysis accuracy. The generalized mission analysis can estimate weight of required energy sources for various energy-propulsion system architectures. However, some cases may require more detailed supplemental analyses. First, an energy source may lose its energy capacity when it is not used for generating power. For example, a liquid hydrogen fueled aircraft that is not equipped with a cryogenic cooler will lose a certain amount of its fuel from boiling-o. Another issue is that specic energy of some energy sources may vary considerably depending on operating conditions. For example, the useful capacity of a battery changes depending on the discharge rate. Furthermore, the amount of battery storage capacity generally decreases over time.
4.4
Generalized Weight Estimation
Equation (39) can be rewritten as follows at take-o: WTO = WE + WPL + WCE (96)
The payload weight, WPL , is usually given as a part of the customer requirements, and the consumable energy weight is determined by Eq. (85). However, empty weight cannot be estimated directly from the information derived from the constraint and mission analyses. It is the authors opinion that the estimation of empty weight is one of the greatest challenges to the sizing of revolutionary aircraft concepts. Very little information on structures and subsystems is available at the time of aircraft sizing, and hence their designs are not yet embodied. It is not possible to employ high delity analysis for the estimation of airframe weight with no corporeal designs. This dilemma also exists for sizing of conventional aircraft, but does not lead to a dire situation because abundant historical data allows the designer to estimate the component weight using just a handful of information about the aircraft with fairly good accuracy. However, such empirical weight equations for revolutionary aircraft 111
are not available and the dilemma cannot be solved simply. One way to estimate the empty weight for revolutionary concepts is to use empirical equations for conventional aircraft with appropriate corrections that account for the repercussions of implementing an alternative energy-propulsion system architecture. In order to establish a relationship between the empty weight and the take-o gross weight, required in the proposed formulation, the empty weight is broken into subgroup weight terms as follows: WE = WE + WPS + WNE + WE (97)
where WE is the empty weight less the weight of installed propulsion systems estimated from a historical database; WPS is the weight of the propulsion system; WNE is the weight of the non-consumable energy source; and WE is the empty weight correction. The fundamental idea behind this equation is that the empty weight of revolutionary aircraft can be computed as the sum of two parts: propulsion system and non-consumable energy that is certainly not available from the traditional database, and others such as structures and subsystems whose weight may still be very close to the trend of the traditional database2 . Therefore, the latter can be obtained by computing the empty weight less the weight of installed propulsion systems from the traditional database with appropriate corrections. This correction is necessary for several reasons. First, using an unconventional energy source may incur signicant impact on structures and subsystems beyond propulsion systems. For instance, if an aircraft is powered by an electric propulsion
Similar methods have been applied to aircraft design optimization problems, in which a portion of the empty weight can be computed by physics-based analyses, and the rest can be estimated only by historical guidance. For example, Muoz and Sparkovsky [131] performed the synthesis and n design optimization of a turbofan engine coupled with an environmental control system (ECS) for a military ghter. The authors estimated the weight of the airframe excluding the weight of an ECS by regression analysis, while they calculated the weight of an ECS using a separate analysis code in order to capture the variation of the ECS weight as per the changes in ECS design variables.
2
112
system that produces thrust with electric motors, engine bleed air is no longer available, which forces the designer to nd a new way of integrating cooling systems into the aircraft. In addition, the aircraft is very likely not to use hydraulic systems to power actuators, which leads to elimination of centralized hydraulic pumps. Furthermore, electric generators that are driven by mechanical power extracted from IC engines are no longer necessary. These changes lead to an unavoidable conclusion of elimination of the accessory gear box, which is a mechanical interface of hydraulic pumps, generators, and engine turbine starters with IC engines. In parallel with this impact propagation, the conventional way of integrating the secondary power system will also be reexamined. In the light of inherent redundancy of electric power generation and conversion processes, conventional APU and EPU systems are likely to be eliminated. Because building distributed power generation and parallel power transformation is relatively easy for electric propulsion systems when compared to conventional IC engines, the same level of or higher reliability may be achieved without additional stand-alone secondary power systems. All of these changes in conventional subsystem integration are very likely to contribute to a reduction in aircraft empty weight, which must be accounted for by the correction weight term, WE . Incorporating alternative energy sources may also incur either positive or negative impact on the airframe weight. For example, if hydrogen fuel is used, an increase in empty weight due to installing a hydrogen storage tank may be signicant. In general, conventional aircraft store liquid hydrocarbon fuel inside structurally integrated fuel tanks whose structures carry load. However, the hydrogen fuel tank, either liquid or gaseous, may not be fully integrated with general structures such as bulkheads, shear webs, and skins, which will increase aircraft empty weight. Furthermore, if liquid hydrogen fuel is used, the low temperature requires the fuel tanks and associated plumbing to be insulated carefully to minimize heat leak into the fuel, thereby limiting boil-o. Not accounted for by regression equations created from traditional
113
Figure 39: AeroVironment WASP [134]
aircraft weight database, the hydrogen tank weight and additional weight provoked by installing the tank must be estimated by appropriate analysis and incorporated into the correction weight term, WE . However, it must be noted that novel approaches of integrating unconventional energy sources may not necessarily incur increases in empty weight. For example, WASP, a micro aerial vehicle developed by AeroVironment under DARPA support, is powered by two lithium-ion battery packs [132]. The conformal spar batteries attached to both sides of the Kevlar wing not only power the electric propulsion system, but also provide structural support to the wing. This design concept is an example of the Multifunctional Structures (MFS) design approach that pursues integrating normally stand-alone functions such as thermal management, batteries, power generation, and electronic subsystems into a composite structure thereby reducing volume and weight [133]. Such benets in saving airframe weight incurred by the use of unconventional energy sources need to be reected to WE via adequate analyses.
114
There is currently a lack of trusted and validated analysis capability for estimating the weight of revolutionary propulsion systems. The way of estimating propulsion system weight entirely depends on the availability of information and analysis models. The crudest way is to compute individual components weight with weight-specic power, a measure of how much power is consumed or produced per the unit mass, of each component. The weight of a component can be computed by its reference power divided by weight-specic power. The selection of the reference power may dier on a case-by-case basis. For example, the weight-specic power of electric motors is usually measured at maximum continuous shaft power. Then, Eq. (97) is rewritten as follows: WE = WTO + WTO + NE WTO + WTO where WE WPS WNE WE = , = , NE = , and = WTO WTO WTO WTO can be estimated from a traditional database, or the following data provided by Torenbeek [104] for subsonic light aircraft: 0.45 for xed gear, 0.47 for retractable gear, 0.50 for utility category, and 0.55 for acrobatic category. By combining Eq. (85), Eq. (96), and Eq. (98), the take-o gross weight equation is given as follows: WTO = WPL 1 NE CE (99) (98)
The take-o gross weight equation, Eq. (99), cannot be solved in a closed form since , , and are also functions of the take-o gross weight. Therefore, an iterative process must be used to solve the equations. The iteration process is very similar to the traditional process presented in most aircraft design books. First, make an initial guess of the take-o gross weight, W TO , and calculate the reference power of each power-path, Pref /WTO , and wing area, S, by simply multiplying the power to weight ratio, Pref /WTO , with the guessed value of the 115
i i
Guess WTO
Constraint Analysis
Mission Analysis
WTO S
Compute Pi o SL and S
Pi o SL WTO
CE
NE
Compute WE , WE , and WPROP
Compute , , and
Empirical or Physics Based Analyses
Compute WTO
WTO =
WP 1 CE NE
No
WTO WTO <
Yes
WTO , S , Pi o SL
Figure 40: Weight estimation process
take-o gross weight, W TO , and dividing W TO by wing loading, WTO /S, respectively. Then, compute the group weights from empirical databases and physics-based analysis tools, if available. Next, compute updated take-o gross weight from Eq. (99). Iterate the whole process until the convergence criterion is satised.
4.5
The Process of AIASM
The fundamental elements that were discussed in the previous chapter are put together in AIASM. The basic structure of the method remains the same as that of the traditional method as depicted in Figure 41 which highlights the primary changes made to the traditional sizing process. First, the information of specic energy and efciencies, in lieu of the traditional engine deck, is used to represent propulsion system performance. Secondly, the historical database is still referred to for the estimation of airframe weight. However, physics-based analysis must supplement empirical analysis. In addition, constraint analysis is changed from thrust-based to power-based, and
116
Aerodynamics
CL CL
Point Performance Requirements
CD
Constraints Analysis
<1 Pref > < Prefn >
Propulsion
Specific Energy
< Prefi >
WTO
WTO S
WTO
WTO S
WTO
WTO S
<i Pref>
Weight Fraction
Efficiency Maps
S
WTO
Mission Analysis
Alt
Notional Concept
Specific Power
dW = kdWCE
WTO
WE , WCE , WP
Weight
Historical Data Empirical Eqs. W TO= (WE) Physics Based Tools
Radius
Figure 41: Overview of AIASM
power is used along the downstream analysis. The power-based constraint analysis is performed with the set of new equations presented in 4.1. Mission analysis is performed with a set of generalized equations formulated in 4.3. The fundamental feature of this method is that, unlike general aircraft sizing tools, it does not require a set of engine deck data that describe thrust (or power) and specic fuel consumption (or fuel ow) at dierent ight conditions and power settings. Legacy vehicle sizing codes estimate the fuel consumption of each mission leg by interpolating the engine deck data around the specic ight condition. However, this traditional method is not appropriate when the power available and fuel consumption depend on more parameters such as power fractions of all power sources and time dependent factors (i.e. current draw for electric battery) as well as ight conditions and power setting. In contrast, AIASM directly calculates the fuel consumption or energy consumption by running the nested propulsion system model at a specic ight condition accounting for all important factors that aect fuel and/or energy consumption. Such dynamic integration is essential for sizing a hybrid power-generation system.
Mission Performance Requirements
117
Similarly, the environment does not require an engine scaling law, the relationship between propulsion system weight and output power (or thrust). Because the technology behind the traditional air-breathing combustion engines has matured, a scaling law for such an engine is well established and its implementation signicantly accelerates the aircraft sizing process without considerable errors. However, generic scaling laws of emerging electric propulsion architectures are not available at the same level of accuracy. Thus, the propulsion system weight must be estimated during the iterative processes of aircraft sizing, which may increase computation time and resources. The delity of a sizing analysis by this method largely depends on the quality of input data. There is no distinguishable dierence in preparing aerodynamic data between the traditional sizing method and AIASM. A range of methods from simple empirical equations to sophisticated computational uid dynamics (CFD) analysis may be employed depending on required analysis delity and available information and resources. In contrast, the inputs for performances and weight of alternative energy-propulsion system architectures can be developed by neither traditional practices using empirical equations nor legacy codes such as NEPP3 (NASA Engine Performance Program) [135] and WATE4 (Weight Analysis of Turbine Engines) [136]. Therefore, appropriate analysis codes need to be acquired on a case by case basis. Due to the scarcity of reliable analysis tools for an alternative energy-propulsion system architectures, a lot of previous studies assumed a constant eciency with ight as well as estimated propulsion system weight simply by multiplying the amount of power required by a representative value of specic power.
NEPP analyzes the one-dimensional, steady state thermodynamic characteristic of an aircraft jet engine. NEPP estimates the performance of the engine in the form of an engine deck as well as detailed thermodynamic analysis results at each station and component. 4 Originally developed by the Boeing Military Aircraft Company in 1979 and improved by NASA and the McDonnell Douglas Corporation, WATE embodies a physical engine by estimating the weight and envelope dimensions of large and small gas turbine engines using a semi-empirical method.
3
118
The equations of both this constraint analysis and the mission analysis of AIASM are described with respect to weight-specic parameters such as power-to-weight ratio, wing loading, and energy weight fraction. As discussed in 2.2.6, the weight specic parameter-based approach to solving an aircraft sizing problem signicantly reduces computational eorts in a human-in-the-loop environment. If the aircraft sizing environment is automated by an optimization tool, however, the actual-valuebased approach may supersede the weight specic parameter-based approach because iteration loops associated with the estimations of aircraft weight can be eliminated and absorbed in the system level optimization loop. Nevertheless, it must be noted that equations described with respect to weight-specic parameters can be still used in the actual-value-based approach with no modication. The bedrock idea of the actual-value-based approach is to use actual values for available power and energy as design variables. Whether the equations representing power balance and energy balance are described in terms of weight-specic parameters or not does not matter for the actual-value-based approach. In contrast, the weight specic parameter-based approach requires the equations described in terms of weight-specic parameters. The bottom line is that the formulation based on weight-specic parameters is applicable both for the weight specic parameter-based approach and the actual-value-based approach.
4.6
Non-dimensional Aircraft Mass (NAM) Ratio
The AIASM will enable designers to size simultaneously and optimize the airframe and revolutionary system propulsion system, and is able to support a wide range of research regarding the design of alternative energy-propulsion system architectures. First of all, the environment may aid an arbiter in selecting the most suitable propulsion system architecture for a given mission. The emergence of various alternative energy sources results in a large combinatorial space for possible propulsion system
119
architectures. Therefore, the selection of the correct propulsion system architecture becomes more challenging than before unless it can be proved that one specic type of architecture substantially outperforms others for most missions. As a case in point, Rohrschneider et al. [56] explore ight system options for the design of a long endurance Mars airplane mission. The study investigates ve dierent alternatives with regard to the propulsion system as well as ve dierent vehicle conguration options.5 Furthermore, such an architectural selection process will require numerous opportunities for decision making, including decisions on energy sources, power generation, conversion, and means of producing thrust. For example, even if fuel cells are selected as the primary power source, a variety of decision making options are still available, for example, type of fuel cells (proton exchange membrane, solid oxide fuel cells, etc.), type of fuel (methane, hydrogen, petroleum, etc.), type of fuel storage (pressurized gaseous tank, cryogenic liquid tank, etc.), and type of electric motors (general AC motor, brushless DC motor, and superconducting motor, etc.). Such a large number of alternatives for each disciplinary design may result in a myriad of combinatorial options in terms of system-level designs, which will bring signicant complications into the conceptual design phase. The problem may be alleviated by using a qualitative assessment technique such as Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) that facilitates downselecting the combinatorial space into a manageable number of promising options based on a subjective ranking of some evaluation criteria. Nevertheless, the capability of quantitative assessment for dierent types of propulsion system architecture is crucial in the selection of the best combination, particularly when one tries to nd a correct
The propulsion system options includes an NTO/MMH bipropellant rocket, a battery powered propeller, a DMFC powered propeller, a beamed solar powered propeller, and a beamed microwave powered propeller. The vehicle conguration options includes a straight wing with a single vertical tail, a straight wing with two vertical tails, a wing-canard, a swept wing with a single vertical tail, and the ARES (Aerial Regional-Scale Environmental Survey) conguration (see Figure 5 in 1.1.5). Their research identied DMFC and a straight wing with two vertical tails as the best options for the on-board power system and the external conguration.
5
120
power mix for hybrid propulsion aircraft. In addition to such cardinal research in which a specic mission identies the best propulsion system, exploratory research in which a specic propulsion system expands available mission space will be also of importance. The history of human technology has shown that needs promote new technology, and, in turn, the technology reveals new needs. As illustrated in Figure 42, IC engines fueled with conventional aviation fuels have produced aircraft which have addressed the shown portions of the mission space. As revolutionary propulsion systems become tangible, however, the technology may open up new feasible and viable mission spaces that the aerospace community has previously abandoned, fettered by ineluctable logic due to obvious limitations of conventional propulsion systems. Furthermore, some alternative propulsion-energy architectures may be more attractive for such an unconventional mission. Therefore, evaluating the potential of an alternative energy-propulsion system architecture must involve investigations within a large area of the mission space that is composed by performance metrics such as ight speed, range, and payload. In addition, evaluating concepts various for alternative propulsion-energy architectures must go along with exploring the mission space in the light of their current capabilities as well as projected capabilities in the near/long term future. At present, the biggest challenge to such exploratory research is the following: it requires a rigorous survey of a possible eet of aircraft for a given energy-propulsion architecture rather than a single representative aircraft. Recently, Soban and Upton [137] have proposed an interactive assessment environment that can conduct a qualitative mapping of new propulsion technologies to their goodness for performing a mission. In order to perform such qualitative mappings, the authors developed new techniques, named IRMA (Interactive Recongurable Matrix of Alternatives) and IQAM (Interactive Qualitative Assessment Matrix), based on well-known qualitative assessment techniques, such as the morphological matrix
121
Fighter
Velocity
Airline Jet
General Aviation Small UAV Helios
Range
Figure 42: Mission Space Exploration
and Quality Function Deployment (QFD). The original objective of this environment is to identify the classes of air-vehicles that could best prosper from new propulsion technologies. Nevertheless, the environment would also be equally useful to compare alternative concepts embracing the options for propulsion system architectures and vehicle congurations for a given mission. However, such a qualitative assessment is vulnerable to subjective judgments by its users. Therefore, there is a need for the capability of mapping the characteristics of a propulsion-energy system architecture into a mission space in a quantitative manner. A simple quantitative method can be developed based on aircraft weight decomposition in connection with the generalized Breguet range equations presented in 4.2. The take-o gross weight can be broken down as follows: WTO = WE + WPS + WNE + WE + WCE + WPL By dividing Eq. (100) with WTO , WPS WNE WE WCE WPL WE + + + + + =1 WTO WTO WTO WTO WTO WTO (101) (100)
The correction term, WE , represents the collective impact on empty weight by integrating an unconventional propulsion system and an energy source, and can be further 122
decomposed into two terms: the change in empty weight due to the integration of an unconventional propulsion system and the change in empty weight due to the integration of an energy source. Assuming that each term is proportional to the weight of the propulsion system and the energy source, respectively, Eq. (101) reduces to the following: WPS WNE WCE WPL WE + (1 + ePS ) + (1 + eNE ) + (1 + eCE ) + =1 WTO WTO WTO WTO WTO (102)
where ePS , eNE , and eCE represent the impact on the empty weight from integrating an unconventional propulsion system, a consumable energy source, and a nonconsumable energy source, all normalized by the take-o gross weight. In the case of conventional propulsion systems, ePS , eNE , and eCE ought to be zero. For simplicity, it is assumed that a single consumable energy source is used. Then, Eq. (102) reduces to the following: WE WPS WCE WPL + (1 + ePS ) + (1 + eCE ) + =1 WTO WTO WTO WTO (103)
The rst term represents the ratio of airframe weight to the take-o gross weight. The airframe weight (WE ) herein excludes the weight of the propulsion system, the energy source, and all components pertaining to the given energy-propulsion system architecture. For example, if liquid hydrogen is used as fuel, the hydrogen tank and cooling system are not accounted as a part of WE . This ratio is reasonably independent of the choice of propulsion system architecture and energy sources, considering structures, a major portion of an airframe, is mainly sized to sustain the aircraft load, which is strongly related to the take-o gross weight. The last term is payload-toweight ratio, often considered a metric of the eectiveness of a transportation system. Higher payload-to-weight ratio means a lighter transportation system for the same weight of payload. As WPL /WTO approaches to zero, WPS /WTO and/or WCE /WTO are allowed to increase.
123
Figure 43: Dualistic relationship between energy weight fraction and propulsion system weight fraction
This equation also provides an insight to the physical limitation of a given energy propulsion system architecture. For given values of WE /WTO and WPL /WTO , the re maining weight fraction (1 WE /WTO WPL /WTO ) must be apportioned between WPS /WTO and WCE /WTO . Therefore, Eq. (103) expresses a boundary of allowable values of the dualistic, dichotomous terms (WPS /WTO and WCE /WTO ), which are depicted in Figure 43. The higher WPS /WTO and the lower WCE /WTO result in a brawnier aircraft. On the other hand, lower WPS /WTO and higher WCE /WTO produce a fatter aircraft. Then, Eq. (103) expresses the linear relationship between the maximum allowable propulsion system weight and energy weight for the energy-propulsion system
124
Alternative Energy-Propulsion System Architecture
Conventional System Architecture
Figure 44: Comparison of an alternative energy-propulsion system architecture and a conventional energy-propulsion system architecture
architecture. The values of ePS and eCE are determined by the selection of an energypropulsion system architecture. Assuming that all coecients are constant for the variation of two variables,
WPS WTO
and
WCE , WTO
the boundary is given as lines as shown in
Figure 44. Note that the gradient of this boundary for the combined weight fractions of the conventional energy-propulsion system architecture is -1, provided that ePS , eNE , and eCE are zero. The slope of the boundary line for an alternative energypropulsion system architecture might dier depending on ePS , eNE , and eCE . Treating Figure 44 as the rst quadrant, another quadrant that maps the energy weight fraction to range capability is juxtaposed as shown in Figure 45. The curve in the newly added second quadrant can be obtained by the generalized Breguet range equations (Eq. (60) or Eq. (62) in 4.2). In addition, the original diagram can be
125
Figure 45: Mapping the energy weight fraction to the mission range
126
Figure 46: Mapping the cruise velocity to the propulsion system weight fraction
horizontally combined with another quadrant that relates the cruise velocity to the propulsion system weight fraction based on the specic power of the propulsion system, as shown in Figure 46. Finally, assembling Figure 45 and Figure 46 complements the development of an integrated visual environment that depicts the allowable mission capabilities in terms of cruise velocity and range for a given energy-propulsion system architecture. The resulting diagram is composed of four quadrants: each representing a design space in terms of aircraft mass, energy, mission, and power space, as shown in Figure 47. The mass space describes the allowable propulsion system mass fraction and energy mass fraction, which are determined by the airframe mass fraction, including the mass fraction of the components concomitant with implementing an energypropulsion system architecture, and a given payload mass ratio. The reduction in airframe weight due to advanced materials or better airframe integration will shift the limit state curve in the diagonal direction, which allows the aircraft to increase the amount of on-board energy sources and/or propulsive power. Alternatively, with a xed airframe mass ratio, the limit state curve may vary with the payload mass
127
Aerodynamics Lighter Propulsion System
y An
d an
m lI Al
em ov pr
Figure 47: Illustration of a notional NAM ratio diagram
ratio. The energy space establishes the relationship between the allowable mission range versus the energy weight fraction. The limit state curves will be improved by the propulsion system and aerodynamic eciencies. The power space depicts the allowable cruise speed versus the propulsion system mass fraction. The limit state curves will be improved by the specic power of a propulsion system and the aerodynamic eciencies of the aircraft conguration. Finally, the mission space integrates velocity and range capabilities, which can be enhanced by improvements of any other spaces. A salient feature of this diagram is that it identies the allowable mission space for a vehicle rather than a single point design. This capability is also useful for the conceptual design of a terrestrial exploration winged platform, in which the allowable 128
Aerodynamics Energy Efficiency
ts en
An Ad d va Im nc pr ed ov M ed a In teri te al gr s at io n
range of the exploration per the selection of the propulsion system and energy architectures is given as an FoM rather than a design constraint. With such applications in mind, mass appears to be a more universal parameter than weight. Without any loss in generality, the rst quadrant can be expressed in terms of mass ratios. In this sense, this visual environment is henceforth named as the Non-dimensional Aircraft Mass (NAM) ratio diagram. In order to demonstrate the usefulness of the NAM ratio diagram, a simple example study has been performed, in which three alternative propulsion-energy architectures are considered for the design of a high altitude unmanned surveillance platform: a) conventional turbo prop engine fueled with jet fuel; b) PEMFC-powered electric propulsion system fueled with liquid hydrogen; and c) triggered isomer heat exchanger engine (TIHE) [83] fueled with isomers of hafnium (178 Hf), denoted as Architecture A, B, and C. The assumptions for each architecture are listed in Table 5. For Architecture A, it is assumed that eCE = 0 and ePS = 0. The value of is estimated based on a simple equation in Ref. [103] that computes from the density ratio. For Architecture B, the propagated impact of introducing the FC-powered electric propulsion system on other subsystems is ignored and the impact of the use of hydrogen fuel is taken into account by assuming eCE = 0.5, which means that the weight penalty on the empty weight equals one half of the hydrogen fuel weight. For Architecture C, it is assumed that the primary impact of introducing an atomic power source on airframe weight is the addition of a radiation shield, whose weight is sensitive to design parameters such as the tolerance of equipment radiation, and the distance from the radiation source. It is assumed that the weight is reasonably proportional to the amount of output power of the TIHE engine, which is reected in the value of ePS = 0.51. The weight data of several aircraft that belongs to this category are listed in Table 6. The empty weight fraction without including the propulsion system as a
129
Table 5: The assumptions of architecture options Architecture ePS eCE Overall eciency at cruise Specic energy (ft) Specic power (ft/sec) A 0 0 0.16 - 0.2 1.786 107 1400 0.27 B 0 0.5 0.38 - 0.43 4.464 107 138 0.3 C 0.51 0 0.35 - 0.5 5.653 1011 421 0.3
Table 6: High altitude unmanned aircraft Predator 2230 1130 166.4 650 450 0.43 0.07 0.29 0.20 Dark Star 8600 4286 459 3314 1000 0.45 0.05 0.39 0.12 Proteus 13700 5900 459 6000 1800 0.40 0.03 0.44 0.13 Global Hawk 25600 9200 1580 14500 1900 0.30 0.06 0.57 0.07
WTO WE WPS WF WPL
(lbs.) (lbs.) (lbs.) (lbs.) (lbs.) WPS /WTO WF /WTO WPL /WTO
part of the vehicles empty weight ranges from 0.3 to 0.45. The payload weight fraction ranges from 0.07 to 0.2. Furthermore, values of L/D at cruise are assumed to range from 25 to 35. In order to encompass the gamut of vehicle design attributes including dierent empty weight fractions, payload fractions, and the aerodynamic eciencies, Monte Carlo simulations were performed with a uniform distribution of the associated variables. A NAM ratio diagram that compares the three architectures is created from two hundred random cases, in which the cruise speed is varied from M 0.3 (290 ft/sec) to M 0.7 (774 ft/sec), as depicted in Figure 48. In the power space of the gure, the three architectures occupy distinct regions.
130
The propulsion system weight fraction of Architecture A ranges from 0.03 to 0.09 for the range of cruise speed. Architecture B has higher values for the propulsion system weight fraction, ranging from 0.25 to 0.61, for the same ight speed, compared with the others. The poorer specic power of fuel cell systems makes Architecture B a heavier propulsion system than others for the same level of power. Furthermore, the maximum allowable cruise speed of aircraft equipped with Architecture B is limited to approximately 650 ft/sec. In the weight space, the three architectures also occupy dierent regions. Architecture B has a cluster at the region of higher power fraction and lower energy fraction, because of its heavier propulsion system and LH2 tank, an aircraft with Architecture B is constrained to carry less fuel. Nevertheless, such an aircraft outperforms the ones with a conventional architecture in the low-speed range as shown in the mission space, because the architecture provides higher overall eciencies of converting the more energetic fuel. The specic power characteristics of Architecture C allow higher fuel fractions when compared with Architecture B. Furthermore, the specic energy of the isomer is far greater than those of other energy sources, which allows virtually unlimited ight.
4.7
Extension to Solar-Powered Propulsion Architecture
Solar-powered aircraft obtain energy from the sun using Photovoltaic (PV) or solar cells as they are often called. This type of semiconductor device converts sunlight into direct current (DC) electricity. By collecting energy from outside of the aircraft, solar-powered aircraft may y with less or no on-board stored energy source. An important aspect of the sizing of solar-powered aircraft is that a solar-powered propulsion system introduces a coupling between the available power and wing geometry, creating an additional constraint in the design space. It is obvious that the maximum available solar power is limited by the wing area since PV cells are generally installed over the wing. In fact, limitation of available power is inherent for other
131
Power Space
0.6 0.5
Mass Space
0.4
0.3
0.2
0.1
0
(ft/sec)
700
600
500
400
300
200
100
0
1.00E+02 0.1 1.00E+03
0.2
0.3
0.4
0.5
0.6
0.7
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
Mission Space
1.00E+10
Energy Space
(nmi)
Architecture A (Turboprop + Jet Fuel)
Architecture B (PEM Fuel Cells + Hydrogen)
Architecture C (TIHE + Isomer)
Figure 48: The NAM ratio diagram of a high altitude unmanned aircraft
132
types of propulsion systems. As discussed in the previous section, the propulsion system weight is limited depending on the maximum allowed total aircraft weight, which underlies the limitation of available power. However, the maximum available power of other aircraft is not directly dictated by wing area. Yet, as long as the coupling does not act as an active constraint on aircraft sizing, the coupling will be of no interest. Under the circumstances that the design power is determined to be considerably less than the maximum available power, the maximum available constraint does not act as an active constraint although the maximum available power is determined by wing area. However, the maximum available power is very likely to be an active design constraint in the design of solar-powered aircraft. The amount of vertically-aligned incoming solar radiation per unit area, often called the solar constant, is 1,352.8 watts per square meter. Both attenuation and radiation angles limit the actual energy conveyed from PV cells to much less than the solar constant. Thus, the available power from sunlight is considerably lower than the achievable power level of conventional IC engines. For example, if state-of-the art PV cells covered 541 square meters of the Boeing 747 wing shown in Figure 49, the system could deliver about 700 HP to a propeller, which equals roughly half of the maximum power of a P-51B/C Mustangs engine. The scarcity of power available from solar radiation leads to a typical integration of PV cells with wings. As shown with both the Pathnder and Helios, PV cells are installed almost throughout the entire wing surfaces, implying that the maximum available power of solar-powered aircraft is directly limited by wing area. In the next two sections, an additional design constraint imposed by the coupling of wing area and power available is derived for two dierent propulsion system architectures: a solar-powered electric propulsion system with or without regenerative fuel cells. In both cases, the constraint due to coupling of the maximum available power and aircraft size is established in terms of the maximum allowable wing loading.
133
P-51B/C Mustang
32 ft 3 in
231 ft 10 in
B747-400
Figure 49: B747-400 vs P-51B/C Mustang
4.7.1
Application to Solar-Powered Aircraft
If the aircraft is powered by solar energy only, the ight is limited to a certain period of daytime when the PV cells provide sucient power. Since the aircraft does not store any energy on-board, an energy balance is not included as a sizing constraint for this type of aircraft. The available power from a photovoltaic panel varies with time of year and latitude. The total available power from the PV panels that covers Sff 100% of the wing, is given as follows [138]: PP V = io SC SSff (sin() sin() cos() cos() cos(a)) (104)
where is the solar attenuation factor, which varies with the location; SC is the PV cell eciency; and is the latitude and is the earth declination angle. varies with 134
the day of the year (dn ), which is based on the vernal equinox (dn = 1 is March 21st), and is given as = 0.4091 sin(2dn /365) (105)
The hour angle (a) is given by the following expression, where i is the instantaneous time of day in hours, a = 2i/23.935 (106)
io is the amount of incoming solar radiation per unit area on the atmosphere boundary at a given time and is given as follows:
2 2 io = iom (rorbm /rorb )
(107)
where rorbm and iom represent the mean distance from the earth to the sun and the amount of incoming solar radiation per unit area on the atmosphere boundary with the mean distance. The distance from the earth to the sun (rorb ) varies throughout the year and is computed by the following equations: rorb = rorbm (1 e2 )/(1 + e cos()) is given as follows: = 2dn2 /365 (109) (108)
where the day number (dn2 ) is based on the data of perihelion for earths orbit, so dn2 = 1 is January 4th. The values for the constants used in the above equations are given below:
The mean orbital radius of the Earth (rorbm ): 1.496 108 km The mean solar intensity at the Earths orbital radius (Siom ): 1352.8 W/m2 The Earths orbital eccentricity (e): 0.017 The Earths mean radius (r): 6.378 106 m
135
250
200
0o Lat, Date 12/21 0o Lat, Date 6/21 x 60o Lat, Date 12/21 60o Lat, Date 6/21
(W/m2)
150
100
50
x
0 0 -50 5 10 15 20 25
time (hr)
Figure 50: Available power per unit wing area for several combinations of geographic locations and dates
The available power from PV cells can then be rewritten as PP V = S (110)
where stands for the amount of electric power produced by the PV cells per unit area of wing, computed by = io SC Sff (sin() sin() cos() cos() cos(a)) (111)
The power available per unit wing area, is plotted against time in Figure 50, in which PV cell eciency, PV cell ll factor, and solar attenuation factor(t) are 0.2, 75%, and 0.7, respectively. The available power-to-weight ratio is given as a function of wing loading as follows: W PP V = / W S 136 (112)
Power-to-Weight Ratio
Landing
Climb Max. Power Available
W P = / W S
Feasible
Wing Loading
Figure 51: Constraint analysis of solar-powered aircraft
The required power-to-weight ratio for performance requirements can be computed by the power-based master constraint equation in 4.1. The available power-to-weight ratio and the required power-to-weight ratio are overlaid in Figure 51, in which the feasible design range of wing loading is determined where
PP V W
Prequired . W
wing loading reduces wing area and thus aircraft weight, the highest available wing loading value is very likely to be the optimum solution. It must be noted that the reference power in Eq. (112) is measured by the output power of the PV panels. The required power must be estimated accordingly. 4.7.2 Application to Solar-Powered Regenerative Propulsion Aircraft
Solar-powered regenerative propulsion is a type of propulsion system capable of regenerating internal energy sources such as fuel cells or an electric battery using solar energy and provides a possibility for virtually perpetual ight. Such a capability can be obtained when the PV cells provide sucient extra energy during the day to regenerate internal energy sources for nighttime ight as illustrated in Figure 52. The on-board regenerative energy storage systems might include a Regenerative Fuel Cell
137
Since higher
PV Cells
Electric Propulsion System
Charge
Energy Storage System (RFC or Battery)
Discharge
Payload
Daytime
Nighttime
Figure 52: Illustration of a solar-powered regenerative propulsion system
(RFC) system envisioned for the AeroVironment Helios or a rechargeable lithium battery demonstrated by the AC Propulsion Solong drone [139]. Unlike aircraft powered solely by solar power, the energy balance throughout the entire 24 hour cycle must be addressed as a critical design constraint. Figure 53 depicts the typical prole of a daily power cycle of a regenerative solar powered propulsion architecture. For the daytime period (Td ), the PV cells provide the power required for ight, payload, and regenerating an on-board energy storage. During the nighttime period (Tn ), the on-board energy storage feeds the electric propulsion system. Td , herein, is dened as a period of time within a day when the PV cells can produce the extra amount of power that can be diverted to regenerate the on-board energy storage system. Therefore, the PV cells subsidize the power demand right before and after the daytime period. The energy balance requirement can then be stated for daytime and nighttime operation. for daytime balance: for nighttime balance: Esun |d = Ef light |d + EStorIN + Epayload |d EStorOU T = Ef light |n + Epayload |n Esun |n (113) (114)
A continuous ight for days or months is now possible when the total amount of daytime excess energy is at least the same as the energy required to regenerate
138
Td
Power/Aircraft Weight
Tn
EStorIn
Tn
EStorOUT
hr
EStorOUT
Total Power Required Power for Flight Power for Payload
Figure 53: Power prole of solar-powered aircraft with a regenerative propulsion system
the energy storage system. Therefore, the diurnal energy balance requirement can be stated as follows: EStorIN EStorOU T rt (115)
where the round trip eciency, denoted as rt is dened as the ratio of total energy output from the energy storage system to the total energy input transferred from PV cells to the energy storage system. The physical meaning of Eq. (115) can be explained graphically with Figure 53. The gridded area that represents the total energy input must be greater or equal than the roundtrip eciency times the shaded area that represents the total energy output from the energy storage system. Similar formulation of the diurnal energy balance has been proposed in previous research [140, 141]. Sharing the crux with previous work, this research attempts to reform the diurnal energy balance as an additional constraint with respect to wing loading.
139
The total energy obtained by PV panels during the daytime is given as follows: Esun = S
Td
dt
(116)
The energy required for level ight during the daytime is the integration of power required as follows: Ef light |d =
Td
D WV dt L +
(117)
Combining Eq.(113)-(117), the daytime and nighttime energy balance are consolidated into the following single equation: S
Td
dt
Td
D WV 1 + PPL dt + + L rt
Tn
D WV + PPL S L +
dt dt
Tn
(118)
Finally, by rearranging Eq.(118) the maximum allowable wing loading meeting the diurnal energy balance is given as follows: W S =
max Td Td D V L +
dt + dt +
1 rt 1 rt
Tn Tn
dt
D V L +
+
PPL W
+
PPL W
dt
(119)
(W/S)max does not involve power-to-weight ratio, appearing as a vertical line in the constraint analysis space as illustrated in Figure 54. This equation cannot be solved in a closed form because D/L, Td , and Tn depend on W/S. Therefore, it requires an iterative process, in which W/S is manipulated until the equation is satised. In addition, there may be a need for another iteration loop to solve the equation. If payload power is given as a xed input, the ratio of aircraft weight to payload power (PPL /W ) in Eq.(119) varies with the aircraft weight, which implies the constraint analysis is coupled with weight estimation. Several approaches can be taken to solve this problem. One possible approach is illustrated in Figure 55. The process begins with estimating the power prole of for a given time of year, latitude, PV cell eciency, and PV cell lling factor. This step is followed by an iteration loop, in which aircraft weight is varied. Note that is insensitive to the sizing parameters, and thus does not have to be involved in the iteration process. 140
Landing Power-to-Weight Ratio
Climb
Wing Loading Max. Allowable Wing Loading
Figure 54: Constraint analysis of solar-powered aircraft with a regenerative propulsion system
The ratio of aircraft weight to payload power (PPL /W ) varies with every update of the aircrafts weight during its iteration process. For the given PPL /W , the maximum allowable wing loading ((W/S)max ) is computed from a nested iteration loop using Eq. (119). Subsequently, the total power to weight ratio P/W can be estimated from the constraint analysis. These analyses lead to estimations of P and S, followed by an update of aircraft weight. These steps are iterated until aircraft weight suciently converges. Rizzo and Frediani [142] derived a similar wing loading constraint due to the diurnal energy balance from their work on solar powered aircraft. Zero lift drag coecients are predicted by a simple equation that includes wing area, wetted area, and at plate friction drag coecient and is given as a function of the Reynolds number. The induced drag coecient is predicted using a simple equation that includes the wing aspect ratio and Oswalds eciency factor. However, the authors did not consider the power consumption by the payload. Another limitation is that the aerodynamic 141
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= + dt + +
W S
dt +
dt
dt
Wguess PPL/ W
Power / Wto (Watt/N)
Td
Tn
Pref/W Pref, S W No
Power Balance
Components Weight
Converge? Yes W, Pref, S
Figure 55: Sizing process of solar-powered aircraft with a regenerative propulsion system
coecients are hard-coded into the wing loading constraint, which prevents designers from using more reliable aerodynamic analysis results when they are available.
142
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CHAPTER V
FORMULATION OF THE PROBABILISTIC AIRCRAFT SIZING METHOD
The architecture-independent formulation presented in the previous chapter paves the way for the development of a generalized aircraft sizing environment that is capable of handling various types of unconventionally propelled aircraft in a deterministic way. Nevertheless, none of the design parameters involved in the formulation can be accurate or certain since aircraft sizing is performed during the pre-conceptual or conceptual design phase, when the least amount of knowledge about the system is available. Therefore, if an aircraft is deterministically sized without adequate design margins, any unforeseen changes to the underlying assumptions about the associated parameters will result in failure of the design to meet any or all of the design requirements [100]. On the contrary, if extraneous design margins are imparted, the cost may increase without commensurate benets of risk mitigation. This antipodal nature of determining design margins has spawned Research Question II in 3.2: how to allocate design margins against probabilistic design constraints intelligently. This chapter presents the probabilistic aircraft sizing method (PASM) as an attempt to solve Research Question II, beginning with discussions related to several approaches to probabilistically constrained optimization problems found in literature in 5.1. This discussion and a numerical example presented in 5.2 reveal that the chance-constrained programming (CCP) is suitable for formulating a probabilistic aircraft sizing problem under uncertainty. In 5.3, PASM is formulated as the form of the CCP, built upon the equations of AIASM, which is followed by an abridged discussion
143
on numerical techniques to solve the resultant problem in 5.4. 5.5 introduces several derivative formulations that can be extended from the original formulation. The last section presents two sensitivity analysis techniques that enhance the capability of PASM.
5.1
Approaches to Probabilistically Constrained Optimization Problems
Fields such as nance, management and industrial engineering have rigorously advanced various methods to solve probabilistically constrained problems. Those methods can be broadly classied into three categories. The rst one is an averaging approach that rst xes the random variables at their mean values or any other representative value, and then solves the resultant deterministic problem. This approach seems eortless or like oversimplifying the probabilistic nature of a system. Nevertheless, it can be said that this method is implicitly exercised for every deterministic optimization problem, provided that in reality there are no truly deterministic problems. The second method is a penalizing approach. In certain situations, constraint violations can be corrected by appropriate compensating decisions. In such circumstances, one would rather investigate more challenging options that result in a better objective function value with an increased chance of violating the constraints. If the costs of the compensation can be quantied in the same currency of the objective function, the constraint violation can be considered a penalty to the objective function often modeled as a recourse function denoted as Q(x, ). The objective of this type of problem is reestablished as minimizing the total expected cost, which is a summation of the original cost function and the cost incurred by constraint violations as follows: min f (x) + E[Q(x, )]
x
(120)
This is the fundamental idea of the penalizing approach, and it has been further 144
developed into what is known as a two-stage or multi-stage stochastic programming [143, 144]. However, the concept of compensation is not appropriate in many applications where safety-relevant restrictions are critical design constraints, or the compensation cost cannot be modeled in the same currency of the original objective function in any reasonable way. The aircraft sizing problem also ts into this category. Under such circumstances, one would like to insist on feasibility as much as possible. However, such an ideal solution may be dicult to obtain in actual problems for several reasons. First, one can hardly nd any decision that would denitely exclude later constraint violations in the design of complex systems. Second, even if designers can identify the worst case scenario, the objective function value may be prohibitive. In general, cost and reliability conict with each other in applications in which the optimum solution is pushed to the constraint boundaries. A more conservative decision must commit a higher cost, but it is impossible to create a perfectly safe design because of unexpected extreme events. On the other hand, it makes sense to call decisions feasible whenever they are feasible with a high enough probability. Chance-Constrained Programming (CCP) is a mathematical formulation of this type of stochastically (or probabilistically) constrained problem. A generic expression for such a probabilistic constraint as an inequality is given as follows: min f (x)
x
(121)
s.t. P [gi (x, ) 0] gi where x and are the decision and random vectors, respectively, and g(x, )i 0 is the ith inequality constraint. The optimum solution to this type of problem can be found at the location of minimizing the objective function inside of the feasible set, all entities of which satisfy the probabilistic constraints with a probability of at least gi . In some cases, the probability level is strictly set from the very beginning (e.g., gi = 0.95, 0.99 etc.) by management decisions. In other situations, the decision 145
maker may only have a nebulous idea of the properly chosen level of gi , aware that higher values of gi lead to higher costs. Under such circumstances, the information of the objective function trends with respect to the target reliability can assist the arbiter. As it turns out, for most cases, gi can typically be increased over quite a wide range without signicantly aecting the optimal value of the problem, until it approaches 1, and then a strong increase in costs becomes evident. In this way, models with chance constraints can also provide insights into a good compromise between cost and safety, while stochastic programming oers a trade-o between cost at present and cost in the future. The chance-constrained approach has a long history, dating back to the work of Charnes and Cooper in linear programming in 1959 [145]. Since its conception, the method has been mainly applied to the elds of civil engineering, industrial engineering, and nance, such as inventory systems sizing [146], ecology model analysis [147], and portfolio selection [148, 149], in which uncertainty enters the inequalities that describe the proper working of a system under consideration. The earliest eort to apply the CCP approach to engineering design problems was made by Rao [150] who considered the engineering structural security of the probabilistic constraints in 1980 [151]. After that, the method has also been applied to structural optimization problems under uncertainty [152, 153, 154, 155] under the name of reliability-based design optimization (RBDO). In the RBDO problem, the probabilistic constraints are often formulated as elemental reliability indices corresponding to various limit states [156], which are essentially equivalent to the target probability in CCP. However, not much research has been performed for the application of the concept of CCP to aerospace vehicle designs so far. Smith and Mahadevan [157] applied the RBDO approach to the design of a second-generation reusable launch vehicle, in which a single probabilistic constraint was considered. However, no literature regarding CCP or RBDO approaches to airplane system designs has been found.
146
2 1.8 1.6 1.4 1.2 14 0.8 0.6
2
12
10
y
8
6
0.4 0.2 0
2 4
0
0.5
1
1.5
2 x
2.5
3
3.5
4
Figure 56: Design space of a deterministic optimization problem
5.2
A Numerical Example of Optimization Under Uncertainty
This section presents the application of the probabilistic approaches presented in the previous section to a numerical example problem to articulate the dierence in the approaches of the methods. A simple deterministic linear programming is given as follows: min f = x + 4y
x,y
s.t. x + 3y 2 x0 y0
(122)
The design space of this problem is illustrated in Figure 56. The optimum solution to this problem is found at x = 2 and y = 0, where the objective function value is 2. Now, it is assumed that the two coecients of x and y in the rst linear constraint
147
are not deterministic but probabilistic, following normal distributions, the mean values of which are their deterministic values, 1 and 3, and the standard deviation values of which are 1. Then the optimization problem is given as the following statement: min f = x + 4y
x,y
s.t. 1 x + 2 y 2 x0 y0 where 1 N (1, 12 ) and 2 N (3, 12 ) Unlike the previous deterministic system, the feasible solution area of this problem is not xed but amorphous. One intuitive approach, a so-called averaging approach, is to x the random variables at their mean value and to solve the resultant deterministic optimization problem, which results in an identical solution of the original deterministic problem. Nevertheless, the solution of the probabilistic problem can only satisfy the constraints by chance. The probability of meeting the constraint at the location of the solution can be computed by converting the linear combination of normal distributions into a standard normal distribution, yielding 0.5. However, this approach is not available when one wants to nd a solution that can ensure a higher probability of success. 5.2.1 Deterministic Solution Sampling (DSS) Method (123)
An alternative approach is the deterministic solution sampling (DSS) method which wraps the optimization process with a Monte Carlo simulation and sequentially solves a deterministic optimization problem which is set up with a set of sampled constraints. A certain number of simulations produce a set of optimum solutions, the distribution of which is shown in Figure 57 along with a set of corresponding objective function values, whose cumulative probability distribution (CDF) is depicted in Figure 58. What is the implication of the two graphs? The distribution of the objective function 148
4 16 3.5 14 3 12 2.5 10
10 16
18 18 16 14 14 12 12 12 10 8 10 8 6 6 4 8 6 4 2 4 14 16
2 8 1.5 14
6
f=3.25
0.5 2 0
y
0
0.5
1
1.5
2 x
2.5
3
3.5
4
Figure 57: Optimum solutions from Monte Carlo Simulation
obtained by DSS method encompasses all possible outcomes of the objective function values from a wait and see type simulation, which assumes that the optimum design solution will and can be achieved after all uncertainty sources are cleared. If one wants a 90% probability of meeting the constraint, then will setting the design variables to those corresponding to 90% of the cumulative probability of the objective function in Figure 58 be the solution? The answer is No, and even if the answer were Yes, this approach does not provide a unique solution. From Figure 58, the objective function values at 90% of the cumulative probability is estimated at 3.25. Two solutions whose objective function value is 3.25 exist as shown in Figure 57. 5.2.2 Two-Stage Stochastic Programming Method
Two-stage stochastic programming can be used when the following assumptions are valid:
The designer can correct constraint violation after all random variables are
149
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
1
2
3
4
5
6
7
8
Figure 58: CDF of optimum values of the objective function of the Monte Carlo simulation
observed.
The cost associated with the compensation is proportional to the shortage to
the constraint.
The objective is to minimize the expected total cost, the sum of the original
cost function and the cost incurred by compensation. Based upon the above assumptions, the given problem is formulated as the following two-stage stochastic programming: min f = x + 4y + E[Q(x, y, 1 , 2 )] x,y c (2 x y) if (2 x y) > 0 p 1 2 1 2 where Q(x, y, 1 , 2 ) = 0 otherwise 1 N (1, 12 ) and 2 N (3, 12 ) Q, a recourse function representing the cost incurred by the compensation, is assumed to be proportional to the minimum amount of the adjustment required to make the 150
(124)
2 1.8 1.6
10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 0 0.5 1 x 1.5 0 1y 0.5 1.5
1.4 1.2 1 0.8 0.6 0.4
E(f)
y
2
0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x
9.426 6.17068 8.77493 5.51961 8.12387 4.86855 7.47281 4.21749 6.82174 3.56642
Figure 59: The total cost function of the two-stage stochastic programming problem
constraint feasible, that is, the penalty coecient (cp ) multiplied by the amount of constraint violation (2 1 x 2 y). The optimum solution to the problem, when cp = 4, is found at x = 0.334, y = 0.584 and the value of the objective function is 3.567. The objective function, f = x + 4y + E[Q(x, y, 1 , 2 )], is plotted in Figure 59. Optimum solutions of this problem are heavily aected by the values of cp as shown Figure 60, which depicts the optimum values of the total cost (x + 4y + E[Q(x, y, 1 , 2 )]) and the design variables varying the value of cp . As cp decreases, the optimum moves toward the point (x = 0 , y = 0), where the probability of failure is the greatest, but the original objective function (x + 4y) is the most favorable. These results are consistent with the real world observation: wherever is less penalty incurred by a constraint violation, a more aggressive decision will appear attractive.
151
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 Penalty Coefficient x y f
Figure 60: Optimum solutions per the value of the penalty coecient
5.2.3
CCP Method
If any compensation is not allowed, the two-stage stochastic programming is not appropriate. Under such circumstances one would rather seek an optimum solution satisfying the probabilistic constraint with a certain level of probability, which is an intrinsic idea of the CCP. The given problem is formulated as follows: min f = x + 4y
x,y
s.t. P[1 x + 2 y 2] x0 y0 where 1 N (1, 12 ) and 2 N (3, 12 ) A linear combination of normal distributions of two random variables, 1 x + 2 y is also a normal distribution whose mean and standard deviation are given as x + 3y and x2 + y 2 . Therefore, P[1 x + 2 y 2] is true, if and only if (1 x + 2 y 152 (125)
2)/
x2 + y 2 z is true, where 1 and 2 stands for the mean value of normal
distributions of 1 and 2 , and z denotes the 100th percentile of the standard normal distribution. Thus, the probabilistic inequality constraints can be converted into a deterministic-equivalent constraint and the probabilistic optimization problem is transformed into an equivalent deterministic optimization problem as follows: min f = x + 4y
x,y
s.t.
(1 x + 2 y 2) x2 + y 2 x0 y0
z
(126)
Using a nonlinear constrained optimization algorithm built in the MATLAB optimization suite, the optimum solution of the above deterministic optimization problem for = 0.9 is found at x = 0.4534 and y = 0.9748 where the objective function is computed at f = 4.3527. Figure 61 shows the deterministic-equivalent constraints for dierent values of . An important observation is that the linear probabilistic constraint is converted into a nonlinear constraint. 5.2.4 Lessons Learned
By DSS method, the objective function value at 90% of CDF is estimated at approximately 3.25. However, the optimum objective function value satisfying 90% of in the CCP problem is found to be 4.2. What causes this discrepancy? In the case of DSS, the values of the design variables corresponding to 90% of the cumulative probability of the objective function are not equivalent to meeting all constraints with 90% chance. In fact, the values of at the two design points yielding the objective function value at 90% of CDF are estimated to be approximately 65%. The bottom line is that the DSS approach is not suitable for obtaining an optimum solution to a probabilistically constrained problem although it is useful for assessing and comparing dierent architectures. 153
2 1.8 1.6 1.4 1.2
12
Ps=0.95
10
Ps=0.90
14 0.8 0.6
2
y
8
Ps=0.85 Ps=0.80 Ps=0.50
0 0.5 1
6
0.4 0.2 0
2 4
1.5
2 x
2.5
3
3.5
4
Figure 61: Equivalent deterministic constraints
The reason for highlighting the incongruousness of the DSS approach for probabilistically constrained optimization problems is that the approach may be intuitively appealing in the practice of aircraft sizing, when a deterministic aircraft sizing environment is ready to go. One would simply hook up the environment to a random number generator and run it! Nevertheless, this is essentially another practice of the DSS approach, and the implications of the output from the simulation would be the same as those of the example study. The penalizing approach is appropriate for the application of probabilistically constrained optimization problems where the expected penalty costs, measuring the expected amount of surplus and/or shortage, can be dened. However, such penalty costs can hardly be dened in an aircraft sizing problem. For instance, how can we quantify the violation of the take-o eld length by 100 ft in terms of pounds or dollars if the objective function is given as aircraft weight or cost, respectively? On the other hand, the CCP approach measures the probabilities of having any shortage
154
in design constraint(s), irrespective of its size [158]. From these observations the CCP approach appears the most appropriate for formulating aircraft sizing problems under uncertainty.
5.3
Standard Form of the Probabilistic Aircraft Sizing Method
The discussions of the rst two introductory sections in this chapter have disclosed that the CCP is the most germane approach to formulating aircraft sizing problems in which the decision maker would compromise the associated risks with cost in a quantitative manner. This section presents a standard form of PASM. For the sake of simplicity, it is assumed that the propulsion system consists of a single power-path and single energy source. After the probabilistic formulation of the simple system is complete, the formulation will be further extended to a more complex system. 5.3.1 Integration with AIASM
A general idea regarding the application of CCP to aircraft sizing problems is contained in Eq. (46) presented in 3.4.2. The equation can be rewritten in terms of Pref and Wenergy for the use of the AIASM formulation developed in the previous chapter. min f
x
s.t. P P
Pref WTO Wenergy WTO
(127)
where Wenergy is the weight of an on-board energy source, physically measured in the production aircraft, and is the required consumable energy weight fraction estimated via mission analysis. The power balance constraint in Eq. (127) must hold for all performance requirements such as take-o eld length, climb, maximum speed, service ceiling, sustained turn, and so forth. If some of the parameters associated with the constraints are 155
subject to uncertainty, the required power to weight ratio is given as follows: WTO i = i , pi , pi S (i = 1, , np ) (128)
where pi stands for a vector of random parameters; pi stands for a vector of xed parameters; np represents the number of performance constraints; and i represents the required power-per-aircraft weight as a function of wing loading and other parameters for ith constraints listed in Appendix A. For example, c1 stands for take-o eld length, c2 stands for climb, and so forth. Many newly developed aircraft, especially military aircraft, are designed to be multiple-mission capable. Usually, one driving mission appears transparently and it can be used as a sizing mission. However, as more aircraft move toward multi-mission capability, a dominant mission will become rarer. The fuel balance constraint is very likely to include a multitude of inequality constraints. Then the required energy weight fraction representing the production aircraft mission performance in the future can be expressed as the following probabilistic form: j = j Pref WTO , , pj , pj WTO S (j = 1, , nm ) (129)
where np represents the number of mission fuel constraints. Then Eq. (26) changes as follows: min f
x
s.t. P P
Pref WTO i , pi , pi S WTO
i j
(130)
Wenergy Pref WTO , pj , pj j , WTO WTO S
where x = [Pref , S, Wenergy ], i = 1, , np , and j = 1, , nm . 5.3.2 Decision Variables as Random Variables
Pref and Wenergy of the three decision variables may be subject to randomness. In the sense that the wing area may vary due to manufacturing tolerances and thermal 156
expansions/contractions in its operation, S possesses inherent uncertainty. However, the impact of such variation on aircraft performance is deemed less signicant compared with other factors. Therefore, S is considered a deterministic variable in this research. Suppose that these two quantities are functions of random parameters given as a vector form, v. Then the random distribution of each parameter is replaced with a deterministic value, vi multiplied by a normalized distribution, i as follows: vi = i vi (131)
The deterministic values, v, are the most likely values selected within the distribu tion of each random variable by the responsible disciplines. Notice that all necessary engineering work of the design team cannot be done or does not have to be done in a probabilistic manner. Describing technical data in probabilistic terms, for example, may bring ineciency and inconsistency to communication and documentation. Therefore, most engineering tasks, especially disciplinary activities, are performed in a deterministic manner. Such deterministic values are the most reliable and latest engineering data which are supposed to be shared by all disciplines in a consistent manner. It follows that the random response of the reference power of a power plant can be expressed as follows: Pref = Pref |v P (132)
where Pref |v is the reference power dened by the deterministic values of the random variables, and P , given as a function of , is the normalized distribution of Pref to Pref |v . The random response of the amount of available energy can be expressed as follows: Wenergy = Wenergy |v Wenergy (133)
where Wenergy |v is the amount of available energy calculated with a vector of the
157
deterministic values of the random variables, denoted as v, and Wenergy is the distri bution of Wenergy normalized by Wenergy |v . By combining Eq. (132), the rst set of constraints in Eq. (130) can be rewritten with respect to v as follows: P WTO P Pref |v i , pi , pi S WTO 0 i (i = 1, , np ) (134)
where np is the number of constraints. Similarly, by combining Eq. (133), the second set of constraints in Eq. (130) can be expressed as follows: P Wenergy Wenergy |v P Pref WTO j , , pj , pj S WTO WTO Standard Form of PASM 0 j (j = 1, , nm ) (135)
5.3.3
One distinguishing feature of the aircraft sizing problem is that the objective function is also probabilistic as well as the constraints. The objective function in Eq. (130) may be the take-o gross weight that has been commonly used for aircraft sizing. Other candidates for the objective function are other metrics, which include total cost (acquisition cost and operating cost) or an overall evaluation criterion that represents an aggregate quantity of multiple criteria. Regardless of the choice, the quantity deemed as the objective function is very likely to involve uncertainty. One way to handle such a probabilistic objective function is a mean value-based approach [157], by which the probabilistic objective function f is deterministically treated by converting it into the expectation of the probabilistic objective function (E[f ]). Such a mean value-based approach is widely used in the current RBDO practice [159]. Finally, the standard form of PASM for an energy-propulsion architecture with a single power-path is formulated as follows: min E[f ]
x
s.t. P [gi (x, )|v 0] gi (i = 1, , np + nm ) where x = [Pref |v , S, Wenergy |v ] 158
(136)
Aerodynamics
CL CL
Point Performance Requirements
Distributions for Uncertainties
CD
Propulsion
Specific Energy
Efficiency Maps
Weight
Historical Data Empirical Eqs. W TO= (WE) Physics Based Tools
Mission Performance Requirements
Target Probability
Figure 62: Overview of PASM
where gi (x, )|v represents ith probabilistic constraint, and represents a random vector that includes p associated with random parameters and associated with random variables. The overall concept of PASM is illustrated in Figure 62. Unlike a deterministic approach, PASM determines the optimum values of the sizing variables with target reliability. The optimization process simultaneously assesses power balance and energy balance to evaluate the sizing constraints, which are given in the form of reliability rather than a deterministic value. It also must be noted that in PASM formulation the energy balance constraint appears explicitly, in contrast to the most classical sizing methods, in which the consideration of fuel balance is embedded in the calculation of the weight of the aircraft [160]. As mentioned in Chapter II, the classical deterministic sizing methods do not distinguish between available fuel quantity and required fuel quantity, assuming
159
Notional Concept
g 1
Specific Power
! &#! &$ %$ "
g 2
g 3
P ref
S
WTO
that the former equals the latter that is determined by the mission analysis. Therefore, fuel quantity is treated as an internal variable and a response, rather than a design variable of a given aircraft sizing problem. In contrast, in PASM, the available fuel (energy) quantity appears as an independent design variable.
5.4
Solution Techniques of CCP and RBDO
Three distinctive approaches to solve the CCP and RBDO problems are found in literature. The rst approach is to convert the probabilistic constraints into an equivalent deterministic function and solve the consequent deterministic optimization problem. The second approach is to approximate the feasible set with sampled constraint functions. However, these two methods are applicable to a few specic types of problems and not applicable to most complex engineering problems, where constraint functions are not given in the form of explicit mathematical expressions. The last approach is to employ a conventional design optimization loop with reliability analysis modules. This approach can be applied to a wide range of problems. 5.4.1 Deterministic Equivalent
As shown in the numerical example in 5.2, probabilistic constraints can be converted into an equivalent deterministic function. A more general form is given as a system of probabilistic constraints: gi (x) = x h, where = (1 , 2 , , n )T is a vector of multivariate normal distributions with = (1 , 2 , , n )T as their mean values and V as the covariance matrix. When a target value (g ) exists for the probability of meeting the probabilistic constraints simultaneously, the deterministic equivalent of the joint probabilistic constraint is given as T x h + 1 (g ) xT Vx where 1 is the inverse normal cumulative distribution function, or so-called Second Order Cone constraints [161]. Another type that can be converted into a deterministic equivalent, known as the right-hand side problem in the context of CCP, has one or a set of constraints given 160
as follows: P [gi (x) i )] gi Then the constraints can be converted into gi (x) F1 (gi ) i (138) (137)
Aerospace system optimization problems including aircraft sizing, in which requirements are uncertain, are very likely to be formulated in this manner. Generally, the metrics related to requirements such as take-o eld length, NOx emission, and noise level are system responses, gi as functions of decision variables, x. When the metrics are included in a system optimization problem in the following fashion, gi must be greater than target values given as uncertain variables, i whose distributions are known, the constraints can be formulated by Eq. (137), which is converted into its deterministic equivalent as Eq. (138). Unlike the previous form, the conversion of Eq. (137) to its deterministic equivalent does not require any information regarding gi whose explicit expression is not available in most applications. This method allows designers to avoid a massive analysis in order to compute the probability of failure of each constraint. However, only a few specic types of CCP problems as presented above can be converted into exactly equivalent deterministic optimization problems. 5.4.2 Constraint Sampling Approach
Suppose that a CCP is given as min f
x
(139)
s.t. P [g(x, ) 0] g and the constraints cannot be converted into a deterministic equivalent. Then, one approximates the convex chance constrained problem Eq. (139) with a deterministic
161
optimization problem that has multiple sampled constraints as follows: min f
x
(140)
s.t. P [gi (x, i ) 0] (i = 1, . . . , N ) In other words, this approach replaces the probabilistic constraints with a large number of sampled constraints. De Farias and Van Roy [162] showed that for a special case of linear constraints, a sample size of N 4n log 1 g 12 1 g + 4 log 1 g 2 , (141)
where log () denotes the logarithm with base 2, ensures that the set of decision vectors feasible for the sampled problem in Eq. (140) is feasible for the chance constrained problem, Eq. (139), with a probability of at least 1 . Calaore and Campi [163] also proposed a similar result. The authors considered general convex functions, and showed that for N 2n ln 1 g 2 1 g + 2 ln 1 g 2 + 2n, (142)
where ln () denotes the natural logarithm, the optimal solution of the sampled problem Eq. (140) is feasible for P [g(x, ) 0] g with a probability of at least 1 . A notable advantage of this approach is that the set {y|gi (y, i ) 0, i = 1, . . . , N } is convex although the feasible set {y|P [g(y, ) 0] g } is not generally convex. 5.4.3 Optimization with Reliability Analysis
The problem stated in Eq. (121) has a complex structure that cannot be converted or approximated to an equivalent deterministic problem in any practical way. Therefore, the optimization process must be combined with a reliability analysis to compute the probability of meeting constraints. The most popular method is to employ a deterministic optimizer with a nested reliability analysis module, whereby the reliability (or probability of failure) at the current design variables set by the optimizer is estimated inside the optimization loop. The design optimization loop (outer loop) is 162
a deterministic nonlinear constrained optimization process and rich literature exists regarding various approaches towards nonlinear constrained optimization problems, including the Method of Feasible direction (MoFD) [164], Sequential Quadratic Programming (SQP) [165], and Sequential Linear Programming (SLP) [165]. The reliability assessment (inner loop) can be performed by simulation based methods such as Monte Carlo simulation or approximated analytical methods. In this approach, each call for constraint functions by the optimizer in the outer loop triggers a reliability assessment, which ends up becoming a computationally intensive process. To alleviate such an impediment by nested reliability analysis, many researchers have developed single-loop RBDO methods, such as the traditional approximation method by Torng and Yang [166]; single-loop single-variable (SLSV) method by Chen et al. [167]; safety-factor approach (SFA) by Wu and Wang [168]; and a sequential optimization and reliability assessment (SORA) method by Du and Chen [169]. Those methods collapse an original two-loop optimization problem into a single-loop optimization problem, which requires fewer reliability assessments. For example, SORA decouples the RBDO process into a sequence of deterministic design optimizations and a set of reliability assessment loops. In each cycle, a deterministic design optimization is performed with a set of constraint functions, the boundaries of which are shifted in the feasible direction based on the reliability information obtained in the previous cycle. Hence, the design is improved from cycle to cycle, and computational eciency is improved signicantly. 5.4.4 Reliability Analysis
The multitude of techniques developed to assess system reliability can be broadly categorized into two groups: random sampling methods and approximated analytic methods.
163
Optimizer
Reliability Analysis
Figure 63: System optimizer integrated with a nested reliability analysis
5.4.4.1 Sampling Based Reliability Analysis Random sampling methods have been dominated by Monte Carlo simulation, which generates a set of random values and computes the corresponding probability of failure using the following equation: 1 N
N
Pf =
I(i ) where I(i ) =
i=1
1 if g (x, )| 0 i i v 0 otherwise
(143)
In the above equation, N is the number of sample points and I is the indicator function. As N increases, the solution asymptotically approaches the exact solution. However, the number of sample points required to estimate small magnitudes of probability of failure is extremely high, which may lead to a computationally intractable problem formulation. This issue may be resolved by employing Quasi Monte Carlo methods such as Importance Sampling methods [170]. 164
5.4.4.2 MPP-Based Reliability Analysis In the application of the RBDO approach to a real engineering problem, the computational eciency is of signicance. Particularly, the eciency associated with computing reliability is crucial because it is the most expensive part of the whole optimization process. Implementing more ecient and faster methods than MCS may greatly accelerate the overall process. Several approximation methods such as the First-Order Reliability Method (FORM) and the Second-Order Reliability Method (SORM) that are based on MPP (Most Probable Point) concepts are widely used, since these methods usually provide superior computational eciency over the standard Monte Carlo method. In general, RBDO methods often formulate inequality constraints in terms of the probability of failure rather than the probability of success as shown in Eq. (121), which leads to min f
x
(144)
s.t. P [g(x, )i 0] (ti ) 0 where is the cumulative distribution function for standard normal distribution and
1 ti is the target reliability index, which is given as Fgi (gi ).
In both FORM and SORM, an MPP of failure (denoted as u ) is found in the standard u-space , where the components of u are standard normal distributions and statistically independent from each other. The standard u-space can be obtained by performing a Rosenblatt Transformation (u = T ()) on the set of random variables. When the random variables are statistically independent, the transformation of to u can be performed by ui = 1 [Fi (i )]. Transformations for several popular distributions are listed in Table 7. The MPP lies in a limit state where gi (u) = 0 and has a minimum distance from
165
FORM
(u ) = 0 u2
SORM
g (u) = 0
=
MPP
u1
Figure 64: Approximation of the limit state function by FORM and SORM [171]
the origin of the u-space, which leads to min
u
u (145)
s.t. gi (u) = 0 The solution to this problem can be found by employing a numerical constrained optimization programming algorithm or tailored iterative methods including those developed by Hasofer and Lind [172] and extended by Rackwitz-Fiessler [173]. There exist two distinct approaches, the reliability index approach (RIA) and the performance measure approach (PMA) for implementing the probabilistic constraints in Eq. (144), which can be further reformulated in two equivalent forms through an inverse transformation as follows [175]: si = 1 [Fgi (0)] ti
1 gpi = Fgi [(ti )] 0
(146) (147)
166
Parameters f () =
2 1 2
PDF exp 0.5
2
Transformation = + u
Normal
= mean
= standard deviation
Lognormal f () = >0
1 2
= mean exp 0.5
= standard deviation
= exp( + u)
2 = ln[1 + (/)2 ]
= ln() 0.5 2 f () = >0 f () = exp[( ) exp(( ))] f () =
1 ba k (k1)
Table 7: Transformation of the u-space to the -space [174]
167
ab exp(0.5u2 )
Weibull
k > 0, = (1 + 1/k)
exp
k
= [ ln((u)] k
1
2 = 2 [(1 + 2/k) 2 (1 + 1/k)]
Gumbel
= + (0.577/) = / 6
=
1
ln[ ln((u))]
Uniform
= (a + b)/2 = (b a)/ 12
= a + (b a)(u)
where (u) =
1 2
where Fgi denotes the cumulative distribution of gi , and si and gpi are the safety reliability indices and the probabilistic performance measure for the ith probabilistic constraint, respectively. RIA replaces the probabilistic constraint in Eq. (144) with the reliability index constraint given as Eq. (146), while PMA uses the performance measure constraints given as Eq. (147). The rst-order safety reliability index si in RIA is obtained by solving the optimization problem given as Eq. (145). The magnitude of si equals u , and its sign is positive if the origin of the u-space is feasible, and vice-versa. A performance measure approach (PMA) proposed by Tu et al. [153] formulates the evaluation of probabilistic constraints as an inverse problem of RIA as given in Eq. (147). PMA searches for rst-order probabilistic performance gpi with the lowest performance function value on a hyper-surface determined by the target reliability index ti within the u-space, which can be formulated as min gi (u)
u
(148)
s.t.
u = ti
where gpi equals the objective function value gi (u) at the optimum. PMA, in general, is more ecient than RIA, especially for high reliability problems [176] due to the simpler constraint for reliability analysis. Moreover, it has been reported that PMA converges with certain problems where RIA diverges.
5.5
Extended Formulation of PASM
The formulation presented in 5.3 provides a basic form of probabilistic aircraft sizing methods applicable for a single random objective function and multiple probabilistic constraints. This formulation can be extended to fulll a dierent need depending on the application. For example, one may need to consider joint probabilistic constraints or multi-objective functions. Furthermore, one may want to nd a robust solution that minimizes the mean value of the objective function as well as its variance. This 168
section brings to light ramications of the standard formulation of PASM adaptive to such a purpose. 5.5.1 Joint Probabilistic Constraints
The probability of meeting all constraints simultaneously must be less than or equal the minimum of the probability level of meeting individual constraints. Particularly, if there is a strong negative correlation between any two hard constraints, the probability of meeting the joint constraints may be signicantly lower than the given probability for individual constraints. Therefore, it is worthwhile to investigate the impact of implementing joint probabilistic constraints by formulating the CCP problem as follows: min f
x
s.t. P [G(x, )|v 0] g where x = [Pref |v , S, Wenergy |v ] and G = [g1 , , gnp , gnp +1 , , gnp +nm ]T
(149)
where G(x, )|v 0 is a system of inequality constraints that the design solution must satisfy simultaneously. 5.5.2 Multidisciplinary Design Optimization
The proposed probabilistic aircraft sizing method can be applied to aircraft design optimization problems in which multidisciplinary design parameters such as wing geometry, tail arrangement, and propulsion system design parameters are included as design variables. This type of problem can be solved by double-loop optimizations, in which the inner-loop optimization is performing a probabilistic sizing for xed design parameters determined by the outer-loop optimization that drives the design parameters to an optimum solution. However, a more ecient approach is to integrate a double-loop optimization problem into a single optimization problem in which the two
169
groups of design variables from the inner-loop and outer-loop optimization processes are combined. In the resultant optimization problem, the design variables include disciplinary design variables, denoted as xd , as well as the original sizing variables. In addition, a number (nd ) of constraints resulting from decoupling the disciplinary analyses may be added. Therefore, the resultant problem can be stated as follows: min E[f ]
x
s.t. P [gi (x, )|v 0] gi (i = 1, , np + nm + nd ) where x = [Pref |v , S, Wenergy |v , xd ]
(150)
Although such eorts to enhance computational eciencies, when the objective and/or constraint functions are evaluated by computationally-expensive analyses such as nite element methods (FEM) or computational uid dynamics (CFD), a probabilistic design approach would be computationally intractable. To alleviate such a problem, some researchers have used surrogate models such as response surface polynomials [177, 178, 179, 175] and neural networks [180, 181] to approximate the objective function and constraint functions with respect to the decision variables as well as random parameters. 5.5.3 Probabilistic Objective Function
The basic PASM formulation seeks the design that minimizes the mean value of the probabilistic objective function responses. There are some alternative methods to handle a probabilistic objective function.
170
5.5.3.1 When a Target Probability Exists An alternative approach suggested by Liu [182] is to impose a target condence level (f ) of the probabilistic objective as an additional constraint as follows: min f
x
s.t. P[f (x, ) f ] f P[gi (x, ) 0] ci (i = 1, , nc )
(151)
where f and c are the predetermined condence levels for the probabilistic objective and constraints, respectively. 5.5.3.2 When a Target Value Exists If a target value for the objective function is given, an approach that maximizes the probability of achieving the target value, as shown in the RDS method, may be appropriate, leading to max P[f (x, ) fallowable ]
x
(152) (i = 1, , nc )
s.t. P[gi (x, ) 0] ci
Nevertheless, it is not common for a target value of the aircraft weight to be given for aircraft sizing. Furthermore, given as a design requirement, the aircraft weight may be incorporated as a constraint with an another metric as its substitute for the objective function. However, if another response such as a monetary metric is selected as the objective function, then this approach is suitable. 5.5.4 Multi-Objective Function
Extension to a multidisciplinary design optimization problem is very likely to call for multiple objectives in many applications. In general, maximization or minimization of a multitude of criteria concurrently in many complex system design problems results in a conict that allows only compromised solutions. Under such circumstances, the conditions for optimality of the given problem are not apparent. 171
For deterministic optimization problems, two largely distinct classes of multiobjective formulation methods known as a priori and posteriori have been developed to aid in multi-objective problem denition [183]. The former, including the weighted sum method [184], goal programming [185], and -constraint method, seeks to create the single scalar substitute that aggregates multiple objectives based on the preferences of the decision makers. Once formulated, this scalar value can be optimized using traditional techniques to nd a single best solution to the multiobjective problem. The most commonly used and intuitive method for converting multiple objectives ([f1 , f2 , , fn ]) into a single objective (f ) is the weighted sum: f = w 1 f1 + w 2 f2 + + w n fn where wi = 1 and wi > 0
(153)
However, such approaches include subjective information, and can thereby be misleading concerning the nature of optimum design [186]. In contrast, the latter requires no preference or goal information before performing an optimization run [183]. Instead, these methods assist the users in forming their preferences by providing information for a set of non-inferior solutions [187]. A non-inferior solution, or so-called Pareto optimum, has the property that an improvement in one objective requires a degradation in at least one other. The set of all Pareto optimal solutions in the objective space creates an ecient frontier of solutions, known as the Pareto Frontier. All solutions on the Pareto Frontier are not dominated by other feasible solutions, which can provide valuable information for decision-making. The Pareto Frontier can be obtained by performing successive optimization runs using one of the a priori aggregation methods and dierent weight vectors1 . When considering uncertainty in multiobjective optimization, computational effort becomes even larger, and as a result very few studies have been performed [190].
Some alternative methods such as multiple objective Genetic Algorithms (MOGA) [188] and Non-dominated Sorting Genetic Algorithm [189] are based on the genetic algorithm.
1
172
Literature review has shown that researchers have attempted to solve a multiobjective optimization problem under uncertainty using existing multiobjective optimization methods combined with the RBDO formulation. For example, El Sayed et al. [191] have investigated a multi-objective RBDO formulation using the nonlinear goal programming method for structural design problems that involves multiple design criteria including structural weight, load induced stress, deection, and mechanical reliability. Barakat et al. [192] formulated a multi-objective RBDO using -constraint for the designs of prestressed concrete beams (PCB). Lian and Kim [193] studied a multiobjective RBDO problem in which bi-objectives are optimized subjected to a probabilistic constraint. They used a two-loop approach. For the outer loop, they used a genetic algorithm and a gradient-based optimizer to facilitate the convergence of the genetic algorithm. For the inner loop, they used Monte Carlo simulations. Such approaches can also be applied to the probabilistic aircraft sizing problems in which multiple objectives F = [f1 , , fn ] are given. In the case of the objectives being subject to uncertainty, the approaches for a single objective subject to uncertainty discussed in 5.5.3 need to be combined. When no target values exist for the objective functions, a Pareto frontier being developed via integration of these methods with the proposed method may lead to a probabilistic method for solving complex system design optimization problems, in which both objectives and constraints are multiple and probabilistic. For example, the integration of CCP and JPDM will provide a suitable solution when one seeks an optimum solution that maximizes the probability of satisfying multiple objectives and simultaneously satises the uncertain constraints with a prescribed probability, which is quantitatively stated as follows: max P[F(x, ) c]
x
(154)
s.t. P[G(x, ) 0] c where F(x, ) stands for a vector of multitude objectives, and c is a vector of criteria for each objective. 173
Joint Objective Probability
Joint Constraint Probability
f 2 (x (i ) , )
Desired Space
g 2 ( x (i) , )
g 1 ( x (i) , )
f1 ( x ( i ) , )
Feasible Space
Figure 65: Simultaneous application of joint probability to the objective space and the constraint space
5.5.5
Robust Design
In general, two distinct approaches toward system design under uncertainty exist: reliability-based design and robust design. The goal of robust design is to minimize the eect of variation under controllable and/or uncontrollable factors without eliminating the sources of variations, while reliability-based design seeks a solution to ensure that system performance meets the pre-specied target with a required probability level [194]. The classical denition of robust design is design at which the variations in performance characteristics are minimal [195]. When it is combined with robust optimization strategies, not only the variation but also the objective function value is of interest. Therefore, implementation of robust design philosophy into a design problem is prone to yield a multi-objective problem for which the objective is to minimize both the mean and deviation of the objective function. When uncertainty is involved in both the objective function and the constraint functions, one may want to have a solution that satises the reliability constraints and ensures the robustness of the objective function. A formulation that minimizes 174
both the mean of merit function and its variance while maintaining feasibility under uncertainty can be rewritten as follows: min f
x
min f
x
(155) (i = 1, , nc )
s.t. P[gi (x, ) 0] ci
Su and Renaud [194] formulated the two objectives in Eq. (155) using the weight sum method, f + wf , where w is a weighting function that trades between performance and robustness to minimize (convex) weighted sums of the dierent objectives for various dierent settings of the weights. Mourelatos and Liang [159] proposed a preference aggregation method to handle the two objectives with the use of indierence points as a means of trading between performance and robustness.
5.6
Sensitivity Analysis
Although it is the end-goal of any optimization problem to seek the optimal solution, sensitivity analysis is also an important tool for designers to gain insights into the complex behavior of the system under study, thus leading to more informed decisions. Sensitivity analysis has been widely applied in engineering design to explore a models response behavior, to evaluate the accuracy of a model, and to test the validity of assumptions [196]. Especially, sensitivity analysis provides information on the rate of change in a systems response(s) due to changes in inputs. This is usually performed by perturbing the input variables one at a time near a given central point, which involves computing the partial derivatives of the responses with respect to the input variables: an action that is known as local sensitivity analysis. Large volumes of publications are available in literature that discuss sensitivity analysis in terms of rigorous mathematical elaborations, and such is well beyond the scope of this dissertation. However, certain sensitivities can be obtained as by-products of applying PASM and result in critical information that guides a designer to a better 175
understanding of the sizing problem. Therefore, it is worth reviewing such sensitivities that include that of the objective function to the target probability and that of the constraint functions to the distributions of random parameters. 5.6.1 Sensitivity of Objective Function to Target Probability
It is well known that a Lagrange multiplier has implications as a sensitivity coecient. Consider a typical constrained optimization problem min f (x)
x
(156)
s.t. gi (x) ci Introducing Lagrange multipliers (), Eq. (156) can be rewritten as follows:
m
min L(x, ) = f (x) +
x i=1
i (gi ci )
(157)
for a suitable set of Lagrange multipliers i , i = 1, . . . , m. A pair (x, ) has to satisfy the following conditions to be a solution to the original constrained problem in Eq. (156):
x L(x , )
=0 (158)
gi (x ) ci 0 i which are known as the Kuhn-Tucker conditions.
From Eq. (157) and Eq. (158), the optimum solution x must meet a necessary condition for optimality 0=
xf m
+
i=1
i
x gi
(159)
The interpretation of this equation implies that at the optimum, the Lagrange multipliers are used to balance the gradients of the constraints and the objective function. With this consideration and Eq. (158), the following relation can be surmised: f (x ) = i ci 176 (160)
This equation states that an ith Lagrange multiplier represents the sensitivity of the objective function to the ith constraint. This sensitivity is known as the LHS (Left Hand Side) sensitivity [197], and its proof can be found in References [198, 199]. The relative magnitude of the ith Lagrange multiplier represents the importance of the ith constraint for this solution. Lagrange multipliers are the by-products of the optimization process in most gradient-based path building algorithms since the KuhnTucker condition species optimality. In the RBDO method, the LHS sensitivity represented by the Lagrange multiplier corresponds to the sensitivity of the objective function to the target probability. This information is particularly useful to an arbiter who wants to trade reliability (probability of success) with the objective function. If the decision maker values each constraint dierently, he or she may want to reduce the objective function value by accepting an increase in the probability of failure of less important constraints. 5.6.2 Sensitivity of Constraint Functions to Distributions of Random Parameters
Another useful analysis for an RBDO problem is exploring the sensitivity of the objective function(s) and constraint function(s) with respect to random parameters. When uncertainty is involved with an input variable, sensitivity analysis does not simply mean computation of a partial derivative with respective to the variable. The random parameters appear in a simulation model in the form of a distribution, rather than a single denite value. The impact of random parameters on system responses is measured by the probabilistic characteristics of the response such as its mean (), variance (), the probability density function (PDF), or the cumulative distribution function (CDF) rather than a point estimation. Correspondingly, a sensitivity analysis under uncertainty needs to be performed on the probabilistic characteristics of a model response with respect to the probabilistic characteristics of model inputs [196]. Therefore, the sensitivity of the objective function(s) and constraint function(s) with
177
respect to random parameters can be obtained by a dierent class of sensitivity analysis, known as probabilistic sensitivity analysis (PSA). Various types of PSA techniques are found in literature. Some techniques provide information on the relative importance of random parameters on system responses, and assist a problem solver in identifying probabilistically insignicant factors. Variance-based methods including the Fourier Amplitude Sensitivity Test (FAST) [200, 201, 202], correlation ratios [203] or importance measures [204], and Sobols indices [205] decompose the total variance of a response by the uncertainty sources (random parameters). The MPP-based method measures the relative importance of random parameters utilizing the information of the gradient with respect to random parameters at MPP. Another widely used category of PSA techniques is the investigation of the rate of change in a probabilistic characteristic of a response Y due to the changes in the probabilistic characteristics of a random input Xi , such as g /i . If an MPP-based reliability analysis is employed in solving the probabilistic aircraft sizing problem, an MPP-based sensitivity analysis can be performed with little or no additional computation. The gradient of a limit state function at MPP is construed as an index of relative contribution of the random parameters to the probability of failure as shown in Figure 66. The components of , decomposed to each dimension of the standard u-space, provide sensitivity indicators of reliability with respect to random parameters, whose mathematical expression is given as [196]:
y (ui ) xi h(xi ) n j=1 2 2 |MPP =
Si =
y (uj ) xj h(xj )
(uMPP ) i 2
(161)
where y is the random performance, is the PDF of the standard normal distribution, h is the PDF of a random variable, i , ui is the standard normal random variable transformed from i , and is the reliability index. It should be noted that
n i=1
Si =
1. Moreover, Si is the directional cosine in the gradient of the limit state at the MPP. 178
u2
MPP u2
Failure
MPP
g (u1 , u2 ) = 0
u1MPP
u1
Figure 66: Illustration of the MPP-based sensitivity measure [196]
The magnitude of Si indicates the relative signicance of the uncertainty in the random parameters to the response. In PASM, this information can assist designers in various ways. Before the optimum is found, the analysis may serve as a supplemental means to check the validity of a model structure. In addition, the results may be used to reduce the dimension of a design problem by eliminating the probabilistically insignicant factors from the PASM problem. Once an optimum solution is found, the information reveals what eorts need to be pursued to reduce the variability of the response and eectively improve the reliability. The MPP-based methods also oer potential improvement on reliability by reducing variability or by improving the mean value in random inputs, Pf /i and Pf /i , respectively [206]. ui = i ui i ui = i ui i
(162)
where /ui is the directional cosine of the vector , and ui /i can be obtained 179
from u = T ().
180
CHAPTER VI
METHOD IMPLEMENTATION
Although the objective of the proposed research is not to provide a specic solution down to the detailed design level of a specic problem, both AIASM and PASM must be veried through appropriate implementation. This chapter presents two example studies as proofs of concept, in which the proposed methods are applied to a fuel cell-powered general aviation (GA) aircraft and a Solar-Powered High Altitude Long Endurance (SPHALE) aircraft with a RFC system. These two studies are selected for dierent purposes. The former can be also performed with traditional aircraft sizing method if an analysis model for the fuel cell power plant can be properly integrated with a traditional sizing environment. Therefore, the comparison between the results from the traditional sizing method and the proposed method is possible. On the contrary, the latter articulates an exclusive capability of the method compared with the traditional methods. Another comparable aspect of these two studies is the incorporation of the analyses for propulsion system performance metrics and weight values. The former study employs a physics-based analysis code specially developed for electric propulsion system architectures powered by fuel cells, whereas the latter uses eciency and specic power data available from literature.
6.1
Fuel Cell-Powered General Aviation
Both AIASM and PASM were applied to a GA aircraft conguration equipped with an all-electric aircraft propulsion system architecture powered by a PEMFC system. The baseline conguration and the sizing mission are similar to those of a Cessna 172 Skyhawk, as illustrated in Figure 67. Point performance requirements are comparable or slightly relaxed compared with those of the Skyhawk. 181
Mission Profile
Idle/Takeoff
Figure 67: Mission prole of electric GA aircraft
The all-electric propulsion system architecture consists of a PEMFC stack, a power management and distribution (PMAD) system, an electric motor, and other accessories, as illustrated in Figure 68. The performance and weight of the propeller are estimated by GTPROP, which is a legacy code originally developed and validated by Hamilton Standard [207]. The performance and weight of the components of the propulsion system architecture are calculated by the design and simulation environment developed by Choi et al. [208]. The design framework consists of two main parts: an on-design analysis routine and an o-design analysis module. The on-design analysis is set up to estimate the overall eciency and weight breakdown of the propulsion system that is designed to produce target thrust or power at a specied ight condition. The o-design analysis estimates the steady-state performance of the propulsion system whose geometric and hardware characteristics were determined by the prior on-design analysis, along the mission prole. At each mission segment, the analysis converges
# " # #" $ # ) )& $ $ & 1 1 &2 &2
Climb
Point Performance
TOFL
! '& % '& % ( ! ( ! '& & 0 '& & 0 ( % 1 ( % 1 5 43 5 43 3 3
Cruise 575 Nm (8K ft) Decent Loiter 45 min (2K ft)
Land/Taxi
182
on the available thrust and overall eciency (fuel consumption) that match the required shaft power for the given ight condition. Figure 69 illustrates the integrated environment of the aircraft sizing code developed with MS EXCEL spreadsheets
and Choi et al.s propulsion system analysis and sizing code. All codes were unied under a common GUI environment using ModelCenter Inc. The sizing reference power (Pref ) of the propulsion system is selected as a design shaft power at cruise (Mach 0.184 and 8,000 feet). Thus, the propulsion system weight must be estimated iteratively in conjunction with aircraft sizing, which demands signicant computational eort. The new environment employs an adaptive technique for developing engine scaling law, which is capable of updating the scaling law for a specic propulsion system during the aircraft sizing process. The technique has allowed the environment to accelerate the process moderately without compromising the accuracy of estimated propulsion system weight. As identied in descendent research [65, 117, 118], the performance of o-the-shelf fuel cells and electric power components is barely sucient to realize a yable fuel cell airplane, mainly because of their low specic power and power density. Dramatic improvements in all sub-system components technology readiness levels must be achieved to yield a feasible and viable aviation solution. Therefore, the weight of the fuel cell based propulsion system was estimated by consulting the hypothetical technology growth scenario reported in Ref.[117]. In addition, 10% of increase in overall eciency of the propulsion system was assumed, resulting in 0.04 lbs/hr/lbs of TSFC at cruise. The relationship between the reference power and propulsion system weight is depicted in Figure 70, which exhibits a strong rectilinear trend. The PEM fuel cells require pure hydrogen as fuel to carry out the electrochemical reaction. Hydrogen fuel can be stored in gaseous, gelled, or liquid form. Gaseous hydrogen fuel can be stored in a number of ways: pressure tank, metal hydride, from Phoenix Integration,
183
H2
Hydrogen Hydrogen Tank
Hum
Air
C*
Motor
PEM Fuel Cell
Air, H2O
Motor
PMAD Batteries Aux
* Compressor
HX
Figure 68: Notional fuel cell propulsion system architecture
Propulsion System
(Developed by Choi) Choi) On-design Off-design Propulsion System Weight Power Lapse Ratio Propeller Efficiency Overall Efficiency Hydrogen Mass Wing Loading Power-to-Weight Ratio Fuel Fraction Hydrogen Tank Sizing
Constraint Analysis Required Power Flight Condition Design Shaft Power Flight Condition
Mission Analysis
Aircraft Sizing
Weight Estimation
Figure 69: Integrated analysis environment
184
Tank Weight
1400 1200 1000 800 600 400 200 0 0 50 100 150 200 250 300
Propulsion System Weight (lbs)
Reference Power (Hp)
Figure 70: Power vs. weight of the electric propulsion system
carbon nanotubes, and glass microspheres. A cryogenic tanker is used for gelled or liquid hydrogen [209]. Among the options, liquid hydrogen was chosen for this study because previous studies on the options for hydrogen fuel storage [115, 119, 209] support the technology as the most promising for aircraft applications. A model that estimates the weight of the liquid hydrogen tank based on the formulation of Chambliss and Kelly [210] was used for this study. The uncertainty sources considered for the GA study include the zero-lift drag coecient, propulsion system eciency, payload, and hydrogen fuel tank weight. These four random variables were assumed to have a uniform distribution as depicted in Figure 71. These distributions have been arbitrarily selected for the purpose of the demonstration. In reality, however, the distributions must be carefully selected by experts from all involved disciplines, based on their knowledge and analyses.
185
6.1.1
Deterministic Solutions
Three dierent deterministic approaches to allocating varying amounts of design margin in the presence of uncertainty were tested. The rst, denoted as D1, represents the voice of experience approach that determines appropriate design margins by an experts engineering intuition. The second, denoted as D2, represents a deterministic optimization approach without any design margin obtained by a numerical optimization process. The last, denoted as D3 is also a deterministic optimization approach, but applies a safety factor to each design variable. The rst approach, D1, follows the steps described in 4.5. The results of the constraint analysis of the electric GA aircraft are presented in Figure 72. The input parameters for the analysis are listed in Table 21 in Appendix E. Assuming that the constraints are not deterministic, one may choose a design point within the feasible region that is removed from the constraint curves. The design point for the aircraft was selected as 11.7 lbs/ft2 of wing loading and 37 Hp/lbs of power-to-weight ratio as marked in Figure 72. With these combinations of wing loading and power-to-weight ratio, the required fuel fraction was estimated at 2.85% by the mission analysis; detailed results are listed in Table 22 in Appendix E. The available fuel fraction was determined to be 3.4%. Such a low fuel fraction is representative of the high energyconversion eciency of the fuel cell power plant and the high specic energy of the hydrogen fuel. The second approach, D2, employs a numerical optimization process to nd an optimum design resulting in the minimum take-o gross weight. During the optimization process the random parameters are set to their reference values, namely i = 1. The solution of D3 was found by using the same optimization algorithm as D2 but with a 10% pre-xed deterministic margin to each design variable. The sizing results of the three deterministic approaches are listed in Table 8.
186
Distribution of normalized values
0.9
1
1.1
0.95
2
1.05
Zero Lift Drag Coefficient
Propulsion System Efficiency
1
3
Payload
1.5
0.9
4
1.1
Factor of Hydrogen Fuel Tank Weight
Figure 71: The distributions of random variables considered in the electric GA study
Power-to-weight ratio (ft/sec)
Wing Loading (lbs/ft2)
Figure 72: Constraint analysis of the electric GA Aircraft, D1
187
! "
V G W FU U B R TS D RQ IH P C GF D EC A B@
# $ & ' % () $ 0 ( & 21 3 ) 4 5 ) $1 # 6 ' & 7 8 0 9
Table 8: Deterministic solutions Power to Weight Ratio (ft/sec) Wing Loading (lbs/ft2 ) Fuel Fraction Take-o Gross Weight (lbs) Wing Area (ft2 ) Propulsion System Power (Hp) D1 37.00 11.70 0.0350 3168 271 213 D2 32.01 11.45 0.0288 2424 212 141 D3 35.57 11.45 0.0323 2877 251 186
6.1.2
Code Verication
It is desired that the proposed methods are validated through their applications to existing aircraft designs before applying them to new aircraft design problems. Validation is distinguished from verication in that the former assesses the degree of representing the real-world while the latter assesses the degree of representing the developers conceptual description. Therefore, in order to validate proposed methods, sucient data of an actual aircraft that uses alternative energy sources are required. Validation of AIASM requires reliable, sucient data of performance specications and aircraft design attributes. However, very limited information of unconventionally powered aircraft can be found in the public domain. Although missing information could be estimated by manipulating or extrapolating available data, it will undermine solidity of the validation. For this reason, validation of the methods was not performed. Instead, the analysis code was veried by comparing mission analysis results of the three deterministic solutions with those obtained by FLOPS. Although the aircraft is fueled with liquid hydrogen, FLOPS can analyze its performance if an engine deck of the propulsion system is provided. An engine deck was developed by using the same propulsion system model. The design shaft power of the engine is 171 Hp at the cruise altitude. The engine can produce 20% more shaft power at sea level because of reduced power take-out for the air compressor due to higher ambient air density. A portion of the engine deck data is pictorially described 188
in Figure 73, which shows the variation of thrust and fuel ow with altitude and mach number at maximum power conditions. The numeric data set also contains thrust and fuel ow for several part power conditions. The range performance of the three deterministic designs were computed by FLOPS, which scales the engine deck based on maximum thrust at sea level static condition. The results were compared with the range values obtained by the newly developed analysis environment in Figure 74. The range values depicted in the gure include both climb and cruise segments. The dierence in range performance of the three solutions was found to be less than 1%. 6.1.3 Probabilistic Sizing
Three dierent approaches including FORM-RIA, FORM-PMA, and MCS-based optimization, denoted as P1, P2, and P3, were employed to obtain the probabilistic solutions to the CCP problem per Eq. (150) in Appendix C. The condence level of the objective function was targeted at 95%, and the probability of failure of each constraint was set to 0.05. In applying FORM, the probability of failure was computed by RIA, whereas f can be obtained by PMA from a nonlinear optimization problem in u-space subject to an equality constraint as max f
(163) u = (f )
1
s.t.
The MCS-based optimization employs the MoFD as the system level optimizer. 10,000 random cases were simulated to compute the probability of failure of the prob abilistic constraints and f , which is given as the value of the objective function that corresponds to the specied percentile. The seed values of the random number generator and the number of sample points (10,000) were xed to alleviate a discontinuity in the probability of failure and its gradients. 189
1200 1000
800 600 400 200 0 0 0.05 0.1 0.15 Mach No. 0.2 0.25
Fuel Flow (lbs/hr)
0 ft 4000 ft 8000 ft
20 19 18 17 16 15 0 0.05 0.1 0.15 Mach No. 0.2
0 ft 4000 ft 8000 ft
Thrust (lbs)
0.25
Figure 73: Thrust and fuel ow of the PEMFC propulsion system
800 700 600 Range (nmi) 500 400 300 200 100 0 D1
AIASM FLOPS
0.6%
-0.5%
0.1%
%difference between AIASM and FLOPS
D2
D3
Figure 74: Comparison of range performance
190
Table 9: Probabilistic solutions Power to Weight Ratio (ft/sec) Wing Loading (lbs/ft2 ) Fuel Fraction Take-o Gross Weight (lbs) Wing Area (ft2 ) Propulsion System Power (Hp) P1 35.52 11.06 0.0341 2936 265 190 P2 35.48 11.05 0.0341 2933 265 189 P3 35.37 11.03 0.0336 2902 263 187
The results of the dierent approaches are listed in Table 9. All of the optimization problems resulted in the same set of active constraints: take-o eld length, climb, and mission fuel. The three methods also yielded very similar optimum solutions. The probability of failure of the constraints of all deterministic and probabilistic solutions was estimated by additional Monte Carlo simulations, as comparatively listed in Table 10. Solution D1 is an example of the downfall of the voice of intuition, which results in an unacceptable level of risk for the take-o eld length constraint while having extraneous design margins for the other probabilistic constraints. This implies that the solution yields disproportionate amounts of reliability to the failure modes, despite yielding considerably heavier aircraft than the RBDO solutions. Solution D2 yields a signicantly high probability of failure for three hard constraints, which is not surprising because the solution is obtained without considering any design margins. Solution D3 possesses more balanced reliability than D1. However, it does not provide the target reliability, since the amount of the xed design margin (10%) was arbitrarily chosen. These observations lead one to surmise that, despite best intentions, the deterministic approach relying on Rules of Thumb may allow a qualied success at best. In contrast, the solutions from the RBDO approaches yield balanced design margins to the hard constraints while minimizing the aircraft weight. The values of Pf for
191
Table 10: Comparison of probability of failure D1 TOFL Climb Cruise Approach LDFL Mission Fuel 0.1473 0 0 0 0 0 D2 0.9879 0.6647 0.1843 0 0 0.813 D3 0.3106 0 0 0 0 0.1077 P1 0.0352 0.0268 0 0 0 0.0258 P2 0.0352 0.032 0 0 0 0.0254 P3 0.0507 0.0505 0 0 0 0.0507
the three hard constraints in FORM-RIA (P1) and FORM-PMA (P2) were smaller than the target value of 0.05. The dierence beyond the tolerance of a constraint violation set for the optimization process (0.001) can be considered the error of FORM provided that the values computed by MCS are suciently accurate. It was observed that FORM constantly underestimates Pf in the example of this particular study. Although FORM possesses such noticeable errors, it demonstrated approximately 5 times the computational eciency of the MCS-based optimization method for this sizing problem. The probability of meeting all constraints simultaneously must be less than or equal the minimum of the probability level of meeting the individual constraints. Moreover, if there is a strong negative correlation between any two hard constraints, the probability of meeting the joint constraints may be signicantly lower than a given probability for individual constraints. In this particular example, the solution of the MCS-based optimization method met joint probabilistic constraints by 89%. Therefore, it was deemed worthwhile to implement joint probabilistic constraints. A CCP problem, in which the six individual probabilistic constraints were consolidated into a joint probabilistic constraint, was formulated based on Eq. (149) presented in 5.5.1. The MCS-based optimization method was used for this study. The results, denoted as P4, are listed in Table 11. Applying the joint probabilistic constraints resulted
192
Table 11: The solution of joint probabilistic constraints Power to Weight Ratio (ft/sec) Wing Loading (lbs/ft2 ) Fuel Fraction Take-o Gross Weight (lbs) Wing Area (ft2 ) Propulsion System Power (Hp) P4 36.01 11.22 0.0340 2992 267 196 % Dierence to P3 1.80 1.72 1.00 3.13 1.39 4.98
in higher required power and more fuel than applying the individual probabilistic constraints.
6.2
Regenerative Solar-powered Aircraft
In addition to the fuel cell-powered electric GA, the proposed methods were applied to the sizing of a solar-powered high altitude long endurance (SPHALE) aircraft with a regenerative PEM fuel cell system as a proof of concept. This analysis served as an excellent demonstration example of the proposed methods because such a regeneratively powered concept cannot be sized with traditional sizing methods as discussed in 4.7.2. 6.2.1 Mission and Conguration
The notional SPHALE aircraft is assumed to operate for more than six months starting on April 1st , at 38 N latitude and a design altitude of 17 km, carrying a payload that weighs 981 N1 (100 Kg) and consumes 1,000 W of power continuously. The mission prole is depicted in Figure 75. The vehicle must be able to climb to 17 km by depending solely on solar energy and then maintain a level ight pattern. During the day, any excess energy from the sun is charged to the RFC system, while at night
In this solar-powered aircraft study, the International System of Units (SI) is used because of its preference in the literature regarding this topic. For the same reason, British units are used in the general aviation study.
1
193
the platform would maintain altitude and its geostationary position by utilizing the stored energy. The aircraft maintains the ight speed that maximizes its loiter eciency (E = CL /CD ). In order to ensure a safe ight during periods of extremely high wind-speed jets, the aircraft must be able to sustain a level ight with a speed of 35 m/s for at least two hours per day. On the rst day starting its service, the aircraft climbs to the loitering altitude by taking advantage of the available solar insolation with tanks full of oxygen and hydrogen. Thus, no excess power is needed to be delivered to the electrolyzer during climb. It is expected that the aircraft will not require access to large numbers of public airports, and no specic rate of climb is required either, upon which takeo and climb performance are considered to be fall-outs. In addition, landing and approach speed are also treated as fall-outs, since the aircraft is very likely to have a very low wing loading, and consequently, a slow approach speed. Therefore, the maximum cruise speed is considered to be the only performance constraint in this study. The HeliPlat shown in Figure 76 is selected as the baseline conguration for this study, because substantial analysis and experimental data for the conguration are available from Ref.[49]. The conguration has a high aspect ratio wing and a boommounted tail arrangement. The aspect ratio and taper ratio of the wing are 31 and 0.32, respectively. Aerodynamic coecients of a quadratic drag polar for the conguration are obtained from regression analysis with CFD analysis data from Ref.[49] as shown in Figure 77. The regressed drag polar curve tightly matches the original wind tunnel data for 0.5 CL 1.6, which is the expected ight range of the aircraft. The coecients of the regressed quadratic drag polar were estimated to be K1 = 0.0141, K2 = 0.0024, and CDo = 0.0134.
3/2
194
Payload (981N, 1000 W)
Decent Climb
Loiter for 6 month @17Km 19.7 m/s for 22 hrs 35 m/s for 2 hrs
Idle/Takeoff
Land/Taxi
Figure 75: The mission prole of the SPHALE
Aspect Ratio: 31 Taper Ratio: 0.31
Boom Mounted Tail
8 brushless electric motors
Figure 76: Baseline conguration of the SPHALE [49]
195
6.2.2
Deterministic Sizing
The airframe weight was computed with the following equation used in the HeliPlat study [49]: Wairf rame = 8.75n0.311 AR0.4665 S 0.7775 (164)
where n is the maximum load factor, which is assumed to be 3.1 as proposed by the HeliPlat study. The avionics system weight was assumed to be xed at 49 N. The weight and eciency of the PV cells was estimated based on the PV cells manufactured by SunPower, which have been used in both the Pathnder and Helios prototypes. The weight of the solar cells was computed by multiplying the weight per unit area (0.81 Kg/m2 ) by the required surface area, and the collective eciency of all panels was assumed to be 20%. The RFC system weight was computed based on its specic energy2 . The specic energy of the state-of-the-art RFC system is reported to be 298 W-Hrs/Kg [211]. Recently, Jakupca and Wendell [211] showed that the specic energy of an RFC based on PEM fuel cells could possibly reach 359 W-Hrs/Kg by packaging the hydrogen and oxygen tanks within a common pressure vessel. For this study, the optimistic 359 W-Hrs/Kg projection was assumed to be representative current technology level. The roundtrip eciency of the RFC system is estimated by employing an electrolysis theory and regression analyses as described in Appendix D. The electric propulsion system includes the motors, inverters, gears, and propellers, whose weight and eciencies are estimated based on the information from the HeliPlat study. The design reference power, Pref , is dened as the maximum continuous output power of the RFC system at the loiter altitude in this study,
Fuel cells operates on an input fuel, and do not run down or need to be recharged, making them more similar to combustion engines in their use rather than batteries [64]. In such a sense, dening the specic energy for a fuel cell system would not make sense. In the case of RFC systems, however, the amount of energy content is part of the system properties, making them more similar to rechargeable batteries. Thus, the specic energy is apposite as a measure of the gravitational energy storage capability of the system. For this reason, the specic energy has often been used to estimate the weight of an RFC system; e.g., Ref. [49, 142]
2
196
which leads + to be the aggregated eciencies of the electric propulsion systems components. Based on the disciplinary data described above, the aircraft was sized via the iterative process illustrated in Figure 55 in 4.7.2. The initial guess of the aircrafts weight was 12,100N. The convergence history pertaining to aircraft weight, payload power-to-weight ratio, and wing loading are depicted in Figure 78 and Figure 79. At every iteration, the 24 hour-cycle energy balance analysis and the constraint analysis are performed to update wing loading and power to weight ratio. Figure 80 shows the 24 hour-cycle energy balance analysis at the convergence point, depicting the variations in solar power, power required for ight, power required for payload, and RFC discharging power, each of which is normalized by aircraft weight. Power required for ight steps up during the rst two hours of the cycle to account for the maximum headwind condition. Figure 81 depicts the constraint analysis of the SPHALE at the convergence point. The total power required is the sum of the power required for level ight and power for the payload. The maximum power required for ight occurs when headwind is at maximum, which establishes the condition of power sizing. The sizing results of the baseline conguration are summarized in Table 12. In order to distinguish this solution from the others in subsequent investigations, it is denoted as D1. Figure 82 depicts the aircraft weight breakdown of solution D1. The airframe and propulsion system account for 39% and 52% of the aircraft weight, respectively. The lift-to-drag ratio at loiter and loiter eciency (CL /CD ) are estimated to be 35 and 44.3, respectively. The actual-value-based approach is preferable for this example, since the existence of a xed payload power signicantly osets the advantage of using weight-specic parameters, requiring a double-loop iteration. The actual-value-based approach employs an optimizer to match the performance to the target values by varying the
3/2
197
2 CD = 0.0141CL2 - 0.0024CL + 0.0134 R2 = 0.9989 1.5
1 CL
Quadratic regression CFD
0.5
0
-0.5 0 0.01 0.02 CD 0.03 0.04 0.05
Figure 77: Comparison of drag polars
12200 12000 11800 11600 Weight (N) 11400 11200 11000
0.092
W TO Guess W TO Calculated Payload/W TO
0.088
0.086
0.084
0.082 10800 10600 10400 1 6 11 16 21 Iteration 26 31 36 41 0.08
0.078
Figure 78: Convergence of weight and payload power-to-weight ratio
198
Payload Power-to-Weight Ratio (Watt/N)
0.09
47.75 47.7 47.65 47.6 47.55 Wing Loading (N/m2) 47.5 47.45 47.4 47.35 47.3 47.25 47.2 1 6 11 16 21 Iteration 26 31 36 41
Figure 79: Convergence of wing loading
4.5 Power per unit aircraft weight (W/N) 4 3.5 Solar Power 3 2.5 2 1.5 1 0.5 0 0 -0.5 hr 5 10 15 20 25 Power Required for Flight Power Required for Payload Total Power Required Fuel Cell Output Power
Figure 80: Power prole of the SPHALE
199
Maximum Allowable Wing Loading
2.0 1.8
Power-to-Weight Ratio (Pref/W, m/s)
1.6 1.4 1.2 1.0
Power required for flight
0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70
N/m2) Power required for flight and payload Design Point
80
90
100
Wing Loading (W/S,
Figure 81: Constraint analysis at the converged solution
design variables (S, Pref , and WH2 ). The design reference power (Pref ) for this case study is referred to as the total amount of power transferred from the PV cells and/or the RFC system to power electronics. The following constraints are considered to ensure the power balance, nighttime energy balance, and diurnal energy balance, which leads to the following nonlinear-constrained deterministic programming problem: min WTO
x
s.t. Pref
DVmax + PPL +
(165)
WH2 |available WH2 |required EStorOU T = rt EStorIN
200
Table 12: Baseline conguration sizing - solution D1 Aspect Ratio Wing Loading (N/m2 ) Power-to-Weight Ratio (m/s) Required RFC System Energy to Weight Ratio (Wh/N) Wing Area (m2 ) Max Power required (KW) Aircraft Weight (N) Required RFC System Energy (KWh) Optimum AoA Optimum Loiter Velocity (m/s) L/D E 31 47.4 1.378 11.3 234 15.3 11,086 125.1 1.6 20.4 35.0 44.3
15.1% 0.4%
6.1%
Payload Airframe
30.8%
Avionics Solar Panel
38.7%
Motors and Props Fuel Cell System
8.8%
Figure 82: Weight breakdown of the design gross weight of D1
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This problem was solved by an SLP algorithm of a commercial optimization package, DOT , included in ModelCenter and its solution was found to be numerically
identical to D1, obtained by the specic parameter-based method. 6.2.3 Impact of Technology Advancement
Solution D1 was obtained based on currently available technologies. However, relevant technologies are rapidly advancing as discussed in 1.1. Therefore, it is worthwhile to investigate the impact of technology improvement on aircraft sizing. This study considered ve technology factors: 1) PV cell eciency, 2) PV cell weight, 3) RFC system specic energy, 4) fuel cell eciency, and 5) propulsive system specic power. The aircraft was re-sized while incorporating a series of improvements in each technology factor, between 10% and 50%, and the resultant aircraft designs are compared in Figure 83. The most signicant reduction in aircraft size occurs by increasing the PV cell eciency and the RFC specic energy, whereas the mass of the power electronics plays a marginal role in comparison3 . Romeo et al. [49] proposed the goals of future technologies in solar-powered HALE aircraft as follows:
PV cell eciency increased to 25% RFC system specic energy increased to 550 KWh/Kg eciency of fuel cells of the RFC increased to 70%
The present research accepts these hypothetical advanced technology assumptions as representative near-term goals, which are consistently applied through the subsequent studies presented in 6.2.4 to 6.2.7. To reect the projected improvement in each technology, the baseline conguration was re-sized, and the resultant design is denoted
Mission requirements also substantially aect the aircraft size. By limiting the time of year the aircraft is required to y at high northern or southern latitudes, a signicant reduction in aircraft size or increase in payload capacity can be achieved.
3
202
12000
11000
Aircraft Weight (N)
10000
9000
8000
7000
6000 0% 10% 20% 30% 40% 50% Technology Improvement (%)
Solar Cells Efficiency Solar Cells Mass per Unit Area RFC System Specific Energy Fuel Cell System Efficiency Propulsive System Specific Power
Figure 83: Sensitivity of technology impact
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Table 13: Baseline conguration sizing with advanced technologies - solution D2 Aspect Ratio Wing Loading (N/m2 ) Power-toWeight Ratio (m/s) Required RFC System Energy to Weight Ratio (Wh/N) Wing Area (m2 ) Max Power required (KW) Aircraft Weight (N) Required RFC System Energy (KWh) Optimum AoA Optimum Loiter Velocity (m/s) L/D E 31 54.7 1.372 12.8 105 7.86 5729.6 73.3 1.6 22.0 36.1 44.3
as D2. The details of D2 are summarized in Table 13. Weight reduction in each weight group by infusing advanced technologies is illustrated in Figure 84, which shows a 46% to 52% reduction in the weight of airframe and propulsion system components, which collectively reduces the aircraft weight by 48%. The weight breakdown of the aircraft is depicted in Figure 85, which identies, in comparison with Figure 82, that reecting the projected technology improvements results in a considerable reduction in the RFC system weight fraction. The solar cells weight fraction did not change considerably because the panels weight decreases in proportion to the amount of reduction in aircraft weight. In contrast, a notable increase in the payload weight fraction was observed to be caused by aircraft weight reduction. 6.2.4 Conguration Optimization
Even when a basic concept is determined, a large number of external conguration options such as the selection of airfoil, platform, and twist distribution for wings and tails may still be altered by designers. As discussed in 5.5.2, aircraft sizing can be performed in conjunction with optimizing disciplinary design variables. For demonstration purposes, the wing aspect ratio was selected as a representative disciplinary
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RFC system
Propulsive system
Solar cells
Airframe 0 10 20 30 40 50 60 70
% Reduction to the baseline
Figure 84: Weight reduction in weight groups by infusing advanced technologies
0.9% 40.8%
13.2% 6.1%
Payload Airframe Avionics Solar Panel Motors and Props
21.0%
Fuel Cell System
18.0%
Figure 85: Weight breakdown of the design gross weight of D2
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variable, because it is generally considered a primary design parameter that trades the aerodynamic eciency and wing weight. As the aspect ratio increases, drag-dueto-lift reduces. Such a tendency can be captured by employing a set of empirical equations. The coecient of drag-due-to-lift (K) can be expressed by: K = 1/eAR (166)
where AR is the aspect ratio, and e is known as Oswalds eciency of the wing of sailplanes, which was approximated as suggested in Ref. [212]. 0.9 AR 20 e= 1.2 0.015AR AR > 20
(167)
The variation in K as a function of aspect ratio, estimated by Eq. (166) and Eq. (167), is illustrated in Figure 86. This gure shows that the values of K decrease as aspect ratio increases, and measurements from the wind tunnel test of the baseline conguration match the empirical equation well. The variation in airframe weight with aspect ratio can be captured by Eq. (164). Figure 86 also shows the impact of aspect ratio on airframe weight for two wings of dierent sizes. The optimum aspect ratio can be found by solving the optimization problem expressed as Eq. (165) in 6.2.2 by adding aspect ratio to the list of design variables. The same analysis tools and environment as used for the previous actual-value-based attempt to solve Eq. (165) were used. The results, denoted as D3, are listed in Table 14. The optimum aspect ratio was found to be 25.6, which results in 92 N of reduction in the total aircraft weight. 6.2.5 Probabilistic Sizing
The impact of technology improvements and wing geometry optimization presented thus far are from a deterministic aircraft sizing perspective; that is, the aircraft was sized without any additional design margins. It was assumed that the uncertainty 206
0.026 0.024 0.022
5000 4500 4000 3500 3000 2500 2000
K1 (Empirical Data [212]) K1 (CFD Analysis [49]) Airframe Weight (S=105 m2 ) Airframe Weight (S=234 m2)
K1 0.02
0.018 0.016 0.014 0.012 0.01 10 15 20 25 Aspect Ratio 30 35 40
1500 1000 500 0
Figure 86: Impact of wing aspect ratio on drag and airframe weight
Table 14: Optimum conguration with advanced technologies - solution D3 Aspect Ratio Wing Loading (N/m2 ) Power-to-Weight Ratio (m/s) Required RFC System Energy to Weight Ratio (Wh/N) Wing Area (m2 ) Max Power required (KW) Aircraft Weight (N) Required RFC System Energy (KWh) Optimum AoA Optimum Loiter Velocity (m/s) L/D E 25.6 52.5 1.432 13.3 107 8.07 5637.6 75.2 1.5 22.0 33.3 41.4
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Airframe Weight (N)
Table 15: Assumed distributions of random parameters Normalized Random Parameter Induced Drag, K1 Maximum Level Flight Speed PV Cell Eciency Airframe Weight RFC System Specic Energy Distribution Normal Normal Normal Normal Normal 1 1 1 1 1 0.01 0.02 0.03 0.02 0.05 LB 0.95 0.9 0.9 0.9 0.833 UB 1.05 1.1 1.1 1.1 1.167
sources listed in Table 16 would play an important role in this design example. All random parameters were modeled as normal distributions normalized by their mean values. The upper and lower bounds (UB and LB) truncating both sides of tails were set to prevent unrealistic, extreme cases. The three constraints included in the deterministic sizing problem per Eq. (165) are considered again, but were rearranged as follows: g1 = Pref |available Pref |required 0 g2 = ERF C |available ERF C |required 0 g3 = WH2 |regenerated WH2 |required 0 Based on the assumptions of uncertainty, the probability of failure for each constraint in deterministic solution, D3 was computed by MCS. Out of 100,000 runs, the probability of failure for all constraints was estimated at 0.5. Such a high probability of failure results from not having considered adequate design margins for the random parameters. The target probability was set at 95% for each constraint. The design objective was established as minimizing the design gross weight (W ), which is also aected by the two random parameters: the airframe weight and the specic energy of the RFC system. In order to address the probabilistic nature of the response, a target condence level (95% for this study) was imposed on the probabilistic objective as an additional constraint as described in 5.5.3.1, which means the 95th percentile of (168)
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the aircraft gross weight responses were used as the objective function value by the system-level optimizer. MCS and FORM were used to estimate the probability of success (reliability) for each design constraint. The probabilities computed by two dierent methods are compared at 30 points randomly selected in the design space. In Monte Carlo simulations, 100,000 cases were generated by Crystal Ball with a xed seed number, and the probability of meeting each constraint was computed. As shown in Figure 87, FORM estimated the reliability with comparable accuracy to the MCS. Figure 88, illustrating the dierences in the 95th percentile of aircraft weight responses computed by the two methods, shows that FORM was also able to estimate the objective function with comparable accuracy to MCS for this case study. It was also observed that FORM produces a higher probability of meeting the constraints and lower values of aircraft weight than MCS for most test points. It must be noted that such a tendency is valid for this particular example, since FORM does not necessarily yield a more optimistic estimation than MCS. It was found that FORM was approximately 4 times faster than the Monte Carlo simulation for this particular example problem and the analysis environment employed. To enhance computational eciency, the problem was solved rst by using FORM for a nested reliability analysis. The solutions are denoted as P1. Subsequently, the problem was solved again using MCS with P1 as an initial point for the optimization process, the solution of which is denoted by P2. The SLP algorithm was used both for solving the system level optimization problem and for the MPP search. The optimization results are summarized in Table 16, showing that the solution obtained by FORM was very close to that obtained by the Monte Carlo simulation. The dierence in the design variables and the objective function values was less than 1%. For further comparison of accuracy of FORM and MCS, Monte Carlo simulations
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0.012 0.01 Difference to MCS 0.008 0.006 0.004 0.002 0 -0.002 0 0.2 0.4 0.6 0.8 1
g1 g2 g3
Probability of Meeting a Constraint
Figure 87: Dierences in probabilities meeting the constraints estimated by FORM to those estimated by Monte Carlo simulations
0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 5600 5800 6000 6200 6400 6600 6800 7000
Difference to MCS
Aircraft Weight (N)
Figure 88: Dierences of the objective function values (95th percentile of the objective function responses) estimated by FORM to those estimated by Monte Carlo simulations
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Table 16: Comparison of the results of probabilistic sizing by FORM and MCS Optimum Solution Aspect Ratio Wing Area (m2 ) Power Available (KW) RFC System Energy (WKh) Power Balance Nighttime Energy Balance Diurnal Energy Balance 95th percentile Mean P1 26.07 120.02 9.27 81.68 Active Active Active 6260 6110 P2 25.93 120.38 9.30 81.89 Active Active Active 6269 6116 -0.14 -0.11 % Dierence 0.53 -0.29 -0.27 -0.26
Constraints
Objective Function (N)
were performed with P1 while varying the number of trials from 102 to 105 . For each trial number, Monte Carlo simulations were repeated ten times while varying the seed number for random number generations to produce dierent sets of random events. The results are depicted in Figure 89, which shows the probabilities of meeting the non-deterministic constraints and the values of the objective function for dierent numbers of trails. The dotted lines and the solid lines represent the probabilities computed by FORM and MCS with one million trials, respectively. The gure also shows that the clusters of Monte Carlo simulations results converge to the solid line as the number of trials increases. The gure shows that FORM slightly overestimates the reliability of the constraints. However, it is observed that some Monte Carlo simulations with less than 105 trials produced higher errors than FORM. Figure 90, which compares the percentage-dierence between the probabilistic solutions (P1 and P2) and deterministic solution (D3), shows that power, wing area, and the amount of RFC system discharge energy of the probabilistic solutions are increased by 6.2%, 12.0%, and 8.9%, respectively, in order to satisfy the target probability of meeting the constraints. Such dierences in design variables can be referred to as the design margins required to mitigate the risk associated with uncertainty.
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Power Balance Power Balance (g1) 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 10 100 1000 10000 100000 1000000 Number of trials
Probability
Nighttime Energy Balance (g2) Hydrogen Fuel
1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 10 100 1000 10000 100000 1000000 Number of trials
Probability
Diurnal Energy Balance (g3) Daily Energy Balance
0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 10 100 1000 10000 100000 1000000 Number of trials
Probability
Aircraft Weight 6290 6280
Weight (N) Probability
6270 6260 6250 6240 6230 10 100 1000 10000 100000 1000000 Number of trials
Figure 89: Monte Carlo simulations with dierent numbers of trials for P1 212
As shown in Figure 90, the design margin required for each design variable is not the same. Therefore, simply adding constant percentage of a design margin to all design variables may result in a non-optimal solution in the sense of CCP and RBDO. Since the design gross weight is aected by the random parameters, the response is given as a distribution for a certain design variable setting. The distributions of the objective function and the constraint functions at P2 are shown in Figure 91 and Figure 92, respectively. The distributions of several other responses of interest at the optimum solution, such as optimal angle of attack, loiter eciency, ight speed, and required power at loiter, are depicted in Figures 114-117 in Appendix E. As mentioned previously, one of the advantages of using CCP is that it allows the decision maker to trade the associated risk with the goal of the objective function. This is particularly useful when the decision maker has only a vague idea of what the appropriate condence level is. This trade-o can be facilitated by a visual relationship between and the value of the objective function, as illustrated in Figure 93. The objective function (the 95th percentile of the design gross weight responses) increases as increases. After exceeds 90%, the expectation of the take-o gross weight rapidly grows, which indicates more cost per reliability. This information assists a decision maker to determine the level of . 6.2.6 Sensitivity Analysis
As discussed in 5.6.1, Lagrange multipliers provide an insight into the sensitivity of the objective function value with respect to the target probability. The Lagrange multipliers at the probabilistic solutions, P1 and P2, are listed in Table 17. Their magnitudes identify that the diurnal energy balance constraint (g3 ) is most important for both P1 and P2. A Lagrange multiplier represents the variation of the objective function per unit change in the constraint function. The Lagrange multipliers listed in Table 17 were obtained by outer-loop optimization processes, in which the constraint
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Aspect Ratio 20.0 (%) 15.0 Weight (Deterministic) 10.0 5.0 0.0 P1 P2 Wing Area
Weight (95th percentile)
Power
RFC System Energy
Figure 90: Dierence of probabilistic solutions, P1 and P2 to the deterministic solution, D3 in percentage
W (N)
Figure 91: PDF of the objective function
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Figure 92: PDFs of constraint functions
215
P1 and P2 7000 6750 6500 Aircraft Weight (N) 6250 6000 5750 5500 5250 5000 0.4 0.5 0.6 0.7 0.8 0.9 1
MCS (95th percentile) MCS (Deterministic) FORM (95th percentile) FORM (Deterministic)
Figure 93: Objective function vs. target reliability
216
30 28
AR
26 24 22 20 130 125
Wing Area (m2)
120 115 110 105 15 10
Power (KW)
5 0
90 Fuel Cell Energy (KWh) 85 80 75 70 0.4 0.5 0.6 0.7 0.8 0.9 1 P1 P2
Figure 94: The optimum values of sizing variables vs. target reliability
217
Table 17: Lagrange multiplier Power Balance, g1 Nighttime Energy Balance, g2 Diurnal Energy Balance, g3 P1 420 889 1895 P2 726 879 1732
functions were given in the form of P[g > 0] > instead of g > 0. Therefore, the sensitivity of the target probability, in terms of percentages, for a constraint can be obtained by dividing its Lagrange multiplier by 100. For example, it is expected that a relaxation of the target probability of g3 by 0.01 (1%) will result in a reduction in the aircraft weight by 19.0 N and 17.3 N for P1 and P2, respectively. In order to assess the accuracy of Lagrange multipliers as sensitivity indices, the local sensitivities of the objective function to the target probability of constraint functions were computed by a nite dierence method, and comparatively listed in Table 18. The rst column lists the Lagrange multipliers of P1 divided by 100. The second column presents the sensitivity of the objective function to the target probability obtained by subsequently solving the optimization problem while reducing the target probability of each constraint by 1% at a time. As it can be seen from Table 18, the sensitivities computed by Lagrange multipliers are in agreement with those obtained by the nite dierence method. It must be noted that Lagrange multipliers provide only local sensitivities near the optimum solution. Nevertheless, they can be obtained as by-products of the optimization process, requiring no additional computational cost. Another important sensitivity analysis was performed by the PSA discussed in
5.6.2. The sensitivities assist engineers in understanding the relative importance of
the uncertainty of the probabilistic objective and constraint functions. The sensitivity of the constraint functions to the random parameters for P1 were evaluated by the MPP-based sensitivity method presented in 5.6. Figure 95 shows the sensitivity
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Table 18: Sensitivity of the objective function per target probability (%) By Lagrange Multiplier 4.2 8.9 19.0 Finite Dierence Method 4.1 7.8 18.3 % Dierence 0.1 1.1 0.7
Power Balance, g1 Nighttime Energy Balance, g2 Diurnal Energy Balance, g3
indices of random parameters for each constraint. This sensitivity analysis reveals the most signicant random parameter for each probabilistic constraint and objective function. The maximum cruise speed, and PV cell eciency were found to be the dominant factors for the power balance constraint and the energy balance constraint, respectively. The energy balance and the objective function were found to be most heavily sensitive to fuel cell stack energy density. Particularly, the power balance constraint and the diurnal energy balance constraint were mostly aected by one dominant parameter, which implies that the reliability of the probabilistic constraint can be substantially improved by eliminating the uncertainty associated with the parameter. The sensitivity for P2 that was obtained by MCS can be obtained by an embedded feature of Crystal Ball . This tool pairs up each assumption rank list with a corresponding forecast rank list and then calculates a Pearson correlation coecient for each pair [213]. Figure 96 depicts the sensitivity of the responses of the objective and constraint functions of Crystal Ball . The results are consistent with those obtained by the MPP-based analysis shown in Figure 95. Figure 97 depicts how reducing the variance of each assumption improves the odds of meeting the non-deterministic constraints. All importance rankings listed in Figure 97 are consistent with those observed from Figure 95. For example, the probability of meeting the power balance constraint was found to be dominated by the maximum cruise speed in Figure 95. Figure 97 shows that reducing its variance
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Power Balance, g1
Nighttime Energy Balance, g2
Diurnal Energy Balance, g3
Aircraft Weight, f
0
K1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Maximum Level Flight Speed
PV Cell Efficiency
Airframe Weight
RFC System Specific Energy
Figure 95: Sensitivity index obtained by the MPP-based sensitivity analysis
Power Balance, g1
Nighttime Energy Balance, g2
Diurnal Energy Balance, g3
Aircraft Weight, f
0
K1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Maximum Level Flight Speed
PV Cell Efficiency
Airframe Weight
RFC System Specific Energy
Figure 96: Sensitivity index obtained by Crystal Ball
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by 30% increases the probability from 0.95 to 0.99, while reducing the variance in other random parameters leads to a paltry improvement in reliability. 6.2.7 Joint Probabilistic Constraints
Application of the joint probabilistic constraints to the probabilistic sizing problem is preferred when the probability of simultaneously meeting the probabilistic constraints is more meaningful. As mentioned in 5.5.1, the probability of meeting all constraints simultaneously must be less than or equal the minimum of the probabilities of meeting individual constraints. In this particular example, the probability of meeting the joint probabilistic constraint at P2, which meets the individual constraints with 95% probability, was computed at only 0.88. This dierence can easily be explained by Figure 98, which depicts the constraint function responses taken from Monte Carlo simulations with the solution. For better visuality, 3,000 random cases out of 100,000 cases are included in the gure. The problem is set up such that g > 0 is the feasible set. Therefore, the cases that are located in the upper right quadrant meet g1 and g2 concurrently. As shown, there are some random events that do not satisfy g1 and g2 simultaneously, while instead, satisfying only one of them. In addition, some cases marked in blue and located in the upper right quadrant fail to meet g3 , although they satisfy the energy balance and power balance constraints. A CCP problem, in which the three individual probabilistic constraints are consolidated into a joint probabilistic constraint was formulated based on Eq. (149) presented in 5.5.1. It was observed that the SLP algorithm often fell into an infeasible region where the gradient of the constraints was computed at zero, thereby terminating the optimization process. It was conjectured that this problem was caused by the intrinsic nature of the SLP algorithm. The SLP algorithm approximates the nonlinear constraints as linear constraints at a current design point and applies a simplex method to obtain an optimum solution in the approximated design space. No optimum point
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Power Balance
1 0.99
Probability PoS
0.98 0.97 0.96 0.95 0.94 0 20 40 60 80 100 Reduction in Variance, in percentage
Nighttime Energy Balance
1 0.99
Probability PoS
0.98 0.97 0.96 0.95 0.94 0 20 40 60 80 100 Reduction in Variance, in percentage
Diurnal Energy Balance
1 0.99
Probability PoS
0.98 0.97 0.96 0.95 0.94 0 20 40 60 80 100 Reduction in Variance, in percentage
K1 Maximum Level Flight Speed PV Cell Efficiency Airframe Weight RFC System Specific Energy
Figure 97: Reliability improvement of probabilistic constraints by variance reduction in random parameters 222
2.0 Not meeting all Meeting all
1.5
1.0
0.5
0.0
-0.5
-1.0 -4 -2 0 2 4 6 8
Figure 98: Illustration of samples of an MCS in the space of power balance - energy balance
may be found when the design space is constrained with one linear constraint. This problem can be resolved either by using an MoFD algorithm or adding individual probabilistic constraints for which target probability is the same as that of the joint probabilistic constraint. The feasible set conned by multiple individual constraints must include the feasible set that meets the joint constraint. Therefore, adding the individual constraints does not aect the optimum solution. Instead, they help to prevent the process from falling into divergence. The sizing solutions of the CCP optimization problem with joint probabilistic constraints are listed with those of the individual probabilistic constraints in Table 19.
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Table 19: Comparison of the solutions of individual probabilistic constraints and joint probabilistic constraints Individual Probabilistic Constraint Design Variables Aspect Ratio Wing Area Power Available Fuel Cell Energy Available Objective Function Constraints (probability) Power Balance Nighttime Energy Balance Diurnal Energy Balance Meeting All 25.9 120.4 9.3 81.9 6268.6 Joint Probabilistic Constraint 27.5 121.7 9.6 83.0 6396.6 % Dierence
5.9 1.1 2.7 1.4 2.0
0.95 0.95 0.95 0.88
0.99 0.99 0.97 0.95
3.9 3.8 2.0 7.6
6.3
Lessons Learned from Implementation Studies
The AIASM has been veried via its application to two example sizing studies. The major ndings obtained from these exercises are summarized below:
The electric GA sizing study showed that the fuel cell-powered electric propul-
sion system architecture could result in a comparable aircraft weight and size to its counterpart, if the projected technology improvements were achieved.
The study identied the benet of the use of a highly ecient electric propul-
sion system along with a high energy content fuel. The estimated fuel fractions ranged from 2.9 to 3.5 percent, which is remarkably low compared with its conventional counterpart. However, the study also revealed that the weight penalty incurred by hydrogen fuel storage could signicantly oset the advantage of the use of a high energy content fuel.
The SPHALE study included technology impact assessments, which identied
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the eciency of the PV cells and specic energy of the RFC as the most dominant factors. Similar technology assessments have been conducted by other researchers. The impact of technology improvements, however, was evaluated in terms of performance measures for a xed design. In contrast, AIASM oers an outstanding capability of assessing the impact on air-vehicle designs rather than performance measures. Such a technology-to-design capability, as opposed to a technology-to-performance one, is preferred for the design of a brand-new aircraft. The PASM was also demonstrated to be capable of apportioning reasonable design margins required to ensure target reliability against probabilistic constraints. Notable lessons learned from the applications of PASM to the two example studies are abridged below:
The GA study demonstrated that PASM could ensure the minimum cost at an
admissible probability of failure, which, in general, is not expected by traditional deterministic approaches.
The study showed that MPP-based methods may produce a considerable error in
estimating reliability although their computational eciencies surpassed those of sampling based methods. However, the MPP-based methods estimated the reliability for the SPHALE problem fairly accurately, which suggests that the MPP-based methods must be used with caution.
Despite their inherent errors, however, the optimum solutions obtained by em-
ploying the MPP-based methods for a nested reliability analysis were found to be in close proximity to those obtained by Monte Carlo simulations. The dierences in the optimum values of wing area and power were less than 2 percent, suggesting that even if their accuracy is not acceptable, MPP-based
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methods might be useful in a scheme of sequential applications of two optimization processes: the rst with MPP-based methods and the second with an MCS method as their nested reliability analysis. The solution to the rst optimization process will serve as a good starting point for the second one.
The present research also demonstrated another extended capability of PASM
simultaneous optimization of disciplinary variables and sizing variables. In addition, it was found that the probabilistic sensitivity analysis enhanced PASM by identifying the relative signicance of selected random parameters. The information may assist designers and decision makers in various ways. For example, the complexity of the problem and computational eorts could be reduced by eliminating insignicant random parameters. Engineering tasks could also be prioritized in order to validate data associated with random parameters.
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CHAPTER VII
CONCLUSIONS AND FUTURE WORK
Literature review presented in Chapter I identies the emergence of alterative energypropulsion system architecture for future aviation as well as the need for an aircraft sizing capability to properly assess such technologies. Literature review continued in Chapter II, however, reveals the shortcomings of traditional aircraft sizing methods in their application to the conceptual design of such unconventionally-powered aircraft. The ndings in these two introductory chapters formed the basis of the research objective outlined in 3.1. By advocating a couple of critical new capabilities, the objectives in turn allowed the derivation of the research questions and hypotheses, which are delineated in 3.2 and 3.3, respectively. In closing, some critical review of the research questions in light of the accomplished thesis work is presented in the next section. The remainder of this nal chapter is devoted to a section that abridges the major contributions from research of this magnitude, which is followed by a discussion of future work.
7.1
Research Questions Answered
The research questions and the hypotheses are restated below, and the answers to each question are given based on the accomplishment of this dissertation. 7.1.1 Research Questions 1
Question 1: How can a generalized aircraft sizing method, independent of the
architecture of energy storage and power generation, be formulated? Architecture-independence demands a holistic approach toward developing a generalized method rather than creating a specic solution to a particular problem. Such 227
a generalized method must be able to capture the implications introduced by the utilization of alternative energy sources and revolutionary propulsion systems into aircraft sizing processes. In order to address this question, the following hypothesis has been proposed.
Hypothesis 1: A generalized aircraft sizing formulation that is independent of
propulsion system architectures and energy sources can be formulated based on the traditional energy-based sizing approach by making the following modications: The propulsion system architecture is modeled as an integration of powerpath(s), each of which is characterized by three parameters: the specic energy of the energy source, specic power, and eciencies of power transfer devices. Fuel is generalized as the source of energy on board a vehicle and is categorized into based on its nature of conversion: consumable and nonconsumable. Aircraft weight is decomposed into more generalized weight groups, which leads to a more general weight dierential equation. This hypothesis propositions that the key to achieving an architecture-independent capability is to generalize the underlying assumptions of traditional sizing methods, which encompass the concepts of fuel, propulsion systems, and weight estimations. Mathematical interpretations of such generalized assumptions are elaborated in 3.4.1, leading to the development of AIASM in Chapter IV. Built upon such generalized assumptions, the method possesses a common structure that is applicable to any energy-propulsion system architecture and is able to capture the impact on aircraft sizing due to the choice of an architecture. Two implementation studies presented
228
in Chapter VI shows that AIASM is able to size revolutionary aircraft powered by alternative energy-propulsion system architectures. 7.1.2 Research Questions 2
Question 2: How can adequate design margins, required for mitigating the risk
associated with uncertainty having minimal impacts on the design objective(s), be quantied in an aircraft sizing problem? This question seeks a method that is capable of quantifying design margins that represent a trade-o between cost and risk for an aircraft sizing problem.
Hypothesis 2: It is possible to determine adequate design margins that result
in a solution satisfying all probabilistic constraints under consideration with a target probability, while searching for a design optimum. This hypothesis contains a supposition that the CCP and RBDO approaches are appropriate for formulating a probabilistic aircraft sizing problem under uncertainty, which is supported by the discussions in 5.1 and 5.2. The implementation of the CCP and RBDO approach into aircraft sizing formulation resulted in PASM that is capable of intelligently determining appropriate design margins as the best compromise between cost and risk. The GA sizing study presented in Chapter VI shows that PASM can nd a better solution than what is obtainable with other, more routine methods.
7.2
Summary of Contributions
The primary objective of this research has been the development of new capabilities that are required for sizing an aircraft powered by alternative energy-propulsion architectures. The major contributions of this dissertation are AIASM and PASM. In
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the course of the development of AIASM, a set of generalized Breguet range equations and NAM ratio diagram have also been developed, which are now summarized as follows.
AIASM
The architecture independent aircraft sizing method (AIASM) has been formulated. Particular emphasis is placed on ensuring generality, as to encompass a variety of emerging alternative energy-propulsion architectures. The method also provides a proper means to size an aircraft propelled by a hybrid energypropulsion system architecture more eectively and eciently.
Generalized Breguet Range Equations
A set of generalized Breguet range equations can estimate the ferry range of an aircraft powered by alternative energy-propulsion architectures with little information pertinent to the aircraft. The set includes three equations, each of which falls under one of the three following categories: 1) an aircraft is fueled with a consumable energy source, and its consumption leads to a change in the aircrafts weight; 2) an aircraft is fueled with a consumable energy source but its consumption does not lead to a change in the aircrafts weight; and 3) an aircraft is fueled with a non-consumable energy source.
NAM Ratio Diagram
Utilizing the generalized Breguet range equations as well as the intrinsic duality between the mass of on-board energy sources and the mass of propulsion systems, the NAM ratio diagram identies performance frontiers in terms of range and velocity, achievable with a given energy-propulsion system architecture. Such a diagram enables a designer to examine rapidly a eet of aircraft designs that are possible with a given energy-propulsion system architecture, hence, easily identifying the mission that appears to be the most appropriate. 230
PASM
Formulated based on the principles of CCP, PASM nds a solution of aircraft sizing variables that ensures the best objective function value with a given target probability of meeting probabilistic constraints under uncertainty. In addition, the probabilistic sensitivity analysis was found to be benecial by providing a greater insight into the relationship between reliability and the distributions of random variables.
7.3
Future Work
Over the course of its development process, a number of secondary research areas were identied. Such areas, if pursued, would enhance the capability of the formulated sizing methods. This nal section summarizes the most important lessons learned as action items for future work. 7.3.1 Comprehensive Sizing Method
The authors aircraft sizing method has its basis in balancing power and energy. However, a successfully sized conguration must achieve one more criterion: a balance between required and available aircraft volume. All three criteria will not always play equal roles in an aircraft sizing process. The relative importance of the three would depend on various parameters such as sizing requirements; conguration shapes; and characteristics of the propulsion systems and energy sources, which include the following four parameters: the specic energy and energy density of the energy source, and the specic power and power density of the propulsion system. For instance, if an aircraft is powered by an isomer energy source whose specic energy is exceptionally higher than that of conventional fuel (approximately 3 104 times greater), the weight of the energy source may become trivial compared to that of the entire vehicle. In such a case, the energy balance will no longer be a major concern during the aircraft sizing process, just as the weight of the cockpit instruments is not much 231
of a concern in conventional aircraft sizing practice. Under such circumstances, it is possible that the volumetric and power demands may begin to take precedence over the energy balance. In general, the consideration of volume in an aircraft sizing process is encapsulated in a baseline conguration, whose volume balance is veried by an o-line internal layout study after it is initially is sized. An internal arrangement drawing, occasionally called an inboard prole, includes the occupied volume of all major components, such as the avionics bay, cockpit, landing gear, engine, passenger compartment, and fuel tank. This graphical layout of all primary components allows designers to compute the amount of usable volume for fuel, which is generally computed by multiplying fuel tanks OML volume by fuel packing factors. The surplus fuel volume - available fuel volume minus required fuel volume - is often considered as the touchstone of volume balance. The putative reason for measuring the surplus fuel volume as the index of volume balance is that fuel tanks of most modern aircraft are integrated with their surrounding structures such as the bulk heads, shear webs, wing skins, and spars to reduce the aircraft weight, which leads to byzantine-shaped fuel tank boundaries - now a striking feature of modern ghters. Therefore, measuring the fall-out volume with the left-over space from packing other components is usually easier than to allotting fuel tanks with the exact amount of required fuel volume. However, such a follow-on volume assessment, performed separately from a numerical sizing process, entails an extra manual iterative step in the design process. A conguration, if sized under consideration of power and energy balance only, may not provide sucient volume for all required systems. If it does not, then the external conguration may have to be modied so that the eciency of the internal packaging increases, or the aircraft may need to be scaled up beyond the minimum size at which both the power balance and energy balance are achieved. Such a human-inthe-loop process, which can require substantial man-hours, retards the convergence of
232
the conguration and stymies designers from investigating alternative congurations. Therefore, an integrated sizing environment that is able to reect volumetric demands on a sizing process concurrently, as illustrated in Figure 99, is desired, especially in the design of a revolutionary aircraft where the implicit volume balance discussed in
2.2 will not work.
Nevertheless, little literature regarding simultaneous volume assessment methods can be found. One of the most signicant hindrances to the development of such an integrated power-energy-volumetric sizing environment is that it is very dicult to construct a parametric model to assess volumetric balance. Often, most internal components have yet to be designed from scratch, and they will not be designed until the later detailed design phase [214]. For this reason, an internal layout drawing at the early stage of development only includes a few major components or simply their estimated volume, while excluding structures and small equipment items such as actuators, environmental control systems, and the ducting and wiring. An experienced aircraft designer assigns a sucient amount of volume to those un-spoken-for items in the aircraft, properly spaced throughout the aircraft [214]. The correct amount and shape of extra space for such items can be hardly estimated in any practical way, but may just look right. As Raymer put it in his seminal paper [214] on volumetric sizing, Such an intuitive measure of merit is dicult to duplicate or teach, and impossible to program. For this reason, example studies of parametric volume assessment found in existing literature guarantee only several major subsystem components. For example, Pouchet et al. [215] developed an aircraft sizing environment that automated the volume assessment process for the design of a hydrogen-fueled jet commercial transport, called a Quiet Green Transport (QGT). The optimization environment includes a conventional sizing code (FLOPS) and two external analysis modules required for assessing volume: one computes the hydrogen tank volume for a given fuel tank geometry, and
233
the other estimates an appropriate fuselage length and diameter to accommodate the hydrogen fuel tank. However, these authors volume analysis was limited to the hydrogen fuel tank. An alternative classical approach to assess volume balance measures overall crowdedness of a given conguration rather than actual volume. Since Caddell [216] started a pioneering use of a volumetric density to determine a reasonable volume allocation in 1969, such density-based volume assessment has been widely used in the industry. The fundamental idea is, in fact, very simple. The aircraft density is computed by the total aircraft volume1 divided by the aircraft weight. Historical data indicates that the density of aircraft in the same class is tightly distributed around an average value, which can be used as a touchstone to judge if a given conguration has adequate volume. The inordinately high value of this density above an average value would indicate that the design is very likely to suer from lack of volume. This approach is relatively easy to implement into an automated sizing environment. Nevertheless, its dependence on empirical data prevents the method from being applied to the design of the alternative energy-propulsion system architecture as it is. In addition, the method is only applicable to allocating the total amount of internal volume. It is still be possible that the sized conguration could not accommodate all components, even if spacious volume was allotted, depending on the shape and the distribution of usable volume. Raymer [214] rened the classical density-based volumetric design approach and developed the Net Design Volume (NDV) method. NDV is dened as the internal volume of an aircraft fuselage, nacelles, and wings, less the volume dedicated to fuel, propulsion, payload, passengers, and crew. As a result, NDV more closely represents the un-spoken-for extra volume. Raymer showed that there is a solid relationship
The volume is usually measured as the dierence between the volume wrapped around by the OML and the volume of all internal diusers.
1
234
between NDV and aircraft weight via a regression analysis of seven modern ghter aircraft. This approach has a similarity to the generalized weight estimation method presented in 4.4 in the sense that both techniques directly estimate a few selected items and rely on empirical information for all un-spoken-for items. Integrating the NDV method with the authors AIASM is seen as a practical way to develop an automated environment that is capable of simultaneously assessing power, energy, and volume balance. 7.3.2 Uncertainty Modeling
The numerical example study presented in 6.2 demonstrates that an improvement in the distributions of a few signicant random parameters may result in considerable improvement in the reliability of associated probabilistic constraints, thereby inuencing the optimum solution. On the other hand, the results accentuate that distributions of the random parameters cannot be arbitrary, and the accuracy of the assumed distributions of random parameters are of signicance. Therefore, such assumptions must be carefully made2 . In general, uncertainty that is most often encountered in engineering can be broadly classied as follows [217, 171]3 :
Aleatory uncertainty
Aleatory uncertainty is due to variability, which is an intrinsic property of natural phenomena or processes, thus it also referred to as uncontrollable uncertainty, irreducible uncertainty, inherent uncertainty, stochastic uncertainty, and
This kind of uncertainty modeling would heavily depend on the amount of available and applicable data for constructing input probability distributions. Oberkampf et al. [217] categorize the types of uncertainty sources into three groups: 1) Strong Statistical Information, 2) Sparse Statistical Information, and 3) Intervals. If uncertainty sources are associated with the improvements in technology, then the methodical logic, which models uncertainty based on the TRL level of associated technology [218] proposed by Kirby, is recommended. 3 In some of the literature, the concept of error refers to numerical uncertainty. In the context of reliability assessment, however, error diers from uncertainty in that it is a recognizable deciency that occurs in any phase or activity of modeling and simulation but that is not due to a lack of knowledge [219, 220].
2
235
variability.
Epistemic uncertainty
Epistemic uncertainty is a potential deciency in selecting the best action in a decision (the action with the highest probability of resulting in the most desirable outcome) due to lack of knowledge or incomplete information, it is thereby also referred to as controllable uncertainty, reducible uncertainty, subjective uncertainty, model form uncertainty, or simply uncertainty. Traditionally, both types of uncertainty have been attempted to be quantied in terms of probability theory. Such an ascendancy has been in question for sometime now as several other mathematical theories have demonstrated their capabilities of characterizing situations under uncertainty [220]. Some of the more popular quantitative theories include the theory of fuzzy sets [221], the Dempster-Shafer theory [222, 223], and the theory of upper and lower previsions [224]. Many of these new representations of uncertainty are able to more accurately represent epistemic uncertainty than is possible with traditional probability theory [225]. In reality, both aleatory and epistemic uncertainty can be present. Recently several researchers have attempted to develop RBDO methods that are able simultaneously to handle both types of uncertainty in engineering problems. Youn and Wang proposed [226] an integration of the Bayesian approach with the RBDO method, and Agarwal et al. [225] attempted to combine the Dempster-Shafer theory, also known as the evidence theory, with an RBDO method. Since an aircraft sizing problem is also very likely to involve both aleatory and epidemic uncertainty, it would thus be worthwhile to investigate the feasibility of adapting such techniques to PASM. 7.3.3 Multi-Stage RBDO with Recourse
Although proving their usefulness for numerous engineering applications, state-ofthe-art CCP and RBDO methods possess a weakness to be used for a large, complex 236
system-level problems such as designing an aircraft. As mentioned previously, aircraft design is an iterative process, progressively moving the focus from the macro to the micro, or from conceptual to detail design. As such, designers do not determine all design variables at one time. Instead, they rst prioritize the variables according to some criteria, and then progressively make decisions on how to mature the product design. In fact, in the conceptual design phase, design engineers are only considering a handful of design variables. All other variables are to be determined by further design studies, analyses, and tests. Through such decisions, the degree of design freedom gradually decreases as illustrated in Figure 100. Along with increased design maturity, the level of uncertainty decreases as the level of knowledge increases via the applications of high delity analyses such as Computational Fluid Dynamics (CFD), the Finite Element Method (FEM), and 6DoF simulations, and experiments such as wind tunnel tests, structural tests, and subsystem tests. Such increased design maturity and accrued observations of unforeseen randomness may reveal some constraint violations. Under such circumstances, designers seek to x the problem in a way that imparts minimal impact on the overall design to prevent schedule and cost overruns. The reality is that most designers would almost always try to nd an opportunity to resolve the design issues with the currently possessed design freedom, rather than start over from the beginning. However, this thesis work, as well as traditional RBDO approaches, does not reect the reality of design progression, which motivates designers to correct their previous decisions and restore the balance between cost and reliability at any phase of the design. In such a sense, a solution obtained by an RBDO method is prone to be overly conservative, even if the assumptions regarding the uncertainty sources are accurate. This issue has not attracted much attention from previous RBDO researchers. One possible reason for it may be that most RBDO research has focused on structural, component-level design problems that are most appropriate in a detail design phase,
237
Point Performance Requirements
Aerodynamics
Constraints Analysis
Weight Estimation
Propulsion
Mission Analysis
Weight Notional Concept
Mission Performance Requirements A/C Scaling Factor and Subsystem Weight
Volume Analysis
Volume
Figure 99: Comprehensive sizing method
Knowledge Design Freedom 100 %
Cost Committed
Actual Cost Expenditure
Uncertainty 0%
Requirements Definition
Conceptual Design
Preliminary Design
Detail Design
Manufacturing
Figure 100: Distribution of knowledge, cost committed, and freedom in the design cycle [227]
238
when follow-on, corrective design eorts are not much expected. However, the issue may be of importance for a probabilistic optimization problem conducted in the early phases of complex systems design, including an aircraft sizing problem. Therefore, further research on this topic seems important to enhance the realism of the method. A new RBDO method, named MSRBDO (Multi-Stage Reliability-Based Design Optimization), that addresses the above issue has been conceptualized in the course of the development of PASM. The method provides a mathematical model that represents sequential decision processes, in which the decision variables are determined progressively based on increased knowledge as well as remaining uncertainty. The formulation involves the following assumptions: 1. Designers have accumulated some knowledge, which also means cumulative reduction in uncertainty has been achieved from the previous work up to present. 2. Designers hold a certain degree of design freedom that can be used to correct the decision that was previously made. 3. Designers utilize the information obtained from previous work to make follow-on decisions for the remaining portion of the design space. The MSRBDO method only determines a partial list of design variables in the rst stage by simulating the whole process upon the above assumptions. The other design variables will be determined in follow-on stages. The designers choice of current decision variables still aects future decisions, thus aecting the objective function value and the probability of meeting constraints also. The simplest case of the MSRBDO formulation is a two-stage problem, in which a group of design variables, denoted as xs1 , is determined at the rst stage. The remaining design variables are to be determined at the second stage, when a fraction of random parameters, denoted as s1 , are observed, but the rest of random parameters, denoted as s2 , are yet uncertain. Then, a two-stage MSRBDO problem can be 239
expressed as min E[f (xs1 , x , s1 , s2 )] s2
xs1
s.t. P[gi (xs1 , x , s1 , s2 ) 0] i s2 where x (xs1 , s1 ) solves s2 if P[g (x , x , , ) 0 | = ] (s2) i s1 s1 s1 s2 s1 s2 i minxs2 f (xs1 , xs2 ) (s2) s.t. P[gi (xs1 , x , s1 , s2 ) 0 | s1 = s1 ] i s2 else maxxs2 P[gi (xs1 , x , s1 , s2 ) 0 | s1 = s1 ] s2
(169)
Figure 101 compares four dierent probabilistic approaches in view of the degree of uncertainty, with which a decision is made, and the degree of design freedom to which the decision commits. DSS simulates the optimum solutions obtainable with all possible random realizations. The decision on the design will be made after all the random parameters have been observed. In a two-stage stochastic programming, decisions are made for the design variables ahead of the random events, but a correction can be made afterward. Similarly, using RBDO and CCP, decisions on all the design variables are made ahead of the random events, but a possibility of later corrections is not permitted. In contrast to the classical RBDO approach, MSRBDO allows a follow-on corrective decision based on additional information obtained from the observations of a partial realization of random parameters. Unlike a two-stage stochastic programming, the decision at the second stage must still be made under (remaining) uncertainty. In order to demonstrate the fundamental concept of the MSRBDO approach, the numeric example presented in 5.2 is re-investigated. Assuming that the design space includes an additional design variable, denoted as z, that represents design freedom, which the designer can retain until the second stage. The objective function and the 240
Design Freedom Uncertainty
100 % 100 %
0% Stage 1
Stage 2
Stage 3
0% Stage 1
Stage 2
Stage 3
DSS
Multi-Stage Stochastic Optimization
100 %
100 %
0% Stage 1
Stage 2
Stage 3
0% Stage 1
Stage 2
Stage 3
CCP/RBDO
Multi-Stage RBDO
Figure 101: Comparison of nondeterministic approaches
241
constraint function are given as f = x + 4y + z and 1 x + 2 y + z 2, respectively. The following scenario is assumed for this example study:
Designer must determine the values of x and y at the rst stage. Designer can determine the variable z after 2 is observed at the second stage.
Variable z is bound to |z| 1.
Designer determines z such that the value of the objective function is mini-
mized while meeting the second-stage target probability, (2nd) = 0.95, of the probabilistic constraint. With the above assumption, the given problem can be formalized as an MSRBDO programming as follows: min E[x + 4y + z (x, y, 2 )]
x,y
s.t. P[1 x + 2 y + z (x, y, 2 ) 2] 0.9, x 0, y 0 where 1 N (1, 12 ), 2 N (3, 12 ), and z (x, y, 2 ) solves if P[ x + y + z 2 | = ] (2nd) 1 2 1 1 min x + 4y + z z s.t. P[1 x + 2 y + z 2 | 1 = 1 ] (2nd) , |z| 1 else max P[ x + y + z 2 | ] z 1 2 1 s.t. |z| 1 The classical RBDO approach yields the following formulation: min f = x + 4y + z
x,y,z
(170)
s.t. P[1 x + 2 y + z 2] 0.9, x 0, y 0 where 1 N (1, 12 ) and 2 N (3, 12 ) 242
(171)
Table 20: Comparison of the results from RBDO and MSRBDO x y z f , E[f ] RBDO 0.229 0.487 0.999 3.176 MSRBDO 0.087 0.650 2.689
The optimum solutions of the MSRBDO approach and the ordinary RBDO approach for the given problem are compared in Table 20. In this particular example, the objective function of the RBDO approach is deterministically dened, while that of the MSRBDO approach is given as a random response. Thus, for the MSRBDO approach, a mean value is listed for the objective function value in the table. The solution from the MSRBDO method is expected to yield a lower value of the objective function. The RBDO approach determines the optimum value of z at the rst stage. In the case of the MSRBDO formulation, z is classied as a second-stage variable, which will be determined by solving a sub-optimization problem upon observation of the random parameter 1 . In other words, z , the solution of the second stage optimization problem, is essentially a function of 2 and plays as a random parameter in the rst stage. The distributions of z and f are depicted in Figure 102. In this particular example, an increase in z raises both the feasibility of the constraint and the value of the objective function, which means a higher value of z yields more design margin with a higher penalty on the objective function. The MSRBDO approach takes advantage of the retained design freedom: z is determined upon the observation of 2 . Another interesting quantity is the probability of meeting the target feasibility (0.95) of the second stage, which can be computed by separate MCSs. The solutions from the RBDO and MSRBDO approaches are expected to meet the target feasibility of the second stage by 0.85 and 0.94, respectively. In an actual engineering
243
Figure 102: Distributions of z and f from the application of the MSRBDO
244
development program, this quantity can be used as an FoM, with which, the program director can decide whether the product design can successfully proceed to the next development stage or not. When the diagram of the MSRBDO in Figure 101 is extended to a multi-stage problem, it will appear as shown in Figure 103. The two dotted lines represent the
Discretized Design Freedom Actual Design Freedom
100 %
Discretized Uncertainty Actual Uncertainty
0%
Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Figure 103: The MSRBDO simulates a decision making process of a complex system design problem
degree of uncertainty and design freedom, which continues to diminish after every stage, where a decision is made based on increased information as well as remaining uncertainty. In an actual aircraft development program, the design evolution is a continuous process, where the degree of uncertainty and design freedom gradually diminishes as depicted by the solid lines in Figure 103, which were originally shown in Figure 100. In this sense, the MSRBDO imitates the decision making process of a complex system design problem, but consolidates the continuous process into a number of discrete moments in time. Although the numerical example shows the potential of the MSRBDO approach,
245
much further research must be performed to assess its eectiveness in practical engineering problems. It is conjectured that one of the foreseeable challenges would be a prohibitive level of computational cost. The MSRBDO determines the design variables of the rst stage by simulating all the decisions that will be made upon the observations of random parameters at every subsequent stage. Thus, the computational eort is projected to a near exponential rate as the number of stages grows.
7.4
Concluding Remarks
Considering the readiness of revolutionary technologies, infrastructure, and economics, the dominance of IC engines consuming conventional aviation fuels will be likely to persist for the next couple of decades. However, this should not lead to the downplaying of the signicance and urgency of the search for alternative-energy ight. Worldwide petroleum supplies will eventually be depleted. If no viable alternative is orchestrated on a timely basis, the economic and environmental ramications could be appalling. Achieving alternative-energy ight is only a part of the odyssey to a post-oil economy that we as world citizens must complete within this century. Fortunately, the arduous journey has already begun, although the destination is yet to be determined. The expedition must be carefully guided by strategic plans that are premeditated by the collective eorts of all societies. It is this authors hope that the work contained in this dissertation could contribute towards a project of such importance and magnitude.
246
APPENDIX A
CONSTRAINT EQUATIONS
A constraint equation for a particular maneuver of an aircraft can be derived from the master equations, given as Eq. (11) and Eq. (52) of the Mattinglys method and AIASM, respectively. Such tailored constraint equations of Mattinglys method and AIASM are listed below. Most of the equations, except for take-o and landing, can be easily derived by simply applying the ight conditions of the maneuver.
A.1
Constant Altitude/Speed Cruise
The condition of constant altitude and speed yields dh/dt = 0, dV /dt = 0, and n = 1. Cruise altitude, h, and cruise speed, V , are given. Under these conditions, Mattinglys Equation TSL WTO = K1 WTO q S CDo + CDR
WTO q S
(172)
+ K2 +
AIASMs Equation
i Pref
WTO
V = +i K1 i
i
WTO q S
+ K2 +
CDo + CDR
WTO q S
(173)
A.2
Constant Speed Climb
The condition of constant speed climb yields dV /dt = 0. Assuming L W , n 1. The rate of climb, dh/dt, and the ight speed, V , are given. Under these conditions, Mattinglys Equation
247
TSL WTO
= K1
WTO q S
+ K2 +
CDo + CDR
WTO q S
1 dh + V dt
(174)
AIASMs Equation
i Pref
WTO
V = +i K1 i
i
WTO q S
+ K2 +
CDo + CDR
WTO q S
1 dh + V dt
(175)
A.3
Horizontal Acceleration
By denition, dh/dt = 0 and n = 1. The requirement is typically described in terms of values of altitude, h, the initial velocity, Vinitial , the nal velocity, Vf inal , and the allowable time for the acceleration, tallowable . Mattinglys Equation TSL WTO = K1 WTO q S CDo + CDR
WTO q S
+ K2 +
1 dV + go dt
(176)
AIASMs Equation i i Pref V K1 = +i WTO i WTO q S CDo + CDR
WTO q S
+ K2 +
1 dV + go dt
(177)
A.4
Sustained Turn
Sustained turn performance is the aircrafts ability to make a turn maintaining constant altitude and ight speed for an extended period of time, which leads to dh/dt = 0 and dV /dt = 0. Under these conditions, Eq. (11) and Eq. (52) can be rewritten as follows: Mattinglys Equation TSL WTO = K1 n2 WTO q S 248 CDo + CDR
WTO q S
(178)
+ K2 n +
L = nW
W V2 W V = g o RC go
Bank angle
RC
W
Figure 104: Forces on aircraft in a sustained level banked turn [100]
AIASMs Equation i i Pref V = +i K1 n2 WTO i WTO q S CDo + CDR
WTO q S
(179)
+ K2 n +
Sustained turn performance is also measured in terms of turn rate () and turn radius (Rc ) rather than load factor (n), (e.g. maximum sustained turn rate, minimum sustained turn radius). From the Pythagorean Theorem, n in Figure 104 is related to and Rc for a given value of V as follows: n= 1+ V go V2 g o Rc
2
(180)
2
n=
1+
(181)
A.5
Service Ceiling
The service ceiling is the density altitude where the ying at the best rate of climb airspeed for that altitude, and with all engines operating and producing maximum continuous power, will produce a certain climb rate, usually 100 feet per minute of climb, which leads to dV /dt = 0 and n = 1. For given h and dh/dt > 0, and CL , Eq. (11) and Eq. (52) become 249
Mattinglys Equation TSL = WTO AIASMs Equation Pref i V = +i WTO i
i
K1 CL + K2 +
CDo + CDR 1 dh + CL V dt
(182)
K1 CL + K2 +
1 dh CDo + CDR + CL V dt
(183)
A.6
Take-o Distance
In general, an aircraft takes-o in the three steps: acceleration, rotation, and transition. The take-o eld length is the total distance to complete the all steps. A.6.1 Take-o Ground Roll
Given that dh/dt = 0, Mattinglys Equation When TSL (D + R), Eq. (11) becomes TSL dV dV = = WTO go dt go ds V which can be rearranged to yield ds = WTO V dV go TSL (185) (184)
Integrating the equation from static where s = 0 and V = 0 to take-o, where s = sG and V = VTO , with the use of representative take-o values of and . sG =
2 WTO VTO TSL 2go
(186)
The take-o velocity (VTO ) is usually dened as the stall speed multiplied by a constant kTO , which is greater than one and specied by regulations or military standards. Vstall is the minimum speed at which the airplane ies at CLmax , then
2 2 2 kTO WTO VTO 2 V = kTO stall = 2 2 CLmax S
(187)
250
Finally,
2 TSL 2 kTO WTO = WTO sG go CLmax S
(188)
AIASMs Equation Pref V dV V dV = +i = +i ds WTO go dt go V which can be rearranged to yield ds = or sG = Assuming that
+ i i
(189)
+ i
WTO
i g P i o ref
V 2 dV
(190)
+ i
i
3 WTO VTO i Pref 3go
(191)
is constant, the equation leads to
2 Pref 2 2 i kTO = WTO 3 + i i sG go i
WTO S
3 2
(192)
A.6.2
Total Takeo Distance
The total takeo distance consists of four parts: (1) the ground-roll distance (sG ), (2) the rotation distance (sR ), (3) the transition distance (sTR ), and (4) the climb-out distance over an obstacle (sCL ). Mattinglys Equation (1) The ground-roll distance (sG ) A simplied version of the equation regarding the ground-roll distance is given as Eq. (188). A more accurate equation can be derived by accounting for the impact of D and R. Then, TSL = WTO TO 1 dV q S + TO + WTO go dt (193)
where TO = CD + CDR TO CL . Integrating Eq. (193) results in sG = (WTO /S) ln 1 TO / go TO 251 TSL TO WTO CLmax 2 kTO (194)
(2) The rotation distance (sR ) The rotation distance can be computed by sR = tR VTO = tR kTO 2/(CLmax )(WTO /S) (195)
where tR is a total aircraft rotation time based on experience (normally 3 seconds), (3) The transition distance (sTR ) The transition distance can be computed by sTR = Rc sin CL =
2 VTO sinCL go (n 1)
(196)
where CL is the angle of climb, which is given as sin CL =
2 Given that n = 0.8kTO ,
1 dh T D = V dt W
(197)
sTR =
2 kTO sin CL 2 WTO 2 go (0.8kTO 1) CLmax S
(198)
(4) The climb-out distance over an obstacle (sCL ) There are two dierent cases, depending on hobs and hTR , each of which is depicted in Figures 105 and 106, respectively. If hobs > hTR , hobs hTR tan CL
sCL =
(199)
hTR =
2 k 2 (1 cos CL ) 2 WTO VTO (1 cos CL ) = TO 2 2 go (0.8kTO 1) go (0.8kTO 1) CLmax S
(200)
Plugging Eq. (194), (195), (198), (199), and (200) into sTO = sG + sR + sTR + sCL , results in a quadratic equation with respect to (WTO /S)1/2 as follows: a WTO S b WTO S 252
1/2
+c=0
(201)
Transition
VCL Climb
CL
Rotation RC V=0 Ground Roll sG sTO VTO sR
VTO hTR sTR sCL hobs
Figure 105: Takeo terminology (hTR < hobs ) [100]
VCL Transition Climb VTO
Rotation V=0 Ground Roll sG sTO VTO sR
RC
CL
hobs hTR
sobs
Figure 106: Takeo terminology (hTR > hobs ) [100]
253
where a= TSL CLmax ln 1 TO / TO 2 go TO WTO kTO 2 2 kTO sin CL 2 kTO (1 cos CL ) 2 1 + 2 2 go (0.8kTO 1) CLmax go (0.8kTO 1) CLmax tan CL (obstacles considered and hobs < hTR ) = ln 1 TO / go TO TSL TO WTO CLmax 2 kTO (202)
(no obstacles considered) b =tR kTO c =STO 2/(CLmax ) hobs tan CL
Therefore, for a given TSL , WTO /S can be obtained from Eq. (201) 2 b + b2 4ac WTO = S 2a
(203)
If hobs < hTR , sTO is given as the summation of sG , sR , and sobs as shown in Figure 106. Note that unlike the previous case sTR is not included in sTO . From Figure 106, sobs is given as sobs = Rc sin obs = where obs = cos1 1 hobs Rc (205)
2 VTO sin obs 2 go (0.8kTO 1)
(204)
Even if hobs < hTR , the same quadratic equation per Eq. (201) as used for the case of hobs > hTR can be used for computing WTO /S for a given TSL , but the coecients are now modied as follows: a= ln 1 TO / go TO 2 kTO sin obs 2 + 2 go (0.8kTO 1) CLmax 2/(CLmax ) TSL TO WTO CLmax 2 kTO (206)
b = tR kTO c = STO
254
In fact, Eq. (204) and (205) cannot be solved without knowing WTO /S, because obs , VTO and Rc are functions of WTO /S. Mattingly et al. did not address this issue in Ref. [100]. The present study employs an interactive scheme to nd the quantities that are interdependent upon one another. First, a value for obs is assumed. Now, wing loading can be computed by Eq. (201) and Eq. (206). Once WTO /S is obtained, VTO and Rc can be computed by Eq. (187) and Eq. (204), respectively. Subsequently, obs can be updated by Eq. (205). These steps are repeated until obs converges. AIASMs Equation Considering
TSL WTO
=
p (V ) Pref V WTO
and dt = ds/V ,
V 2 2
(V ) Pref = V WTO
+ i
TO
S V dV + TO + WTO go ds
V dV go
(207)
ds =
(V ) Pref V WTO VTO
+ i
TO V 2
2
S WTO V g
(208) TO (209) TO
SG
sG =
0
ds =
0
dV
TO V 2 S 2 WTO
+ i
(V ) Pref V WTO
When the reference power, Pref is given as the shaft power, then + i is simply the eciency of a propeller P , which varies with V . When VTO be assumed to linearly vary with V as follows: + i (V ) = P (VTO )V (210) V (Pmax ), VTO can
where P (VTO ) is the propeller eciency at take-o speed (VTO ). The take-o ground roll is given as (WTO /S) ln 1 TO / sG = go TO i P (VTO ) Pref TO i WTO
i
CLmax 2 kTO
(211)
The other distances, sR , sTR , sCL , and sobs can be computed by the same equations as Mattinglys method.
255
A.7
Landing Distance
The total landing distance (sL ) consists of three parts: (1) the distance to clear an obstacle of given height (sA ), (2) a free roll traversed before the brakes are fully applied (sFR ), and (3) braking roll (sB ) as illustrated in Figure 107.
Vobs
Approach
hobs
VTD
Free Roll
VTD sFR sL
Braking
V=0
sA
sB
Figure 107: Landing terminology [100]
(1) Distance to clear an obstacle of given height (sA ) sA = 2 go (CD + CDR ) WTO S
2 2 kobs kTD 2 2 kobs + kTD
+
CLmax 2hobs 2 2 (CD + CDR ) (kobs + kTD )
(212)
where hobs is the height of the obstacle and the velocity at the obstacle is Vobs = kobs VSTALL (2) Free roll traversed before the brakes are fully applied (sF R ) sFR = tFR VTD = tFR kTD (3) Braking roll (sB ) TSL = WTO where L = CD + CDR B CL . sG = (WTO /S) ln 1 + L / go L 256 B () TSL WTO CLmax 2 kTD (216) L 1 dV q S + B + WTO go dt (215) {2/(CLmax )}(WTO /S) (214) (213)
Then, the constraint curve of TSL /WTO and WTO /S for landing distance performance can be obtained in a similar way that for take-o distance performance is derived. If thrust reversers are not equipped in the aircraft, that is = 0, then landing distance no longer depends on thrust, yielding a second order algebraic equation for WTO /S, which is applicable in both thrust-based and power-based approaches. The equation was used for developing landing distance constraints in the GA study presented in 6.1.
257
APPENDIX B
WEIGHT FRACTION EQUATIONS
B.1
Mattinglys Equation
The ratio of the nal to the initial weight for a mission leg can be computed as follows:
Constant Speed Climb
exp
Horizontal Acceleration
ct h V 1u
(217)
exp
Climb and Acceleration
ct (V 2 /2g) V 1u
(218)
exp
ct (h + V 2 /2g) V 1u
(219)
Constant Altitude/Speed Cruise
exp
Constant Altitude/Speed Turn
ct V
D L
s
(220)
exp ct n
Loiter
D L
2N V g n2 1 D L
(221)
exp ct n
Warm-up
t
(222)
1 ct
Take-o Rotation
TSL WTO TSL WTO
t
(223)
1 ct
tR
(224)
258
B.2
AIASMs Equation
As depicted in Figure 108, the energy weight fraction for the sth mission leg ultimately depends on (s1) , (s) and CE (NE for a non-consumable energy source). The expressions for (s1) and CE are provided as Eq. (80), Eq. (81), and Eq. (71).
(s) (s) (s)
jGl
uTGl uT
Figure 108: Equations associated with generalized mission analysis
In addition, general expressions for (s) are given as Eq. (74) for positive excess power and Eq. (77) or Eq. (78) for zero excess power. The aerodynamic coecients and ight conditions of the equations are assumed to be constant at some representative values so that the integration can be accomplished explicitly. The resulting (s) for dierent cases are listed as follows:
Constant Speed Climb
h 1u 259
(225)
Horizontal Acceleration
(V 2 /2g) 1u
Climb and Acceleration
(226)
(h + V 2 /2g) 1u
Constant altitude/speed cruise
(227)
D L
Constant altitude/speed turn
s
(228)
D L
Loiter
2N V g n2 1
(229)
D L
t
(230)
260
APPENDIX C
WEIGHT-SPECIFIC PARAMETERS AS DECISION VARIABLES
If power to weight ratio ( Wref ), energy weight fraction( TO
P
Wenergy ), WTO
and wing loading
TO ( WS ) are chosen as the design variables in the replacement of power, fuel, and thrust,
the distributions of the random variables are not given in a explicit form. Each of the random variables can be decomposed into a static decision variable and a random distribution by the similar process as presented in 5.3.2. The random response of take-o gross weight can be expressed as follows: WTO = WTO |v WTO (231)
where WTO |v is the take-o gross weight calculated with the deterministic values of the random variables, v, and WTO is the actual take-o gross weight distribution normalized to WTO |v . By combining Eq. (132) and Eq. (231), the rst set of the constraints in Eq. (130) can be rewritten with respect to the deterministic values, v, as follows: P Pref WTO |v i WTO |v , pi , pi |v 0 (i = 1, , np ) WTO WTO S (232)
where np is the number of constraints. Similarly, by combining Eq. (133) and Eq. (231) the second set of the constraints in Eq. (130) can be expressed as follows: Wenergy Wenergy P Pref WTO |v j |v , WTO |v , pj , pj WTO WTO WTO WTO S 0 (j = 1, , nm ) (233)
Finally, a PASM problem whose design variables are weight-specic variables can be
261
Pref |v WTO
Solution Space
Wenergy | v WTO
g 3
g 1
g 2
W TO | v S
Figure 109: Design space of probabilistic aircraft sizing
formulated as follows: min E[f ]
x
s.t.P [gi (x, )|v 0] gi where x = Pref WTO Wf |v , |v , |v WTO S WTO
(234)
The probabilistic constraint, gi (x, )|v 0 is given either Eq. (232) or Eq. (233). A notional design space of a PASM problem established as Eq. (234) is illustrated in Figure 109. The three-dimensional design space includes two power-balance constraints and the single energy-balance constraint for simplicitys purpose.
262
APPENDIX D
ELECTROLYZER MODEL AND ROUND TRIP EFFICIENCY
The RFC system considered in the SPHALE aircraft consists of an electrolyzer component and a fuel cell component as illustrated in Figure 110. Excess energy from PV cells during the daytime provides the electricity to decompose water into hydrogen and oxygen by the process of electrolysis in the electrolyzer. The stored energy is discharged via a redox reaction process of PEM fuel cells at night time. The electrolyzer begins to electrolyze water to produce hydrogen above the calculated theoretical open circuit potential, 1.23V at 25C. The rate of hydrogen ow can be determined by electrochemistry as follows [228]: mH2 = Incell MH2 2F (235)
where I is the current (A), ncell is the number of cells in the electrolyzer, MH2 is the molar mass of hydrogen (2g/mol), F is Faradays constant (96,485C/mol), and m is a mass ow rate in grams per second of hydrogen. Therefore, hydrogen production is proportional to the current applied to the load. The voltage applied to the electrolyzer can be modeled using Vstack = ncells 1.23 + RT ln 2F I I0 + IR0 (236)
where 1.23V is the theoretical open circuit voltage at 25C; R is the gas constant 8.314; T is the temperature in Kelvin (298K); is the charge transfer coecient (approximately 0.325 from the experimental data); I is the current; I0 is the cathode exchange current (approximately 0.5mA); and R0 is the cell resistance (0.0107).
263
Electrolyzer
Charge Oxygen
Water Hydrogen Fuel Cells
Figure 110: Illustration of a regenerative fuel cell system
The electrolyzer eciency is given by the simple relationship electrolyzer = 1.48ncells Vstack (237)
Since the electrolyzer eciency is aected by voltage, the number of cells, and operating temperature, an accurate estimation can be obtained through a comprehensive simulation with detailed models that are able to capture the voltage and the current behavior and environmental changes such as pressure and temperature. However, a simple model that is acceptable for the conceptual design phase can be constructed through a regression analysis. Figure 111 depicts the current and hydrogen mass ow versus the voltage for two dierent numbers (four and ten) of cell stacks. The hydrogen mass ow versus power for the two dierent stacks is plotted in Figure 112, which shows a strong linear relationship between the hydrogen mass ow 264
Discharge
25 20 Current (A) 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 Voltage (V) 4 cell stack 10 cell stack
Hydrogen Mass Flow (gram/sec)
0.0025 0.002 0.0015 0.001 0.0005 0 0 2 4 6 8 10 12 14 16 18 20 Voltage (V) 4 cell stack 10 cell stack
Figure 111: Performance of the electrolyzer model
265
9 8 Hydrogen Mass Flow (gram/hr) 7 6 5 4 3 2 1 0 0 100 200 P (Watt) 300 400 y = 0.0207x 2 R = 0.9959 4 cell stack 10 cell stack Linear (4 cell stack) Linear (10 cell stack) y = 0.0207x 2 R = 0.9959
Figure 112: Hydrogen mass ow versus power
and input power. Regression analyses for the two cases yield an identical linear equation as follows1 : mH2 = 0.0207P (gram/hr) The total discharge energy of the fuel cell system is f c H2 g mH2 dt = f c H2 g 0.0207 P dt
TD
(238)
(239)
TD
The total input energy from PV cells to the fuel cell system is P dt
TD
(240)
Therefore, the roundtrip eciency is given as follows: rt = 0.0207f c H2 g = 0.4895
1
(241)
Note that this equation holds only for the specic temperature 25C. The numeric constant must be revised by another regressions at dierent operating temperatures.
266
which is comparable with 0.42 estimated by Jakupca et al. [211].
267
APPENDIX E
SUPPLEMENTAL CHARTS
This appendix includes additional charts developed in the course of the example studies presented in Chapter VI.
Table 21: Inputs of the constraint analyses of the GA study
TOFL
Climb
8,000 0.99 0.185 150 490.2
Cruise
10,000 0.97 0.185 483.0
Approach
0 0.96 0.09 516.7 2.5
LDFL
0 0.96 0.09 516.7 2.0 1.15
kTD (Touch Down Speed Ratio to Stall Speed) kCL (Climb Speed Ratio to Stall Speed) or kAPP (Approach Speed Ratio to Stall Speed) (Friction Coefficient) tROT (Rotation Time of Take-Off, sec) tFR (Free Roll Time of Landing, sec) hobs (Obstacles Height, ft) STO (Take-off Field Length) SL (Landing Field Length) (Air Density, slug/ft3) (Temperature Ratio) (Density Ratio) (Pressure Ratio) a (Speed of Sound, ft/sec) V (Speed, ft/sec) V (Speed, knot) q (Dynamic Pressure) K1 (K' + K'') K2 ( -2K"(CLmin)2 ) CDo (CDmin +K''CLmin2)
Air Properties & Thrust Lapse Rate
Velocity & Dynamic Pressure
Drag Coefficients
CD (Drag Coefficient) CDR (Drag Coeeficient of Protuburances) CDRc (Drag Coeeficient of Drag Parachute) TO ( = CD+ CDR - TOCL)
Others
g (Gravity Constant, ft/sec2)
For Field Performance Only
User Inputs
PA (Pressure Altitude, ft) (Weight Fraction to TOGW) M (Mach Number) For Constant Speed Climb Only Climb Rate (ft/min) T (Tempreture, R) C (Take-off Speed Coefficient) k Lmax (Maximum LiftRatio to Stall Speed)
TO
0 1 0.1 554.7 2.0 1.2 1.2 0.05 3.0 35.0 1370.0 1.2000 0.002223 1.06940 0.9351 1.0000 1154.4 115 68 14.8 0.0565 0.0136 0.0281 0.1560 0.0694 0.1560 32.17
1.3
1.3 0.18 3.0 50.0 1500
(Installed Full Throttle Power Lapse)
1.1000 0.001868 0.94500 0.7860 0.7428 1085.2 201 119 37.7 0.0565 0.0136 0.0281
1.0000 0.001755 0.93125 0.7385 0.6877 1077.2 199 118 34.9 0.0565 0.0136 0.0281
1.0000 0.002386 0.99614 1.0039 1.0000 1114.1 96 57 11.0 0.0565 0.0136 0.0281
1.0000 0.002386 0.99614 1.0039 1.0000 1114.1 96 57 11.0 0.0565 0.0136 0.0281 0.1273 0.2178 0.0000 0.1273 32.17
268
q Weight Fraction 0.99911 0.99996 1.00000 0.99988 0.99982 0.99994 0.99868 0.99865 0.99865 0.99862 0.4 10.1 10.0 0.52 0 0 95.83 95.83 48.52 0 6338 0 2000 0 48.52 10.00 10.00 45.00 95.83 95.83 0.00 0.00 0.00 0 0 0 0 0 0 0 0.31 0.31 0.31 0.31 0.31 0.31 0.40 0.56 0.56 10.3 8.3 8.2 8.2 8.2 8.2 8.2 9.4 10.5 10.5 0.4 0.4 0.4 1 1 1 1 1 1 1 1 1 0.99855 0.99849 0.99837 0.99825 0.99809 0.99412 0.99016 0.98620 0.98226 0.97832 0.97438 0.97371 0.97330 0.99862 0.99855 0.99849 0.99837 0.99825 0.99809 0.99412 0.99016 0.98620 0.98226 0.97832 0.97438 0.97371 0.97330 0.97149 0.99997 0.99997 0.99997 0.99994 0.99993 0.99988 0.99988 0.99983 0.99602 0.99601 0.99601 0.99600 0.99599 0.99598 0.99931 0.99959 0.99814 ft. 0.0 2.4 9.6 10.6 11.4 11.3 11.4 11.6 11.8 11.8 11.7 11.5 11.1 22.3 37.4 37.4 37.4 37.4 37.4 37.4 28.7 20.4 20.4 2000 5000 8000 8000 0 8000 0 48.52 8000 0 48.52 8000 0 48.52 95.83 8000 0 48.52 95.83 7500 1445 1.34 2.02 6.7 6500 1000 0.97 1.00 9.3 1.05 5517.5 966 0.93 0.96 9.4 1.02 4759 553 0.53 0.55 9.4 1.00 10.2 4209 551 0.53 0.54 9.5 0.99 10.2 0.4 3798.5 274 0.25 0.26 9.8 0.99 10.2 0.3 3533.5 264 0.25 0.25 9.8 1.01 10.2 0.3 3269 274 0.26 0.26 9.9 1.03 10.1 0.3 0.99871 0.99868 2862 546 0.51 0.51 10.1 1.03 10.1 0.3 0.99877 1795.5 1594 1.49 1.45 10.2 1.03 10.1 0.3 0.99895 0.99877 0.99871 500 1030 0.86 0.79 12.1 1.11 9.9 0.3 0.99907 0.99895 0 0 0.05 4.50 3 0.5 0.99907 0.99907 0 126 0.51 0.23 0 2 0.5 0.99911 0.99907 0 0 15 0 1.00000 0.99911 ft. min. mi. deg.
Alt.
ze
t
s
CL
L/D
u
i
f
s01
Warm up
s02
Take-o Accel.
s03
Take-o Rotation
s04
Climb
s05
Climb
s06
Climb
s07
Climb
s08
Climb
s09
Climb
s10
Climb
Table 22: Mission analysis of the electric GA
269
s11
Climb
s12
Climb
s13
Climb
s14
Climb
s11
Cruise
s12
Cruise
s12
Cruise
s13
Cruise
s13
Cruise
s13
Cruise
s14
Descent
s15
Loiter
s16
Loiter
70 Not meeting all 60 50 Daily Energy Balance 40 30 20 10 0 -4 -2 -10 Nocturnal Energy Balance 0 2 4 6 8 Meeting all
70 Not meeting all 60 50 Daily Energy Balance 40 30 20 10 0 -1.0 -0.5 -10 Power Balance 0.0 0.5 1.0 1.5 2.0 Meeting all
Figure 113: Illustration of samples of an MCS (P2)
270
Figure 114: PDF of total power required at loiter (W)
Figure 115: PDF of angle of attack at loiter
271
Figure 116: PDF of loiter velocity (m/sec)
Figure 117: PDF of loiter eciency
272
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VITA
Taewoo Nam was born in Busan, South Korea in 1969. He attended Seoul National University and graduated with his Bachelor of Science in 1993. He remained at Seoul National University to pursue his Master of Science in Aerospace Engineering. After receiving his MS in 1995, he started working for Samsung Aerospace Industries as a structural analysis engineer. He was assigned to a research project, Experimental Analysis of Bird Strike Characteristics, which was funded by the Korean government. In 1996, he participated in the development of the Korea Air Force T-50 and A-50 Advanced Jet Trainer as a conguration design engineer. He continued to work for the program after Korea Aerospace Industries Ltd. was established in 1999 with the consolidation of Samsung Aerospace, Daewoo Heavy Industries, and Hyundai Space and Aircraft Company. In 2002, he joined the Aerospace Systems Design Laboratory at the Georgia Institute of Technology to pursue his doctorate. He has been an active member of a subtask group of the University Research, Engineering and Technology Institutes (URETI) program. His group is responsible for formulation and development of physics-based methods for analysis and design of revolutionary concepts, architectures, and technologies. His area of interest is development of sizing and synthesis methods for revolutionary concepts that use alternative energy sources including zero-emissions aircraft and regenerative solar-powered aircraft. He has also worked on intelligent design margin allocation for risk mitigation. He currently resides in Alpharetta, Georgia with his wife Joung Yun, daughter Jihye, and son Jiung.
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