Morphing Wing Fighter Aircraft Synthesis/Design Optimization

Transcription

1 Morphing Wing Fighter Aircraft Synthesis/Design Optimization Kenneth Wayne Smith Jr. A thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science In Mechanical Engineering Dr. Michael von Spakovsky, Chair Dr. Walter O Brien Dr. Michael Philen Dr. David Moorhouse January 16, 2009 Blacksburg, Virginia Keywords: exergy, morphing wing, optimization, aircraft, decomposition Copyright 2009 by Kenneth Wayne Smith Jr. All other copyrighted items used by permission

2 Morphing Wing Fighter Aircraft Design/Synthesis and Optimization by Kenneth Wayne Smith Jr. Abstract This thesis presents results of the application of energy-based large-scale optimization of a two-subsystem (propulsion subsystem (PS) and airframe subsystem-aerodynamics (AFS-A)) air-to-air fighter (AAF) with two types of AFS-A models: a fixed-wing AFS-A and a morphingwing AFS-A. The AAF flies 19 mission segments of a supersonic fighter aircraft mission and the results of the study show that for very large structural weight penalties and fuel penalties applied to account for the morphing technology, the morphing-wing aircraft can significantly outperform a fixed-wing AAF counterpart in terms of fuel burned over the mission. The optimization drives the fixed-wing AAF wing-geometry design to be at its best flying the supersonic mission segment, while the morphing-wing AFS-A wing design is able to effectively adapt to different flight conditions, cruising at subsonic speeds much more efficiently than the fixed-wing AAF and, thus yielding significant fuel savings. Also presented in this thesis are partially optimized results of the application of a decomposition strategy for large-scale optimization applied to a nine-subsystem AAF consisting of a morphing-wing AFS-A, turbofan propulsion subsystem (PS), environmental controls subsystem (ECS), fuel loop subsystem (FLS), vapor compression/polyalphaolefin loop subsystem (VC/PAOS), electrical subsystem (ES), central hydraulics subsystem (CHS), oil loop subsystem (OLS), and flight controls subsystem (FCS). The decomposition strategy called Iterative Local-Global Optimization (ILGO) is incorporated into a new engineering aircraft simulation and optimization software called iscript which also incorporates the models developed as part of this thesis work for the nine-subsystem AAF. The AAF flies 21 mission segments of a supersonic fighter aircraft mission with a payload drop simulating a combat situation. The partially optimized results are extrapolated to a synthesis/design which is believed to be close to the system-level optimum using previously published results of the application of ILGO to a five-subsystem AAF to which the partially optimized results of the nine-subsystem AAF compare relatively well. In addition to the optimization results, a parametric study of the morphing AFS-A geometry is conducted. Three mission segments are studied: subsonic climb, subsonic cruise, ii

3 and supersonic cruise. Four wing geometry parameters are studied: leading-edge wing sweep angle, wing aspect ratio, wing thickness-to-chord ratio, and wing taper ratio. The partially optimized AAF is used as the baseline, and the values for these geometric parameters are increased or decreased up to 20% relative to an established baseline to see the effect, if any, on AAF fuel consumption for these mission segments. The only significant effects seen in any of the mission segments arise from changes in the leading-edge sweep angle and wing aspect ratio. The wing thickness-to-chord ratio shows some effect during the subsonic climb segment, but otherwise shows no effect along with the taper ratio in any of the three mission segments studied. It should be emphasized, however, that these changes are made about a point (i.e. synthesis/design), which is already optimal or nearly so. Thus, the conclusions drawn cannot be generalized to syntheses/designs, which may be far from optimal. Also note that the results upon which these conclusions are based may very likely highlight a weakness in the conceptual-level drag-buildup method used in this thesis work. Further optimization studies using this dragbuildup method may warrant setting the thickness-to-chord ratios and taper ratios rather than having them participate in the optimization as degrees of freedom (DOF). The final set of results is a parametric study conducted to highlight the correlation between the fuel consumption and the total exergy destruction in the AFS-A. The results for the subsonic cruise and supersonic cruise mission segments show that at least for the case when the AFS-A is optimized by itself for a fixed specific fuel consumption that there is a direct correlation between the fuel burned and total exergy destruction. However, as shown in earlier work where a three-subsystem AAF with AFS-A, PS, and ECS is optimized, this may not always be the case. Furthermore, based on the results presented in this thesis, there is a smoothing effect observed in the exergy response curves compared to the fuel-burned response curves to changes in AFS-A geometry. This indicates that the exergy destruction is slightly less sensitive to such changes. iii

4 Acknowledgements This masters thesis is the product of a significant amount of work by myself and my advisor, Dr. Michael von Spakovsky, as well as cooperation with an industry partner, TTC Technologies Inc., and the Air Force Research Laboratories, specifically Dr. David Moorhouse and Dr. Jose Camberos. The work was sponsored through a joint award of an SBIR Phase II project to Virginia Tech and TTC Technologies. I would firstly like to thank Dr. von Spakovsky for providing the opportunity to study this topic, his sponsorship, his mentorship, his guidance, and his passion for his beliefs, all of which I learned greatly from throughout my time as a graduate student at Virginia Tech. Secondly, I would like to thank Dr. Ken Alabi of TTC Technologies Inc. for helping me so much with learning their software and putting up with my seemingly numerous mistakes as the code progressed through the stages of development and integration. Thirdly, I would like to thank the members of the defense committee: Dr. Michael von Spakovsky, Dr. Walter O Brien, Dr. Michael Philen, and Dr. David Moorhouse for serving on my committee and providing feedback on my thesis draft. Fourthly, I would like to thank all my friends who helped me get through graduate school and pulled me away from my work when necessary (or sometimes more than necessary), those in Chi Alpha, you know who you are, and my new friends in Maryland who are seeing the aftermath of finishing this project. Fifthly, I would like to thank my family who has supported, encouraged, commiserated, and advised me throughout this project. To my parents, Kenneth Sr. and Danlynne, for the work ethic, endurance, and stubbornness to finish a project of this length that you imprinted in me, I thank you as well. But mostly, I would like to thank my Lord and Savior, Jesus Christ. For the strength, endurance, ability, opportunity, family, friends, direction, love and direction, Father, I thank You. Proverbs 16:9 iv

5 Table of Contents Abstract... ii Acknowledgements... iv Table of Contents...v List of Figures... IX List of Tables... XII Nomenclature... XV Chapter 1 Introduction Morphing-Wing Aircraft Aircraft System/Subsystem Synthesis/Design Modeling and Simulation Large-Scale Optimization and Mission Integration Decomposition for Large-Scale Optimization The Use of Exergy Analysis Thesis Objectives Chapter 2 Literature Review Benefits and Design Challenges for Morphing Aircraft Study of Morphing-wing Effectiveness in Fighter Aircraft Airframe Subsystem Aerodynamics (AFS-A) Propulsion Subsystem (PS) Most Important Results from Butt (2005) Decomposition Strategies for Large-scale Aircraft Synthesis/Design Optimization Effects on Aircraft Synthesis / Design of Different Objective Functions Exergy Methods for the Development of High Performance Vehicle Concepts Integrated Mission-Level Analysis and Optimization of High Performance Vehicle Concepts Chapter 3 Model Description and Synthesis/Design Problem Description Problem Definition Airframe Subsystem Lift and Drag Mission Analysis v

6 3.2.3 Weight Fraction Model Calculation of W TO Morphing-wing Considerations AFS-A Exergy Model Propulsion Subsystem PS Layout and Station Definitions PS Thermodynamic Model Thrust and Performance Calculations PS Exergy Model Environmental Controls Subsystem ECS Layout and Definitions ECS Thermodynamic Model ECS Exergy Model Vapor Compression / PAO Subsystem VC/PAOS Thermodynamic Model VC/PAOS Exergy Model Fuel Loop Subsystem FLS Thermodynamic Model Oil Loop Subsystem OLS Thermodynamic Model OLS Exergy model Central Hydraulic Subsystem CHS Thermodynamic Model CHS Exergy Model Electrical Subsystem ES Thermodynamic Model ES Exergy Model Flight Controls Subsystem FCS Thermodynamic Model FCS Exergy Model vi

7 Chapter 4 Large-scale System Synthesis/Design Optimization Problem Definition and Solution Approach AAF Aircraft System Synthesis/Design Optimization Problem System-Level Optimization Problem Definition Need for Decomposition Iterative Local-Global Optimization (ILGO) Approach Local-Global Optimization (LGO) ILGO Approach System-Level, Unit-Based Synthesis/Design Optimization Problem Definitions Subsystem Integration and Coupling Functions AFS-A System-Level, Unit-Based Synthesis/Design Optimization Problem Definition PS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition ECS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition VC/PAOS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition FLS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition OLS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition CHS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition ES System-Level, Unit-Based Synthesis/Design Optimization Problem Definition FCS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition Optimization Decision Variables and Variable Constraints Solution Approach iscript Scripting Language and Optimization Chapter 5 Results and Discussion Two-Subsystem Optimization Results Nine-Subsystem Results Preliminary Synthesis/Design Analysis vii

8 5.2.2 Projected Optimum and Comparison Parametric Study of the Morphing-Wing AFS-A Chapter 6 Conclusions/Recommendations References Appendix A Fan Performance Map Code viii

9 List of Figures Figure 1.1 Lockheed hunter-killer morphing aircraft concept (Bowman, Sanders, Weisshar, 2002) Figure 1.2 NASA Oblique-Wing Demonstrator, the NASA AD-1 (Dryden X-Press, 1979)... 3 Figure 2.1 Effect of morphing on the synthesis/design space of thrust to weight (T/W) vs wing loading (W/S) Figure 2.2 Mechanization of the mission adaptive wing (MAW) trailing edge Figure 2.3 Example of the benefit of mission adaptive wing (MAW) technology Figure 2.4 Mission Profile by segment or leg (Mattingly, Heiser, and Daley, 1987) Figure 2.5 Subsystems and subsystem coupling functions (Rancruel, 2002) Figure 2.6 Evolution of the gross take-off weight, fuel weight, AFS-A weight, and PS weight at different points of the iterative local-global optimization (ILGO) approach (Rancruel, 2003) Figure 2.7 Variation of vehicle specific thrust and exergy destruction rate with fan bypass ratio and turbine inlet temperature for a fixed compressor pressure ratio of 8 for the supersonic penetration mission segment (Periannan, 2005) Figure 2.8 Variation of vehicle specific thrust and exergy destruction rate with fan bypass ratio and compressor pressure ratio for a fixed turbine inlet temperature of 1700 K for the supersonic penetration mission segment (Periannan, 2005) Figure 2.9 Variation of vehicle specific fuel consumption and exergy destruction rate with fan bypass ratio and compressor pressure ratio for a fixed turbine inlet temperature of 1700 K for the supersonic penetration mission segment (Periannan, 2005) Figure 2.10 Optimum gross takeoff weight with and without AFS-A DOF for objectives 1, 2, and 3 (Periannan, von Spakovsky and Moorhouse, 2008)..34 Figure 2.11 Optimum fuel weight with and without AFS-A DOF for objectives 1, 2, and 3 (Periannan, von Spakovsky, and Moorhouse, 2008)..34 Figure 2.12 Hypersonic vehicle configuration (Markell, 2005) Figure 2.13 A physical representation of the forebody and inlet component of the hypersonic vehicle along with design and operational decision variables that govern the flow characteristics throughout the inlet (Markell, 2005) Figure 2.14 Propulsion subsystem components and airframe subsystem (Brewer, 2006) IX

10 Figure 2.15 Total scramjet vehicle mission (Brewer, 2006) Figure 3.1 Supersonic fighter aircraft mission from the RFP found in Mattingly, Heiser, and Pratt (2002) Figure 3.2 Free-body diagram of the aircraft (Rancruel, 2002) Figure 3.3 Engine system layout (Rancruel, 2002) Figure 3.4 Engine Station Definitions (Periannan, 2005) Figure 3.5 ECS layout and components (Muñoz and von Spakovsky, 2001) Figure 3.6 ECS station definitions (Rancruel, 2002) Figure 3.7 Geometric parameters of the offset-strip fin (Muñoz and von Spakovsky, 1999) Figure 3.8 VC/PAOS layout and station definitions (Rancruel, 2002) Figure 3.9 Schematic of the fuel loop subsystem (Rancruel, 2002) Figure 3.10 Oil loop subsystem schematic Figure 3.11 Notional central hydraulics subsystem layout (simplified) Figure 3.12 Notional electrical subsystem schematic (simplified) Figure 4.1 Physical decomposition of a 2-unit system Figure 4.2 Multi-level optimization resulting in a set of nested optimizations Figure 4.3 An example of three subsystems and their associated coupling functions Figure 4.4 Notional flow diagram of the application of the ILGO decomposition strategy to the two-unit system of Figure Figure 4.5 Aircraft subsystem interactions and coupling functions Figure 4.6 Diagram of optimization problem solution approach using ILGO Figure 5.1 Sensitivity analysis of morphing-wing effectiveness for different wing- and fuelweight penalties (Smith et al., 2007) Figure 5.2 Total exergy destruction plus fuel exergy loss for each of the nine subsystems after the first ILGO iteration Figure 5.3 Variation of the mission segment fuel burned with variations in aspect ratio and the sweep angle for mission segment 4 (subsonic climb at Mach from a 20,000 ft to 41,700 ft altitude) Figure 5.4 Variation of the mission segment fuel burned with variations in aspect ratio and the sweep angle for mission segment 5 (subsonic cruise at Mach at a 41,700 ft altitude) X

11 Figure 5.5 Variation of the mission segment fuel burned with variations in aspect ratio and the sweep angle for mission segment 17 (supersonic cruise at Mach 1.5 and 30,000 ft altitude). 155 Figure 5.6 Variation of the mission segment fuel burned with variations in thickness-to-chord ratio for mission segment 4 (subsonic climb at Mach from a 20,000 ft to 41,700 ft altitude) Figure 5.7 Variation of the mission segment exergy destruction with variations in aspect ratio and the sweep angle for mission segment 5, subsonic cruise at Mach at 41,700 ft altitude Figure 5.8 Variation of the mission segment exergy destruction with variations in aspect ratio and the sweep angle for mission segment 17, supersonic cruise at Mach 1.5 and 30,000 ft altitude XI

12 List of Tables Table 2.1 Mission Segment Definition and Description Table 2.2 AFS-A aerodynamics and model equations Table 2.3 Mission specifications (Rancruel, 2003) Table 2.4 Comparison between the optimum ATA and the aircraft proposed by Mattingly, Heiser, and Daley (1987), (Rancruel, 2003) Table 2.5 Comparison of the optimal combustor models (Markell, 2005) Table 2.6 Optimal decision variable values for the energy and exergy based optimizations of a scramjet engine with fixed thrust (Markell, 2005) Table 2.7 Optimal design decision variable values for the single segment optimizations (Markell, 2005) Table 2.8 Optimal operational decision variable values for the partial mission (Markell, 2005) Table 2.9 Optimal vehicle fuel mass flow rate comparison (Markell, 2005) Table 2.10 Mission segment details (Brewer, 2006) Table 2.11 Mission design and operational decision variables for the inlet, nozzle, combustor, and airframe (Brewer, 2006) Table 2.12 Samples of results populations for sparse and dense optimal solution spaces. Note that the very large numbers (i.e. E+15) represent infeasible solutions (Brewer, 2006) Table 2.13 Optimal objective function results (Brewer, 2006) Table 3.1 Air-to-air fighter (AAF) mission segments and details Table 3.2 Master flight equation and governing flight equations Table 3.3 Lift and drag equations for the AFS-A Table 3.4 Mission segment model equations Table 3.5 Weight fraction model equations Table 3.6 Main subsystem weight calculations and W T Table 3.7 AFS-A exergy destruction rate equations Table 3.8 Low-bypass turbofan engine station definitions Table 3.9 Diffuser and nozzle equations Table 3.10 Fan and high pressure compressor equations Table 3.11 Burner and afterburner calculations XII

13 Table 3.12 High and low pressure turbine equations Table 3.13 Turbine cooling mixer and exhaust mixer equations Table 3.14 Thrust and engine performance calculations Table 3.15 Inlet and nozzle drag and installed thrust equations Table 3.16 PS exergy model equations Table 3.17 Thermodynamic model of the ECS (Periannan, 2005) Table 3.18 Geometric and heat transfer models of the compact heat exchangers Table 3.19 ECS exergy destruction rate equations Table 3.20 VCPAOS model equations Table 3.21 Fuel loop subsystem thermodynamic model equations Table 3.22 OLS pump work equations Table 3.23 OLS heating load equations Table 3.24 OLS exergy destruction equations Table 3.25 Actuator flow estimation calculations Table 3.26 Central hydraulic subsystem heating load equations (Majumar, 2003) Table 3.27 CHS subsystem exergy destruction equations (Bejan, 1996) Table 3.28 Fighter aircraft power generation/empty weight estimate Table 3.29 Electrical subsystem generator work Table 3.30 ES heating load model equations Table 3.31 ES exergy destruction model equations Table 3.32 Flight controls subsystem weight equations (Raymer, 1999) Table 3.33 FCS actuator electrical power and fluid power requirements Table 3.34 FCS exergy destruction equations Table 4.1 Number of coupling functions associated with each subsystem Table 4.2 AFS-A fixed-wing design and operational decision variables and inequality constraints Table 4.3 AFS-A morphing-wing design and operational decision variables and inequality constraints Table 4.4 AFS-A mission decision variables and inequality constraints Table 4.5 PS design and operational decision variables and inequality constraints XIII

14 Table 4.6 ECS optimization synthesis / design and operational decision variables and inequality constraints Table 4.6 FLS optimization decision variables and inequality constraints Table 4.8 VC/PAOS optimization synthesis / design and operational decision variables and inequality constraints Table 4.8 OLS optimization operational decision variables and inequality constraints Table 5.1 Comparison of the optimum morphing-wing gross takeoff weights with a 6x wingweight penalty and the optimum fixed-wing gross takeoff weight (Smith et al. 2007) Table 5.2 Optimal fixed- versus morphing-wing AAF configuration and performance parameters for the subsonic cruise and the supersonic penetration mission segments (Smith et al. 2007) Table 5.3 AAF subsystem weights and the percentage of AAF empty weight after the first ILGO iteration Table 5.4 AAF Subsystem percent weight reduction versus ILGO iteration (Rancruel, 2002). 149 Table 5.5 Projected AAF subsystem weights versus ILGO iteration based on the ILGO progression from Rancruel (2002) Table 5.6 Extrapolated nine-subsystem AAF gross takeoff weight and empty weight versus ILGO iteration based on the ILGO progression from Rancruel (2002) Table 5.7 Extrapolated AAF subsystem system-level optimum weights after seven ILGO iterations along with the percentage of AAF empty weight Table 5.8 Extrapolated subsystem optimum weights versus the optimum subsystem weights from Rancruel (2002) Table 5.9 Baseline AAF configuration and performance for mission segment 4, 5, and XIV

15 Nomenclature A Area, aspect ratio FCS flight controls subsystem AR inlet area FLS fuel loop subsystem AAF air-to-air fighter g gravitational constant AFS A airframe subsystem - aerodynamics GA genetic algorithm b wing span ILGO iterative local-global optimization B h horizontal tail span K cb landing gear cross beam factor BCA best cruise altitude K dw delta wing factor BCM best cruise Mach K mc BCLM best climb Mach 1.45 for mission completion required, 1 otherwise K tpg landing gear tripod factor C D drag coefficient K vs sweep wing factor C D0 parasite drag coefficient K vsh C L lift coefficient L lift, wing length C Dwave wave drag coefficient L a C fe skin friction coefficient L m CHS central hydraulic subsystem L n Constant = if variable sweep wing, 1.00 otherwise Electrical routing distance (ft) from generators to avionics to cockpit length of main gear length of nose gear D drag mass flow rate DOF degree of freedom M Mach number ECS e environmental controls subsystem Weissinger span efficiency N c N en number of crew number of engines ES electrical subsystem N gen number of generators Ex des exergy destroyed N l F w fuselage width at horizontal interception N s XV length of nose gear number of flight control systems

16 N u Number of hydraulic utility functions (typ. 5-15) N z ultimate load factor Vstall q dynamic pressure VC / PAOS V velocity stall velocity vapor compression/pao loop subsystem Q flow rate, heating load W weight, work OLS oil loop subsystem Wl ORS optimum response surface Wdg length of main gear design gross weight P pressure W Empty aircraft empty weight P TO power takeoff from PS PAO polyalphaolefin W TO aircraft takeoff weight PS R propulsion subsystem specific gas constant, additional drag Greek S planform area of the wing angle of attack, fan bypass ratio S cs total area of flight control surfaces design bleed air ratio S ht horizontal tail area taper ratio, shadow prices S ref reference area of wing LE leading edge sweep angle S wet wetted area of wing pressure ratio, pi sfc specific fuel consumption temperature ratio t T time thrust, temperature T inst installed thrust T t total temperature T req thrust required t / c thickness to chord ratio v valve ratio XVI

17 Chapter 1 Introduction The energy-based economy is driving a new technology to the aircraft industry: morphing-wing technology, or the ability of an aircraft to change the shape of its wings during flight, is being researched heavily by both the military/government and the private aircraft industry. The goal of the application of morphing technology is to develop an aircraft that can adapt or change its aerodynamic performance to fly dissimilar missions or dissimilar mission segments within a mission more efficiently than a fixed-wing aircraft. A prime platform for investigating morphing technology is the fighter aircraft. Currently designed fighter aircraft are being used for multiple roles, depending on the branch of the military or even country in which the aircraft is being used. The designer of a fixed-wing aircraft would find the dissimilar roles and requirements of the aircraft to be a design challenge, to say the least. Even more of a challenge would be to design an aircraft that can perform all the roles required in the most efficient manner, which has been proven to be impossible (e.g. a subsonic high-endurance reconnaissance aircraft can not perform the role of a supersonic fighter aircraft more efficiently than the fighter aircraft, nor vice versa). Morphing technology allows a system designer to design an aircraft that can adapt to its flight conditions in order to meet the performance requirements in the most efficient manner. 1.1 Morphing-Wing Aircraft Morphing wings is the new catch phrase in the aerospace research industry today. Morphing technology employed in aircraft wings has been proven, at least on a conceptual level, to allow aircraft to outperform their fixed-wing counterparts over the entire mission in fuel savings due to drag reduction and improved lift-to-drag ratios. Morphing wings can imply the ability of an aircraft to change its aerodynamic performance from a very simple morphing, such as flaps or slats, to a more extensive morphing such as variable wing length, sweep, and chord lengths. An example of morphing wing aircraft is the Lockheed Martin Unmanned Air Vehicle (UAV) concept which came out of the DARPA Morphing Aircraft Structures program is shown in Figure 1.1. This concept has the ability fold its wings, effectively changing the flight characteristics (by varying the wetted area and aspect ratio) of the aircraft extensively to allow a 1

18 mission that could include reconnaissance, loiter, and attack/low observeability configurations in the same vehicle, encompassing the requirements of a hunter-killer mission. The hunter-killer mission would involve first searching for and identifying the target, then destroying it with the shortest delay in time between identification and destruction as possible. This type of mission would normally be performed by a package or group of aircraft specializing in certain portions of the hunter-killer mission. The Lockheed Martin UAV concept allows the package to be eliminated, as the concept can perform all tasks in the hunter-killer mission due to the ability to morph its wings. Figure 1.1 Lockheed hunter-killer morphing aircraft concept (Bowman, Sanders, Weisshar, 2002). 1 This variable geometry wing can rotate on a pivot, which significantly reduces the drag of the wing in high-speed flight. The wing leading edge is near perpendicular to the flight direction for takeoff and high endurance mission segments for maximum lift, but is rotated (or swept back) to effectively reduce the induced drag and parasitic drag (due to wave drag) for supersonic flight. Other examples of morphing wings have been investigated for the purpose of building a single aircraft that can perform the duties of a group of aircraft. One such example is 1 Reprinted with permission from author. 2

19 the oblique-wing concept. This concept was investigated and tested under the DARPA Oblique Flying-Wing Project with the NASA AD-1 (Ames Dryden AD-1).(Curry and Sim, 1982; Dryden X-Press, 1979) which is shown in Figure 1.2. Figure 1.2 NASA Oblique-Wing Demonstrator, the NASA AD-1 (Dryden X-Press, 1979) 1.2 Aircraft System/Subsystem Synthesis/Design The synthesis/design process of the AAF is driven by the mission flown which defines the customer requirements for the aircraft to be developed. The mission flown in this thesis work is very similar to Rancruel (2003) and Butt (2005) but with some changes which are discussed at length in Chapter 3. The flow chart of the synthesis/design process is shown in Figure 1.3. The first synthesis/design stage is the conceptual synthesis of the design. Viable solutions to the requirements are developed and analyzed for cost, feasibility, manufacturability, etc. in this stage. Synthesis, design, and operational variables are also investigated at this level to verify the feasibility of the design. The conceptual synthesis stage has the most variation in model design and performance as often the requirements are adjusted at this step as well, depending on the level of technology required / available. The best solution for the requirements is eventually chosen at this step based on multiple performance measures (e.g. cost, weight, ease of manufacturing, performance, etc.) The next stage is the preliminary synthesis/design. The configuration of the best solution from the conceptual synthesis is frozen at this step, and more detailed testing and major subsystem design commences on the solution. Databases of analytical data and performance testing are compiled at the preliminary synthesis/design step as well. For example, the engine 3

20 sizing, weight, compressor and turbine stages, bypass ratios, etc. would be determined at this stage of the synthesis/design process. Figure 1.3 Aircraft synthesis/design stages (Raymer, ; Rancruel, 2002). Following the preliminary synthesis/design stage is the detailed design stage. Actual subsystems would be prototyped and set up for manufacturing, and subsystem performance in the detailed design stage verified. The performance of the total system would be verified / estimated again at this point as the subsystems are prototyped and tested. Tooling for manufacturing would also be developed at the detailed design stage. The final synthesis/design stage is manufacturing. Hopefully, after the first production model is made, the design still meets all the requirements that were originally posed by the customer. Often this is not the case and the synthesis/design needs to be revised slightly (or perhaps drastically) to meet the original performance requirements. The prevalence of the total system not meeting original performance requirements is evidence that a better synthesis/design process/method may be needed. 2 Copyright 2006 by Daniel P. Raymer. Reprinted with permission from author. 4

21 Finally, each synthesis/design step often has a different design group synthesizing/designing and optimizing a subsystem and its components somewhat independently from all the other design groups. A more integrated synthesis/design/optimization process might reduce the number of occurrences of the final synthesis/design not meeting the original requirements. 1.3 Modeling and Simulation An aircraft can be viewed as a system of subsystems. Each subsystem is a component or group of components that can be logically separated from the rest of the components that make up the system. Subsystem boundaries may be determined using a variety of criteria such as logical physical boundaries, thermodynamic boundaries, time boundaries, etc. After the subsystems are clearly defined, a model of each of the subsystems is developed. A subsystem model can take on many different forms: a thermodynamic model, a geometric model, an aerodynamic model, a kinetic model, etc. These models can be analytical, empirical, or semi-empirical and can be zero-d or lumped-parameter models or 1-D, 2-D, or 3-D computational fluid dynamics (CFD) model. Typical of the types of subsystem are geometric, aerodynamic, and thermodynamic used in subsystem modeling, simulation, and large-scale optimization are models with lumped parameter distributions. A lumped parameter distribution indicates that properties that would normally have an infinite distribution of values are represented by an averaged value. For example, the temperature of the air in a turbofan engine would have an infinite distribution of temperatures throughout the engine; however, the temperatures are lumped or averaged to one temperature at a given station in the engine and calculated as such in the model equations. Typically, each subsystem of an aircraft is defined as a group of components that perform a given function. For example, the engine or propulsion subsystem (PS) provides the thrust to the aircraft and power to other subsystems and is easily separable, thermodynamically from the rest of the aircraft. Thus, it is considered a subsystem. The airframe subsystem (AFS) is responsible for providing the aerodynamics required to fly the mission as well as house the other subsystems. The fuel loop subsystem (FLS) consists of all the components associated with the fuel tank and a set of associated heat exchangers which help condition the fuel and deliver it to 5

22 and from the engine. Other subsystems are defined in a similar fashion and each has a range of operating conditions and parameters for the function it must perform. Once the subsystem components and configurations have been defined, every subsystem model (lumped parameter or otherwise) must be written in an engineering software language to represent the model equations which simulate the subsystem behavior. The coded model must then be validated to verify that it produces operating parameter values close to previously published ones or which from engineering experience seem reasonable. After validation, each subsystem is integrated with the other subsystems. Integrating a subsystem involves defining each inter-subsystem interaction by assigning appropriate variables between subsystems. For example, the thrust required by the airframe subsystemaerodynamics (AFS-A) to fly a specific part of the mission segment is assigned as a required thrust from the engine. Thus, the thrust required by the AFS-A is an interaction between the engine and AFS-A. When all the subsystems are integrated, the modeling of aircraft is complete and the code ready to use to simulate the aircraft s behavior for a given design point. The simulation gives feedback as whether or not the aircraft has been designed in a way that enables it to successfully fly at that design point (usually the most difficult part of the mission to fly, i.e. supersonic flight), and if so, how well the design flies at that design point compared to other syntheses (configurations) / designs. When multiple simulations are ran and compared to each other to find the best configuration / design, it can be said that the system is being optimized. Conventionally, this is done by trial and error and not by using large-scale optimization. Furthermore, conventionally, this process of synthesis / design of fixed-wing aircraft is performed only at the design point and off-design operation is simply verified. However, large-scale optimization would greatly enhance this process as would inclusion of the off-design mission segments directly in the synthesis/design process along with the optimization. This requires a mission integrated approach which is in fact a necessary approach when applying morphing-wing technology to an aircraft. This is discussed along with large-scale optimization in the following section. 6

23 1.4 Large-Scale Optimization and Mission Integration When mathematical optimization is applied to the process of finding the best system synthesis / design of a complex system, it is called large-scale optimization. To find this best or optimum system requires a performance metric or optimization objective function by which numerous syntheses / designs can be compared. For example, an optimization objective for a power plant synthesis / design may be to maximize the power output or to minimize the operating cost of the power plant. The optimization takes place by varying the system synthesis / design and operating parameters and then evaluating system performance by running a system simulation with the given synthesis / design and operating parameters. The independent system synthesis / design variables are called the synthesis / design decision variables, while the independent system operating variables are called operational decision variables. The variables are called decision variables because they are being varied, or decided upon, during the optimization to find the best system configuration and design. A complete system optimization often requires that both the synthesis / design and operational decision variables be varied to find the optimum system synthesis / design as knowledge of system behavior during operation as well as the system s physical characteristics are needed to find the optimum system. The highly integrated and tightly coupled (i.e. everything influences everything else) nature of synthesizing / designing an aircraft typically requires that both synthesis / design variables and operational decision variables participate in the optimization as is done in this thesis research. Unfortunately, a complete system optimization for even a single design point can prove to have a very large computational burden depending on the complexity and size of the system being optimized. This creates a need for efficient and effective large-scale optimization algorithms and additional tools and methods to manage the computational burden of optimizing such a large system. Adding morphing-wing technology to an aircraft further increases the complexity of the optimization problem. To fully investigate the benefits of morphing-wing technology, a single design point optimization is not sufficient. Aircraft performance must be evaluated over the entire mission since morphing wing technology is intended to improve the off-design performance of the aircraft. If the aircraft is being synthesized / designed with only the design flight conditions in mind, morphing-wing technology cannot show any benefit over a fixed-wing aircraft. Of course, mission integrated optimization compounds the computational burden 7

24 significantly as additional operational decision variables are required for each operational condition. In reality, this would require a dynamic, real-time optimization. Computationally, this adds an additional computational burden to an already computationally challenged problem. However, for purposes of synthesis / design, this dynamic behavior can be approximated by a quasi-stationary approach. This is done by splitting the entire aircraft mission as defined, for example, in Mattingly et al. (2002), into mission segments (see Figure 1.4). Separating the mission into segments is a form of time decomposition which is common to aircraft synthesis / design and makes the aircraft modeling quasi-stationary instead of dynamic. Operational decision variables are then applied to each of the mission segments for each of the subsystems, and the entire aircraft is optimized over the entire mission. In a sense, the aircraft no longer has a design point per se but is instead optimized for its overall performance in the entire flight envelope, from takeoff to landing. Mission integrated synthesis/design is absolutely necessary for studying the effects of morphing-wing technology in fighter aircraft; however, mission integration synthesis / design and optimization creates a huge computational burden. The computational burden increase is primarily due to an increase of degrees of freedom (DOF) or the number of variables the optimization algorithm has free to decide upon for system simulation. An additional problem is the non-linear response of aircraft performance with respect to the decision variables. This and the large number of DOF as well as the fact that these may include a mix of discrete and continuous variables limits the optimization algorithms that can be used, as many are unable to find the overall aircraft optimum due to the variable search method used. A solution to the dilemma of too many DOF which does not require a reduction in their number is the use of decomposition strategies. A solution to the difficulties caused by the nonlinearities and the mix of discrete and continuous variables is the use of heuristic optimization algorithms (.e.g., genetic algorithms, simulated annealing, etc.) or the use of hybrid heuristicnonheuristic optimization algorithms (e.g., surrogate, model-based optimization, etc.). A nonheuristic algorithm is typically a gradient-based algorithm such as, for example, sequential quadratic programming (SQP), generalized reduced-gradient (GRG), steepest descent, etc. A discussion of the decomposition strategies and optimization algorithms used in this thesis research appears in Chapter 4. However, a brief discussion of decomposition for large-scale optimization is given in the following section. 8

25 1.5 Decomposition for Large-Scale Optimization The size of a complete aircraft system optimization problem may require a decomposition strategy to handle the huge (and perhaps prohibitively so) complexity of the optimization. There are several different types of decomposition strategies (Frangopoulos, von Spakovsky, and Sciubba, 2002) among which physical decomposition is a major player. Most of the physical decomposition strategies in the literature can be characterized as Local-Global Optimizations (LGOs). With LGO, the system is physically decomposed into a set of subsystems or units each of which is characterized by its own set of decision variables. Each set of decision variables is strictly local, i.e. affects in the main system only its corresponding unit. In a complex system, however, there is always another set of decision variables which is not strictly local and which acts at the so-called system level to affect some or all of the units. It is this division which leads to a multi-level optimization problem in which at the system level an optimization occurs with respect to its set of decision variables, while at the unit level individual optimizations of each unit are carried out with respect to each individual set of unit-level decision variables. To maintain the integrity of the overall system optimization, i.e. one which takes into account all of the decision variables simultaneously, the unit-level optimizations must occur many times within the system-level optimization, resulting in a set of so-called nested optimizations. Although such a set can only approximate the so-called single-level optimization which occurs without decomposition, it nonetheless may do so quite closely. However, the computational burden is much greater than with the single-level optimization and may in fact be prohibitive. To address this last problem as well as a number of others discussed in Chapter 4, a physical decomposition strategy called Iterative Local-Global Optimization (ILGO) has been developed and applied by Muñoz and von Spakovsky (2000a,b,c,d, 2001a,b) and Rancruel and von Spakovsky (2006, 2004a,b, 2003a,b) to eliminate this problem of nested optimizations. ILGO does this by eliminating the system-level optimization and, thus, the need for nested optimizations and does so by bringing the system-level information down to the unit-level. This is done by maintaining subsystem interactions via a set of coupling function and shadow price pairs, which measure the effect that changes in the coupling functions have on the system objective function. Elimination of the nested optimizations reduces much of the computational burden of LGO and has the further advantage of permitting the subsystem optimizations to take 9

26 place simultaneously. ILGO, coupling functions, and shadow prices are discussed at more length in Chapter The Use of Exergy Analysis An emerging analysis tool for aircraft systems is exergy or available energy analysis. The available energy of a system is defined as the largest amount of energy that can be transferred from a system to a weight in a weight process while bringing the system to a mutual stable equilibrium with a notional reservoir (Gyftopoulos and Beretta, 1991, 2004). The usefulness of this analysis technique is realized in multi-component, highly-coupled energy systems where identifying and correcting performance losses is made difficult using traditional energy balance methods. Energy balances treat all forms of energy as equivalent, without differentiating between the quality (ability to produce useful work) of energy crossing the system boundary. Hence the energy from a high temperature source is treated in the same way as the energy rejected to a low temperature sink. Energy balances do not provide information about internal losses. For example, an energy balance for an isolated system in a not stable equilibrium state shows that the process the system undergoes incurs no losses. This, however, is not true! In fact, most of the causes of thermodynamic losses in thermal, chemical, and mechanical processes such as heat transfer across finite temperature differences, mixing, combustion, and viscous flow cannot be detected with energy balances since these losses are not associated with a loss of energy (which can neither be created or destroyed) but instead with a decrease in the quality of the energy. An exergy or available energy approach overcomes these deficiencies since exergy accounts for this loss in quality. The rate of loss of exergy internal to the system (i.e. the rate of irreversibilities or entropy generated) provides information about the true inefficiencies of the system. Hence an exergy analysis of a multi-component system such as the propulsion subsystem (PS) of an aircraft indicates the extent to which its components contribute to the inefficiency of the overall aircraft system. Unlike the energy method which is based on the First Law of Thermodynamics alone, the concept of exergy and irreversibility is based on both the First and Second Laws of thermodynamics. The irreversibility of a system, sub-system, or component can be found by doing an exergy balance which combines unsteady or steady state balances of mass, energy, and entropy into a single balance of exergy. The overall exergy destruction rate which is 10

27 directly proportional to the overall entropy generation rate and which appears in this single balance can be determined directly from this balance. To determine the individual exergy destruction rates which comprise this overall rate, a set of independent phenomenological equations relating the exergy destruction rates to the various irreversible phenomena present in a given process must be used. Since the intention of this thesis research is to investigate morphing wing feasibility using modeling, simulation, mission integration and optimization, exergy analysis is an additional tool which can be employed for finding locations in the aircraft system needing the most improvement and for categorizing the internal and external losses which occur in and out of the aircraft. In other words, an exergy analysis can provide information about which components or set of variables may be good candidates for further optimization or re-optimization and is useful for setting up guidelines for process improvements since no matter what aerodynamic, thermodynamic, kinetic or geometric phenomena are being modeled, a common quantity, i.e. the exergy, can be used as the measure of process improvement or performance. 1.7 Thesis Objectives The overall goal of this thesis work is to investigate the feasibility of applying morphingwing technology to an air-to-air fighter (AAF). The feasibility is investigated by modeling, simulating, analyzing, and optimizing a morphing-wing fighter aircraft and comparing it to a fixed-wing fighter aircraft. The aircraft model is an expansion by four subsystems on the fivesubsystem model developed and implemented by Rancruel (2003) and is intended to bring the number of subsystems up to that of a complete aircraft system. It, thus, consists of nine subsystems including the following: airframe subsystem-aerodynamics (AFS-A), propulsion subsystem (PS), environmental controls subsystem (ECS), fuel loop subsystem (FLS), vapor compression/polyalfaolephin subsystem (VC/PAOS), oil loop subsystem (OLS), central hydraulic subsystem (CHS), electrical subsystem (ES), and flight controls subsystem (FCS). The first five of these exist in the aircraft model of Rancruel (2003) while the latter four are researched, developed, implemented, and validated here. Two additional subsystems exist in both aircraft models, namely, the fixed and expendable payload subsystems. However, neither involves synthesis/design or operational degrees of freedom (DOF) even though each plays a role in the synthesis/design process. All the subsystems used are the same between the 11

28 morphing- and fixed-wing models with the exception of additional DOF for the morphing AFS- A which are included to account for the wing geometry changing during flight. To achieve the overall goal outlined above, the first objective of this thesis research involves the research and development of four conceptual-level, lumped-parameter subsystem models (thermodynamic and geometric) and extensive modification of the other five highfidelity, lumped-parameter subsystem models (aerodynamic, thermodynamic, and geometric) which have already been fairly well documented in a previous Ph.D. dissertation (Muñoz, 2000) and M.S. thesis (Rancruel, 2003) and in publications by Muñoz and von Spakovsky (2000a,b,c,d, 2001a,b) and Rancruel and von Spakovsky (2006, 2004a,b, 2003a,b). The subsystems are as listed above. Note, however, that the AFS-A involves separate fixed-wing and geometry morphing aerodynamics models. These along with the PS, ECS, FLS, and VC/PAOS are modified from previous work and implemented in iscript, which is a software language and tool set being developed specifically for the purpose of aircraft mission integrated synthesis/design modeling, analysis, and large-scale optimization by TTC Technologies, Inc., (TTC) as part of a U.S. Air Force AFRL Phase II SBIR in which Virginia Tech is a participant. The remaining four subsystems, the OLS, ES, CHS, and FCS, are developed and implemented specifically for this thesis work. The second objective of this thesis research is to integrate all of the subsystems into an overall aircraft system capable of flying an entire fighter aircraft mission. This requires defining a complete mission able to thoroughly exercise the capabilities of both the fixed- and the morphing-wing aircraft. It also requires assigning the proper wing- and fuel-weight penalties due to morphing-wing technology (both explained in Chapter 3). A third objective is to apply large-scale optimization using physical decomposition (i.e. ILGO) to the mission integrated synthesis/design of both the fixed- and morphing-wing fighter aircraft and then analyze and compare the results. The main comparison for the fixed- versus the morphing-wing aircraft is a sensitivity study of the effects of the wing weight and fuel weight penalties associated with the morphing technology as was done in Butt (2005) but with a much more detailed and complete aircraft system. The final or fourth objective is a study of the morphing-wing parameters that provide the highest payoff in terms of fuel savings for three mission segments: subsonic climb, subsonic cruise, and supersonic cruise. The effect of varying the wing parameters on the fuel burned is 12

29 compared to the exergy destruction plus exergy loss due to unburned fuel for the AFS-A to see a correlation in trends. The results and conclusions drawn are then compared to those given in Periannan, von Spakovsky, and Moorhouse (2008). 13

30 Chapter 2 Literature Review Much research has been done in the areas of morphing-wing technology, exergy analysis, and optimization, and how they relate to aircraft synthesis / design. This chapter reviews some of the previous work done in this area in each of these areas. 2.1 Benefits and Design Challenges for Morphing Aircraft Moorhouse, Sanders, von Spakovsky, and Butt (2005, 2006) illustrate the potential benefits of applying morphing wings to an aircraft as well as the design challenges that are yet to be overcome in applying morphing technology to aircraft. The aerospace industry has shown a history of taking a current technology and expanding it to yield the next generation of aircraft. Often the new designs end up looking like the old designs albeit with new materials or more sophisticated electronics. The current aerospace customer needs a more affordable aircraft with expanded mission capabilities. Technologies are already being applied to wings to allow wing shape and, thus, aerodynamic performance to change depending on the requirements of the flight conditions. An early example of wing morphing, i.e. wing twisting, was used by the Wright brothers for roll control and is being re-invented by the active aeroelastic wing (AAW) program. Currently, variable wing sweep or swing wings are employed on many fighter aircraft to allow better cruise endurance but without sacrificing high velocity flight performance. Low order shape control is being used on aircraft in the form of Fowler flaps and ailerons. These technologies have been hugely successful, enabling aircraft to reduce stall speeds, increase lift, etc. to perform mission segments more successfully than otherwise may have been possible. However, more extensive shape control or morphing aircraft structure is desired to allow drastic wing planform area and aerodynamic performance changes during flight. The benefits of applying morphing wings to a fighter aircraft were investigated with a model and a set of optimizations by Butt (2005; see Section 2.2 for details on the investigation). The optimization results show that drastic morphing of an aircraft wing, including wing sweep, wing length, and root and tip chord lengths could allow a fighter aircraft to use significantly less fuel than a fixed-wing counterpart even for a much heavier morphing-wing aircraft. A point perhaps not obvious to a designer is the fact that design constraints, which are determined by the 14

31 mission requirements, are relaxed due to the ability to morph the aircraft wings. Figure 2.1 shows an example of an aircraft synthesis/design space with the dark shaded region. However, when morphing technology is used on the aircraft, the design space is expanded to include the cross-hatched region. Figure 2.1 Effect of morphing on the synthesis/design space of thrust to weight (T/W) vs wing loading (W/S) (Periannan, von Spakovsky, and Moorhouse, 2008). The increase in the design space indicates that morphing wings allow the aircraft to perform the same mission as a fixed-wing counterpart, but with more flexibility in the design of the wing loading and thrust to weight ratio. Morphing-wing technology certainly shows very promising results for implementation on paper; however, building a morphing-wing presents new challenges in design, implementation, and control. The difficulty is how can the morphing technology be mechanized? The model in Butt (2005) uses mission segment-by-segment morphing or a single configuration for the entire mission segment. Would real-time shape control provide more benefit? One concept for realtime shape control is the mission adaptive wing (MAW). An example of a trailing edge MAW is shown in Figure 2.2. The MAW allows the wing lift and drag to be dynamically adjusted (or morphed in real time ) during flight to outperform a fixed-wing counterpart. An example of such behavior could be observed in a sustained turn. Figure 2.3 shows the benefit of employing MAW technology to maximize the lift/drag ratio to allow a much smaller turning radius over that of a fixed-wing aircraft. The MAW shows great promise as does the morphing-wing aircraft in Butt (2005). Additional design risk is added when real-time morphing is implemented; furthermore, the level 15

32 Figure 2.2 Mechanization of the mission adaptive wing (MAW) trailing edge (Periannan, von Spakovsky, and Moorhouse, 2008). of aircraft integration required to successfully implement morphing technology is substantially higher than what traditional design methods dictate today. The authors believe a new design methodology must be used: the vehicle is considered a device in which all components are Figure 2.3 Example of the benefit of mission adaptive wing (MAW) technology (Periannan, von Spakovsky, and Moorhouse, 2008). optimized to minimize exergy consumption at the system level. The authors assert three things must be changed in traditional design methods. The first is in the philosophy of how structures are designed. Wing structures must be designed for deformation characteristics from the start of 16

33 the design. The second change is that a better understanding of how to optimize the distribution of sensors and actuators in addition to structural properties, such as mass and stiffness, needs to be developed. The final change is to investigate the scalability of actuators and the degree of deformation from a systems level perspective that is best suited to the design problem. The exergy-based concept design method proposed by the authors shows good promise in solving this third item as exergy is a metric common to all systems. The exergy-based objective function for optimization requires an integration of the entire vehicle during the design process rather than the traditional design assumption that minimum weight is the best design. There are other challenges in employing morphing wings that must be overcome. The example in Figure 2.3 should show that even for a simple model, non-linear design and analysis is required in order to avoid the significant error which comes from assuming linearity. Flight control issues of a wing capable of shape changing to modify roll characteristics, drag, and even perhaps wing loading profiles would pose a significant design challenge. Previously, control gains were set based on wind-tunnel testing with a re-design if flight testing showed improvement was needed. The current design method is to use a neural network to actively adjust gains based on actual in-flight aerodynamic performance. Perhaps this technology could be employed to control morphing wings as well or would it be trying to catch up to the constantly changing flight characteristics (forming too great of a computational burden)? These authors as well as others have clearly shown that employing adaptive structures in the next generation of aircraft will yield significant benefits in operating cost and performance for aircraft that perform significantly dissimilar mission segments. They have also shown that current design methodologies will be unable to accomplish the level of integration required to successfully design (and eventually manufacture) a morphing-wing aircraft. Non-linear design methods and an exergy-based optimization metric may be required to reach this level of integration and some questions need answering as well. How fast do the wings need to change shape? If they change slowly, will traditional flight control subsystems be sufficient for flight control? If they change quickly, what will be the flight controls needed to compensate for aerodynamic instability during shape change? These questions and design challenges must be answered and overcome to continue to re-invent and extend the adaptive structures that started with the Wright brothers. 17

34 2.2 Study of Morphing-wing Effectiveness in Fighter Aircraft Previous work has been done in the area of morphing-wing technology at Virginia Tech (Butt, 2005). An afterburning turbojet propulsion subsystem (PS) and the aerodynamic aspects of a morphing airframe subsystem (AFS-A) are used to perform a feasibility study of employing morphing-wing technology in a supersonic fighter aircraft flying a mission as depicted in Figure 2.4 taken from Mattingly, Heiser, and Daley (1987). Butt separates the entire mission (originally 14 segments) into 22 segments to allow a more accurate estimation of aircraft behavior over the entire mission (e.g., separates a subsonic-supersonic acceleration into 3 segments: subsonic, transonic, and supersonic). The mission flown is a subset of the 22 mission segments in that segments 3-21 are what are actually flown. The mission segments are detailed in Table 2.1. Figure 2.4 Mission Profile by segment or leg (Mattingly, Heiser, and Daley, 1987). Table 2.1 Mission Segment Definition and Description. Mission Segments No. Name Description 1 Warm up 1 minute, military power 2 Take off 5 min at idle, take off + roll rotation 3 Subsonic Acceleration 1 Accelerate to climb speed 4 Subsonic Acceleration/Climb Climb at BCM/BCA 5 Subsonic Cruise 1 Cruise until cruise + climb range = 280 km 6 Combat Air Patrol/Loiter Patrol for 20 min and 9150 m and Mach for best endurance 7 Subsonic Acceleration 2 Accelerate to Mach Transonic Acceleration 1 Accelerate to Mach Supersonic Acceleration 1 Accelerate to Mach Supersonic Penetration Mach 1.5 for 185 km 11 Combat Turn 1 Mach 1.6, one 260 degree 5g turn at max power 18

35 No. Name Description 12 Combat Turn 2 Mach 0.9, two 360 degree 5g turns at max power 13 Subsonic Acceleration 3 Accelerate from M = 0.9 to Transonic Acceleration 2 Accelerate from M = 1 to M = Supersonic Acceleration 2 Accelerate from M = 1.2 to M = Deliver Expendables Deliver 2 AMRAAM, 2 AIM-9L, and ½ ammunition 17 Escape Dash M = 1.5 for 46 km 18 Supersonic Climb Climb to BCM/BCA from M = 1.5 to M = Transonic Climb Climb to BCM/BCA from M = 1.2 to 1 20 Subsonic Cruise 2 Cruise at BCM/BCA for range of 278 km 21 Loiter Loiter at 9150m for 20 min at M for best endurance 22 Landing Land Airframe Subsystem Aerodynamics (AFS-A) The aerodynamics model is developed from Raymer (1999), Mattingly, Heiser, and Pratt (2002), Andersen (1998), and Nicolai (1975). Table 2.2 displays the airframe subsystem aerodynamics (AFS-A) calculations and model equations. Table 2.2 AFS-A aerodynamics and model equations. Component Variable Description Model Equation Master T Thrust 2 d Flight V T D V W h Equation dt 2g V Velocity Drag D Drag D qcds C D Drag coefficient C D K where C 2 1CL CD0 " 2 D 0 C D K C min L min_ drag Induced Drag K 1 Parasitic Drag C D0 Induced drag factor (Nicolai, 1975) Parasitic Drag Coefficient Subsonic: Supersonic: Subsonic: Supersonic: K 1 K C 1 ear 1 1 C L D0 C D0 C C D min fe S S C wet ref fe C S S wet ref Dwave Wave Drag C Dwave Coefficient of drag due to shock waves CDwave D / q wave S 19

36 Component Variable Description Model Equation where for M = 1 to 1.2: D q wave D 2 q S H 9 A 2 2 l max 2 D q wave Wave drag efficiency factor and for M > 1.2 D q wave E WD M LE D q S H where D q S H D q SearsHaack Exergy E x despar E x desind Exergy destruction rate due to parasitic drag Exergy Destruction Rate due to Induced Drag T D V Ex 0 0 des Par T where: D qc S E x 0 D0 T D V i T D qc S des Ind 0 where: i Di The master flight equation is derived from a force balance on the aircraft. The drag on the aircraft, D, must be overcome by the thrust, T. The drag coefficient equation for an uncambered wing is also used in the model and the Wessinger span efficiency from Nicolai (1975) is used to calculate the induced drag factor, K 1. The lift curve slope, C L, is determined from supersonic normal force curve slope charts presented in Raymer (1998), Nicolai (1975), and Anderson (1999) where, for a high-performance uncambered aircraft CL min_ drag is approximately zero. The parasitic drag coefficient, C D0, takes on two different forms depending on whether or not the aircraft is in the subsonic or supersonic flight regime. C fe is the equivalent skin friction coefficient and is chosen to be as recommended for an air force fighter (Raymer, 1998). S wet and S ref are the wetted areas of the aircraft and the exposed wing area, respectively; and C Dwave is the coefficient of drag due to the formation of shock waves during supersonic flight. C Dwave is calculated using a correlation for the Sears-Haack body wave drag where E WD is an empirical wave drag efficiency factor which ranges from 1.8 to 2.2 for supersonic fighter aircraft, M is the flight Mach number of the aircraft, and LE is the leading-edge sweep angle. 20

37 The exergy destruction rate in the AFS-A is due to the parasitic and induced drag losses, and the equations are also shown in Table 2.2. The morphing AFS-A has a total of 4 synthesis/design and 72 operational decision variables because for each of the 18 mission segments flown, 4 wing parameters are varied including wing sweep, wing length, root chord length, and tip chord length. The fixed-wing AFS-A had a total of 4 synthesis/design decision variables Propulsion Subsystem (PS) The propulsion subsystem (PS) used in Butt (2005) is from Saravanamutto, Rogers, and Cohen (2001). A schematic of an afterburning turbojet engine is shown in Figure 2.5. The turbojet PS has 3 synthesis/design decision variables which are the compressor design pressure ratio and the design corrected mass flow rates for the compressor and turbine. There are a total of 44 operational decision variables for the PS: 18 compressor pressure ratios, 18 burner fuel/air ratios, and an additional 8 afterburner fuel/air ratios. Fuel Afterburner Nozzle Air Burner Diffuser Compressor Shaft Turbine Products Burner Figure 2.5 Schematic of a single-spool turbojet engine with afterburner Most Important Results from Butt (2005) The results of the large scale optimization show that the benefits of wing morphing are very promising. The morphing aircraft take-off weight was established by multiplying the fixedwing weights and take-off fuel by penalty factors due to the morphing actuator weights and power requirements. Figure 2.6 shows a sensitivity study of the morphing-wing fuel consumption with respect to weight and fuel penalty factors. The optimum fixed-wing fuel consumption is also displayed. 21

38 Figure 2.6 Morphing-wing and fixed-wing fuel consumption comparison (Butt, 2005). Figure 2.6 shows the wing weight penalty factors as well as the fuel consumption penalties. For example, a morphing-wing aircraft that has wings 4 times the weight of the fixed-wing and carries 25% more fuel at take off for actuator power uses approximately 2900 lb of fuel, which shows an improvement of 10% over the fixed-wing aircraft at 3200 lb of fuel. Figure 2.6 indicates that even for very unreasonable morphing-wing weight and actuator power consumption penalties (up to a factor of 7 for wing weight and a factor of 2 for the fuel weight) a better-performing fighter aircraft can be attained by employing morphing technology. 2.3 Decomposition Strategies for Large-scale Aircraft Synthesis/Design Optimization The work of Muñoz (2000) illustrates multiple optimization algorithms as well as different autonomous decomposition strategies for optimizing highly coupled, highly dynamic energy systems. The optimization algorithms discussed and used include the following gradientbased algorithms: the Method of Feasible Directions (MFD) and Sequential Quadratic Programming (SQP) and the following nongradient-based optimization algorithms: simulated annealing (SA) and Genetic Algorithms (GAs). The decomposition discussed and utilized include time decomposition and physical decomposition. The latter in the literature can generally be described as Local-Global Optimizations (LGO). An exception to this, is the strategy developed by Muñoz (2000) and Muñoz and von Spakovsky (2000, a, b, c, d; 2001 a, b) called Iterative Local-Global Optimization (ILGO) which is unique and addresses the major 22

39 drawback to LGO which is its large computational burden. Decomposition methods allow the designer to create more complicated system models than would otherwise be possible for use with large-scale optimization. LGO works by physically decomposing a system into a set of units for which a set of unit-level optimization sub-problems are established within the context of a reduced (i.e. smaller than the original) system-level problem. This results in a set of a nested optimization problems (i.e. the unit-level within the overall system-level) as well as a system-level optimum response surface (ORS) which can be constructed implicitly or explicitly on the basis of the many unitlevel interactions (or coupling functions) between units. A search of this surface results in the global optimum solution for the system. Figure 2.7 illustrates the system-level ORS for a two-unit system as well as the unit-level ORS generated by the sub-problem optimizations which in turn result in the system-level ORS and the large computational burden with which LGO is strapped. In this figure, u 12, and u 21 represent the coupling functions between units 1 and 2; z 2 is the set of local or unit-level decision variables used in the sub-problem optimization of unit 2; and z 2, z 2, and z 2 are the objective functions for the unit- and system-level optimization problems. The superscript indicates either a unit- or system-level optimum. Now, in contrast to LGO, ILGO significantly reduces the computational burden by eliminating the need for nested optimizations. It does so by embedding the system-level optimization at the unit-level and in the process transforms the unit-level optimization subproblems into so-called system-level, unit-based optimizations. This eliminates the need to implicitly or explicitly construct the system-level ORS which ILGO accomplishes by using ORS slope information in the form of shadow prices as well as changes in the coupling functions to move towards the system-level optimum. It is these shadow prices and coupling function changes which embed the system-level information at the unit-level. 23

40 Figure 2.7 Unit- and system-level optimum response surfaces (ORSs; Rancruel, 2003). Muñoz (2000) and Muñoz and von Spakovsky (2000 a, b, c, d, 2001 a, b) apply ILGO to the optimization of a supersonic aircraft consisting of the following integrated subsystems: the propulsion subsystem (PS), the environmental controls subsystem (ECS), the airframe subsystem-aerodynamics (AFS-A), the expendable and permanent payload subsystems (EPAY and PPAY), and the equipment group (EG). These authors show that ILGO can be used as an effective physical decomposition strategy for large-scale optimization by comparing the results from a mission-integrated synthesis/design optimization of the supersonic aircraft with and without the use of ILGO. The results of this comparison show that the final system optimums are within 0.5% of each other. Note, however, that in order to make this comparison, the actual aircraft optimization problem with 153 degrees of freedom (109 for the ECS, 44 for the PS, and 0 for the AFS-A, EPAY, PPAY, and EG) had to be reduced down to one with only 52 degrees of freedom (38 for the ECS, 14 for the PS, and 0 for the AFS-A, EPAY, PPAY, and EG) in order to be able to solve the problem in a reasonable time frame without decomposition. Thus, for the comparison, the optimizations with and without decomposition are based on this latter number. A more involved application of ILGO and further validation of the effectiveness of this approach is found in Rancruel (2003) and Rancruel and von Spakovsky (2006, 2004 a, b, 2003 a, b). In their work, a more complete fighter-aircraft model with five subsystems with optimization 24

41 degrees of freedom (DOF) and three subsystems without DOF is optimized. A list of the subsystems included is as follows: PS modeled with a modern performance code for on- and off- design performance (FAST by Honeywell); ECS bootstrap type subsystem; FLS modeled as a transient subsystem; Vapor Compression / Polyalphaolefin Loops Subsystem (VC/PAOS) a vapor compression refrigeration cycle with high and low temperature heat exchanger networks; AFS-A - the only non-energy based subsystem; and, EPAY, PPAY, and EG involving no optimization DOF. ILGO is used to physically decompose the system in order to be able to do the large-scale synthesis / design of the tactical fighter aircraft, and optimization involving 493 DOF. ILGO permits what had never been done before for highly dynamic, non-linear systems, namely a close approach to the theoretical condition of thermoeconomic isolation (TI) (Frangopoulos and Evans, 1984; von Spakovsky and Evans, 1993). TI is defined as the condition under which each subsystem (or unit resulting from physical decomposition) of a system can be optimized independently of the other subsystems and yet still result in a system optimum identical to what would be attained without decomposition. For a more complete discussion, the reader is referred to Rancruel (2003). Furthermore, as shown in Muñoz (2000), ILGO can be effectively used for the large-scale optimization of energy systems consisting of several subsystems and many DOF, while Rancruel (2003) shows that ILGO can also be extended to non-energy based subsystems such as the AFS-A. By including the AFS-A with DOF in the aircraft synthesis / design optimization process, overall aircraft performance can be improved. The mission profile used in the synthesis / design optimization is the same as shown in Figure 2.4 and used in Butt (2005). However, the mission is split up into segments slightly differently than in Butt. The mission segment details are shown in Table 2.3. The decomposition of the model is shown in Figure 2.5 along with the coupling functions for the aircraft model used by Rancruel. The AFS-A uses the component buildup method for parasitic drag estimation detailed in Raymer (2000). The supersonic parasitic drag is estimated 25

42 Phase 1 Table 2.3 Mission specifications (Rancruel, 2003). Description Warm-up and take off, field is at 600 m pressure altitude with T=310 K. Fuel allowance is 5 min at idle power for taxi and 1 min at military power for warm-up. Take-off roll plus rotation must be 450 m on the surface with a friction coefficient = V 1.2 V Accelerate to climb speed and perform a minimum time climb in military power to best cruise 2 mach number and best cruise altitude conditions (BCM/BCA) 3 Subsonic cruise climb at BCM/BCA until total range for climb and cruise climb is 280 km 4 Descend to 9150 m 5 Perform combat air patrol loiter for 20 min at 9150 m and a Mach number for best endurance 6 Supersonic penetration at 9150 m and M = 1.5. Range = 185 km Combat is modeled by the following: Fire 2 AMRAAM missiles Perform one 360 deg., 5 g sustained turn at 9150 m, M=0.9 7 Accelerate from M = 0.8 to M = 1.6 at 9150 m at max. power Fire 2 AIM-9Ls and ½ ammo Conditions at end of combat are M = 1.5 at 9150 m 8 Escape dash, at M = 1.5 and 9150 m for 46 km 9 Using military power, do a minimum time climb to BCM/BCA 10 Subsonic cruise climb to BCM/BCA 11 Subsonic cruise climb at BCA/BCM until total range from the end of combat equals 278 km 12 Descend to 3000 m 13 Loiter 20 min at 3000 m and a Mach number for best endurance 14 TO STALL Descend and land, field is at 600 m pressure altitude with T = 310 K. A 2 s free roll plus braking distance must be 450 m. Runway has a friction coefficient = V 1.15 V using a correlation to the Sears-Haack body wave drag. The drag due to lift or induced drag is estimated using the leading edge suction method also detailed in Raymer (2000). The thrust requirements from the PS for each mission segment is determined using the master flight equation (see Table 2.2) modified to account for individual subsystem drag due to the ECS air requirements and the thermal management subsystem (TMS) which includes the VC/PAOS and FLS. The empty weight of the aircraft is estimated using the component group weight method from Raymer (1987) which was developed by studying previously built fighter aircraft and applying regression analysis to the data. The ECS is a bootstrap type ECS similar to what is used in the F-16. The ECS provides conditioned air to the pilot in the cockpit and the air-cooled avionics. This subsystem requires bleed air from the PS and has two bleed ports: high and low pressure. Both bleed ports cannot be active at the same time. The bootstrap-type ECS is defined as such because the compressor is driven off its own process airflow rather than some other means of compressor power (such as electrical power or shaft power from the PS). TD STALL 26

43 Figure 2.5 Subsystems and subsystem coupling functions (Rancruel, 2003). The aircraft also uses compact heat exchangers with off-set strip fins for the thermodynamic model of all the heat exchangers; the F-16 uses the same type of heat exchangers. Kays and London (1998), Shah and Webb (1982) and Shah (1981) provide the thermodynamic models for the heat exchanger configurations considered here. The VC/PAOS cools the portions of the avionics that require liquid cooling and uses the FLS as a heat sink as well as a ram air heat exchanger as a cooling mechanism. The FLS is used as a heat sink for the rest of the aircraft subsystems including the Central Hydraulic Subsystem (CHS) and the Oil Loop Subsystem (OLS). Most of the fuel needed to cool the hydraulic subsystem and oil loop subsystem is burned by the engine, but if any excess fuel is needed for cooling duties, it is cooled in a ram air/fuel heat exchanger before being returned to the fuel tank. The evolution of optimal weight with respect to the ILGO iterations for the takeoff weight, AFS-A weight, fuel weight, and PS weight are shown in Figure 2.6. Notice that the slope of the weight vs. ILGO iteration curve becomes nearly zero as the ILGO iterations increase 27

44 for all the subsystems displayed. This indicates an optimum has been reached, even possibly the global optimum. Figure 2.6 Evolution of the gross take-off weight, fuel weight, AFS-A weight, and PS weight at different points of the iterative local-global optimization (ILGO) approach (Rancruel, 2003). Finally, the optimum aircraft configuration is compared to the aircraft design proposed by Mattingly, Heiser, and Daley (1987) to validate the results. The comparison is shown in Table 2.4. It is clear that the ILGO approach yields a superior aircraft to the one proposed by Table 2.4 Comparison between the optimum ATA and the aircraft proposed by Mattingly, Heiser, and Daley (1987), (Rancruel, 2003). Optimum Mattingly, Heiser, and Dailey (1987) W, (lb) 22,396 23,800 TO W, (lb) 7,194 7,707 Fuel W TO / S ref, (lb/ft 2 ) T SL / W TO S, (ft 2 ) ref T SL, (lb) 25,306 30,226 W, (kg) 3,100 4,200 AFS 28

45 Mattingly, Heiser, and Daley (1987). Furthermore, the results of Rancruel (2003) are reasonable and the models behave as they should and are, thus, validated. The objective functions used by Rancruel include minimize gross takeoff weight and fuel consumption. Total cost was also defined as a system-level objective function; however, the work of Muñoz (2000) shows that cost is linearly related to gross takeoff weight and is, thus, not explicitly used as an objective function. More work has been done with optimizing different objective functions in supersonic fighter aircraft applications. The thesis work of Periannan (2005) investigates multiple objective functions including exergy destruction minimization and the effects of the overall optimum found with respect to allowing non-energy based subsystems to participate in the optimizations. 2.4 Effects on Aircraft Synthesis / Design of Different Objective Functions The work of Periannan (2005) and Periannan, von Spakovsky, and Moorhouse (2008) investigated five different objective functions, or figures of merit, for the analysis of a fighter aircraft. The comparison is constructed to study the differences, if any, in the optimum vehicle found for each objective function. The aircraft subsystem is assumed to consist of three subsystems: a PS, an AFS-A, and an ECS. Initially, only the energy-based subsystems are included with DOF in the optimization, namely, the PS and ECS. The AFS-A participates in the optimization but without DOF. These optimizations are carried out for four of the five following objectives (i.e. 1, 2, 4, and 5): 1. Minimize the gross take-off weight: W TO W W W W (2.1) E PS ECS FUEL 2. Minimize the exergy destruction in the PS and ECS plus exergy lost due to unburned fuel loss: Ex obj 2 Ex DEStot _ PS Ex DEStot _ ECS Ex FuelLoss (2.2) 3. Minimize the exergy destruction in the PS, ECS, and AFS-A plus exergy loss due to unburned fuel loss: Ex obj 2 Ex DEStot _ PS Ex DEStot _ ECS Ex DEStot _ AFS A Ex FuelLoss (2.3) 4. Maximize thrust efficiency Wthrust thrust (2.4) m LHV fuel fuel 29

46 where 5. Maximize thermodynamic efficiency thermo W W thrust max ExDEStot _ PS ExDEStot _ ECS ExFuelLoss 1 (2.5) W 30 Max W max is the maximum work rate the PS could provide if no other sources of losses existed in the aircraft (an ideal system) or W Max W Thrust Ex DESTot _ PS Ex DESTot _ ECS Ex FuelLoss (2.6) Note that there is no AFS-A term included for the exergy destruction; however, it is included in a second set of optimizations which include not only these AFS-A losses but AFS-A DOF as well. These optimizations are discussed later in this section. The PS used in all of these optimizations is a low-bypass afterburning turbofan. The model equations are based on Mattingly, Heiser, and Daley (1987). Both the AFS-A and the ECS are based on Rancruel (2003) and Muñoz (2000). All modeling for the AFS-A and PS is done using the gproms dynamic modeling environment. The ECS is modeled using a C code coupled to the gproms models for the other two subsystems. gproms is also used for all optimizations. It uses a gradient-based approach, i.e. a sequential quadratic programming (SQP) algorithm developed for mixed integer nonlinear programming (MINLP) problems. The PS and ECS used by Periannan (2005) and Periannan, von Spakovsky, and Moorhouse (2008) are described at length in Chapter 3 of the present thesis so their descriptions will not be repeated here. The AFS-A model equations are similar to those described in Chapter 3. The only source of irreversibility modeled in the AFS-A is the parasitic drag or zero-lift drag given by: Ex DEStot _ AFS T0 DParasiticu (2.7) A T where T 0 is the dead state temperature (set to the sea-level ambient temperature), parasitic drag, u is the aircraft velocity, and T is the ambient temperature of the aircraft. D Parasitic is the Prior to the optimizations, a parametric exergy analysis is performed on three components in the PS to determine the effect of specific decision variables on the exergy-based objective functions. The compressor pressure ratio, fan bypass ratio, and turbine inlet temperatures are varied to see their effect on the exergy destruction rate, specific thrust, and specific fuel consumption (SFC). Four different mission segments are examined including

47 warm-up/takeoff acceleration (mission segments 1 and 2), climb (mission segment 5), and supersonic penetration (mission segment 8). Only the parametric study for the supersonic penetration mission segment is presented here. Figure 2.7 shows the effects on the PS exergy destruction rate and specific thrust with the variation in the three PS design decision variables. Figure 2.7 Variation of vehicle specific thrust and exergy destruction rate with fan bypass ratio and turbine inlet temperature for a fixed compressor pressure ratio of 8 for the supersonic penetration mission segment (Periannan, 2005). Figure 2.7 proves to be very informative as to the behavior of the PS with respect to the exergy-based objectives. Notice that the exergy destruction rate for a given specific thrust (and compressor ration) generally decreases with decreasing turbine inlet temperature and bypass ratio. The trade-off that this implies is that the lower bypass ratios result in better, more efficient PS designs provided the turbine inlet temperature is lowered as well. Furthermore, mission segment 8, the turbine inlet temperature has less of an effect on exergy destruction than the bypass ratio. This, however, may not be the case for other mission segments as shown in Periannan (2005). Figure 2.8 shows the response of the exergy destruction rate and specific thrust to changes in compressor pressure ratio and fan bypass ratio. The fixed parameter is the turbine inlet temperature, which is set to 1700 K. It can clearly be seen that a higher pressure ratio is beneficial for the overall performance of the PS since the highest pressure ratio results in the highest specific thrust and lowest exergy destruction rate. Furthermore, the lower the losses (i.e. the rate of exergy destruction), the smaller the vehicle with higher specific thrust, which has the consequence of reducing the cost of the aircraft and the total fuel consumption. 31

48 Figure 2.8 Variation of vehicle specific thrust and exergy destruction rate with fan bypass ratio and compressor pressure ratio for a fixed turbine inlet temperature of 1700 K for the supersonic penetration mission segment (Periannan, 2005). Finally, Figure 2.9 shows the effects on the PS exergy destruction rate and specific fuel consumption variations with bypass ratio and compressor pressure ratio. The obvious conclusion from this figure is that the highest pressure ratio and lowest bypass ratio produce the best performing PS. However, the requirements for thrust may demand that more airflow be moved by increasing the bypass ratio above the minimum constraint. In addition, note that at higher compressor pressure ratios, the effects of bypass ratio on the exergy destruction rate and specific fuel consumption is less than at lower pressure ratios. Upon completion of the exergy analysis, Periannan (2005) and Periannan, von Spakovsky, and Moorhouse (2008) return to the comparison of the optimization results for the various objectives. As mentioned above, objectives 1, 2, 4, and 5 are compared to show which, if any, of the objective functions yield a better overall vehicle. Recall that for this set of optimizations only the PS and ECS have DOF. Figure 2.10 shows the optimum gross takeoff weight yielded for each of the objective functions. All three runs for each objective function are shown to give some confidence that a global optimum was found. The first optimum for each objective function being higher than the subsequent two runs indicates that for the first optimization of each objective that a local optima may have been found rather than the global 32

49 Figure 2.9 Variation of vehicle specific fuel consumption and exergy destruction rate with fan bypass ratio and compressor pressure ratio for a fixed turbine inlet temperature of 1700 K for the supersonic penetration mission segment (Periannan, 2005). optimum. However, the second two runs being nearly identical indicate that the global optimum was likely found for all four objective functions. The result that none of the objective functions yield a better aircraft is due to the fact that although the AFS-A participates in the optimization, it has no DOF. In Butt (2005) and Brewer (2006), the thrust efficiency objective function consistently yields the poorest design in terms of performance and fuel usage, the one for a supersonic fighter and the other for a hypersonic vehicle. The difference in these optimizations is that AFS-A DOF are included. The next set of optimizations conducted in Periannan include AFS-A DOF. Only objective functions 1 and 3 are used. Using objective function 5 instead of 2 allows the inclusion of exergy losses (equation (2.7)) due to the AFS-A. Comparisons of the optimum results include those with and without AFS-A DOF. Figures 2.10 and 2.11 summarize these results. For example, the former shows that optimizing using objective 3 with AFS-A DOF reduces the gross takeoff weight by about 4.5% over the optimum aircraft found without AFS-A DOF and with objective 1. This difference is even more pronounced, as seen in Figure 2.11, where the fuel weight is reduced by 9.8%. Even in a comparison with the optimum vehicle found using objective 1 with AFS-A DOF, objective 3 with AFS-A DOF produces an optimum vehicle with a 5.8% reduction in fuel weight. 33

50 Gross Takeoff weight in kg With AFS-A DOF Without AFS-A DOF objective 1 objective 3 objective 2 Figure 2.10 Optimum gross takeoff weight with and without AFS-A DOF for objectives 1, 2, and 3 (Periannan, von Spakovsky and Moorhouse, 2008) Fuel Weight in kg With AFS-A DOF Without AFS-A DOF objective 1 objective 3 objective 2 Figure 2.11 Optimum fuel weight with and without AFS-A DOF for objectives 1, 2, and 3 (Periannan, von Spakovsky, and Moorhouse, 2008). Finally, although the work of Periannan (2005) and Periannan, von Spakovsky, and Moorhouse (2008) also show the strengths of the gradient-based optimization in terms of speed, a weakness of this type of algorithm is the need of generating several feasible but very different initial points with which to start the optimizations. In a complex large-scale optimization 34

51 problem with many DOF, this can be a rather daunting task, to say the least. Even a single starting point is problematic. The need for several very different ones is due to the susceptibility of such algorithms to getting stuck at local optima. Genetic or hybrid genetic-gradient based optimization algorithms are examples of how these difficulties can be overcome, and the following sections provide illustrations of the application of the former to a hypersonic vehicle using both exergy and non-exergy based objective functions. 2.5 Exergy Methods for the Development of High Performance Vehicle Concepts The renewed interest in hypersonic vehicles has fueled new design technologies to be employed in hypersonic vehicles. The work of Markell (2005) illustrates applying exergy analysis and exergy destruction minimization to the hypersonic vehicle. Markell developed a 1- D hypersonic vehicle model and a partial 3-segment hypersonic mission and a detailed exergy model for his thesis work. Two objective functions are compared in the work: an exergy based objective function and a more traditional objective function, namely, the thrust efficiency. The hypersonic vehicle consists of two subsystems, a propulsion subsystem and an airframe subsystem. The propulsion subsystem consists of a inlet, combustor, and nozzle component. Markell performed multiple optimizations on just the propulsion subsystem for a single mission segment and also optimized the entire vehicle for the partial three-segment mission. The 1-D hypersonic vehicle is shown in Figure Figure 2.12 Hypersonic vehicle configuration (Markell, 2005). The forebody serves as the means of compressing the incoming airflow with oblique shock waves forming at the leading edge of the vehicle. The inlet further compresses the incoming air with more oblique shocks forming off subsequent turning angles before entering the combustor / cowl area of the propulsion subsystem. The forebody and inlet design is extremely important in hypersonic vehicle flight due to no other means available for compressing the air 35

52 entering the combustor. The forebody design is also instrumental in maximizing the mass capture into the combustor as well. Figure 2.14 shows a diagram of the design variables associated with the forebody and inlet of the hypersonic vehicle. The operational decision variable shown in the figure is the angle of attack, or. Energy exchange is also included in the forebody and inlet model to modify the flow to maintain shock-on-lip conditions which maximizes the mass capture area into the combustor when the forebody and inlet must operate at off-design conditions. Figure 2.13 A physical representation of the forebody and inlet component of the hypersonic vehicle along with design and operational decision variables that govern the flow characteristics throughout the inlet (Markell, 2005). The exergy model for the hypersonic vehicle is somewhat involved, as losses are tracked for friction, shock waves, combustion, and mixing throughout the vehicle. The inlet exergy destruction has two contributors to total exergy destruction: friction and shocks. The entropy rate due to friction in the inlet is given by: S irr fric m Pt R ln Pt 2 1 where m is the mass flow rate, R is the specific gas constant, and P t2 Pt 1 is the total pressure ratio in the inlet. The entropy generation rate due to the three oblique shocks in the inlet is expressed by: S irr i 1 shocks 13 Ti1 Pti1 c pi ln R ln (2.9) m 3 Ti Pti where the three oblique shock temperature and pressure properties are required to find each oblique shock entropy rate. The total irreversibility rate for the inlet then becomes: 36 (2.8)

53 S S S (2.10) irrinlet irrfric irrshocks13 The combustor is a constant area, hydrogen-fueled combustor in which the flow is constrained to stay supersonic, defining a scramjet-type propulsion subsystem. The flow can become subsonic if the combustor length is too long, inlet area is too small, or the initial combustor flow Mach number is too low (designated M 4 in Figure 2.14). The mixing equations account for incomplete combustion but do not include species dissociation due to the computational burden required to track dissociation. Heat loss from the combustor through the combustor walls is also tracked using the Eckert Reference Enthalpy Method (Heiser and Pratt, 1994). The total irreversibility rate for the combustor is then S irrcomb S S S S (2.11) irr fric irr ht irr inccomb irr mix where the terms to the right of the equals are the entropy generation rates due to irreversible losses resulting from friction, heat transfer, incomplete combustion, and mixing, respectively. The nozzle heat losses are modeled using the Eckert Reference Enthalpy Method (Heiser and Pratt, 1994), with an adjustment to the overall heat transfer rate, as it is known to over predict the nozzle heat transfer rate. Half the heat transfer rate calculated by the Eckert method is used to account for the nozzle plume separating from the vehicle in flight as suggested by Riggins (2003). The frictional losses are calculated with a skin friction coefficient suggested by Riggins (2004). Including the mixing and heat transfer in the nozzle, a control volume of the nozzle an entropy generation of: where S irrnozz m mix j 1 y j s j7 j 1 y j s j6 P Q 7 wall R log P (2.12) 6 Tw y j and s j indicate the mole fraction and entropy of constituent j and the subscripts 6 and 7 indicate the nozzle entrance and exit conditions, respectively. The airframe subsystem-aerodynamics (AFS-A) is modeled to account for pressure wave forces, center of gravity, as well as frictional forces from skin friction. A simple diamond-airfoil wing and elevons are also modeled with the AFS-A to provide additional lifting surface and to balance vehicle moments during flight. Shock expansion theory is employed to calculate the drag and lift of the diamond airfoils (Anderson, 2001). The total drag on the vehicle is converted 37

54 to a frictional force, F f, on the AFS-A which yields the total entropy generation rate due to friction, i.e. F S fric u f irr (2.13) Tt where u is the speed of the aircraft and T t the local skin temperature. The total AFS-A entropy generation rate including shock losses is: S S S (2.14) irraero irrfric irrshocks13 The combustor is initially optimized for a single mission segment to compare to a published hypersonic combustor model from Riggins (1997). The model in Riggins (1997) was presented for a single mission segment optimization, so it was necessary to only optimize a single mission segment for validation. The results show good agreement (see Table 2.5), so the combustor model is considered validated. The model is then used to develop a number of parametric studies which are not repeated here. The x(m) in Table 2.5 represents the station in the vehicle. Zero meters is the inlet entrance, five meters is the combustor entrance, six meters is the nozzle entrance, and eleven meter is the engine exit. M, T, P, and u are the Mach number, temperature, pressure, and velocity of the flow in the vehicle, respectively. Table 2.5 Comparison of the optimal combustor models (Markell, 2005). Riggins Model (1996) Markell (2005) x (m) M T (K) P (N/m 2 ) u (m/s) M T (K) P (N/m 2 ) u (m/s) Next the optimum scramjet engine is determined for a fixed thrust based on two different objectives. The first is an energy based figure of merit called overall the overall efficiency, which is defined as Tu0 m where 0, T, u 0, 0 th p f h (2.15) pr m f, h pr, th, and p are the overall efficiency, engine thrust, vehicle velocity, fuel mass flow rate, fuel enthalpy, and thermodynamic and propulsive efficiencies, respectively. 38

55 The second objective function is the exergy based optimization objective function that uses concepts from both the 1 st and 2 nd laws of thermodynamics. This objective function is defined as T0Sirr 1 (2.16) T Qrelease 0 1 Tt where is the thermodynamic efficiency, T 0 is the dead state temperature (see Gyftopolous and Beretta, 1991), S irr is the (irreversible) entropy generation within the system, Qrelease is the heat loss to the environment, and T t is the temperature of the system at the point of heat loss. The results for these two optimizations are shown in Table 2.6. The two different objective functions produce nearly the same optimum scramjet engine in that the fuel mass consumed m f, is nearly identical between the two optima. The variables in Table 2.6 are shown on the hypersonic vehicle in Figure 2.4. Table 2.6 Optimal decision variable values for the energy and exergy based optimizations of a scramjet engine with fixed thrust (Markell, 2005). Obj. Function X fb (m) X cowl (m) X ramp (m) fb ( ) ) nozz( Obj Obj (m) % m f ( kg / s) 2 ( m ) L comb cowl Obj Obj veh The partial mission flown consists of three segments: a Mach 8 cruise for 1000 nm, acceleration and climb from Mach 8 to Mach 10 in less than 90 seconds, and finally a Mach 10 cruise for 1000 nm. Initially, each mission segment was optimized individually to find the best hypersonic vehicle for a given mission segment based for each of three separate objectives: thrust efficiency, exergy destruction rate, and the exergy destruction rate plus the rate of the exergy loss. The results of these optimizations are shown in Table 2.7. Notice that the exergy destruction objective function produces combustor lengths at or near the minimum of 0.5 m. This is due to a deficiency in the second objective function which ignores the rate of fuel exergy lost out the back end of the engine. Nonetheless, the results for all three objectives show that the combustor is the largest source of exergy destruction. 39

56 Table 2.7 Optimal design decision variable values for the single segment optimizations (Markell, 2005). Mach # des Obj. Funct. X fb (m) X cowl (m) X ramp (m) fb ( ) () nozz() L comb (m) % cowl E x des E x des E x E x des 0 fuel loss E x des E x E x fuel loss The final comparison made in Markell (2005) is between optimizations to determine the optimal hypersonic vehicle that can fly all three mission segments of the partial mission optimally. The optimal operational decision variable values for the three mission segments are shown in Table 2.8 and each vehicle optimal fuel mass flow rate is compared in Table 2.9. The interesting result is that the third objective function produces a vehicle that flies at shallower angles of attack,, and has a lower fuel mass flow rate, m f, than either of the other two Table 2.8 Optimal operational decision variable values for the partial mission (Markell, 2005). Objective Function 2 () ( m ) S wing E x des E x E des x fuel loss Table 2.9 Optimal vehicle fuel mass flow rate comparison (Markell, 2005). Objective Function 0 E x E des x fuel loss optimizations. In Table 2.8 the optimal vehicle for the third objective function requires much less effective wing area, S wing, to fly the mission than that for the optimal vehicle based on the 40

57 first objective. In Table 2.9, the fuel mass flow rates required are compared and show that the third objective (one of the exergy-based objectives) yields a better performing hypersonic vehicle with an overall savings of 6.5% fuel over entire mission. The work of Markell lays the framework for Brewer (2006), who uses the 1-D hypersonic vehicle to fly an entire hypersonic mission. 2.6 Integrated Mission-Level Analysis and Optimization of High Performance Vehicle Concepts The work of Brewer (2006) involves the construction (in collaboration with Markell, 2005) of a 1-D hypersonic vehicle model and using a genetic algorithm (GA) to find the hypersonic vehicle configuration that flies a Mach 6 to Mach 10 flight envelope in the most efficient manner. All the hypersonic vehicle designs found to date in the literature are based on a single mission segment or flight condition. The vehicle design in Brewer (2006) is based on a mission which includes cruise, acceleration / climb, deceleration / descend, and turn mission segments. This thesis work furthermore includes a comparison between three objectives: thrust efficiency maximization (the traditional propulsion subsystem design optimization objective), minimization of fuel mass consumption (the traditional weight-based design optimization objective), and the minimization of exergy destruction plus fuel exergy loss (the non-traditional design optimization objective). As in Markell (2005), the 1-D hypersonic vehicle model consists of two subsystems: the propulsion and airframe subsystems. Irreversible loss mechanisms modeled to account for exergy destruction include losses due to shocks, friction, heat transfer, mixing, and incomplete combustion. The airframe modeling includes trim and force accounting, while the inlet includes the modeling of energy addition of subtraction to or from the flow in order to maintain shock-onlip operating conditions at all operating points. The hypersonic vehicle is shown in Figure 2.5. The mission flown during the vehicle design optimizations is shown in Figure 2.16 and the corresponding flight details are given in Table The hypersonic vehicle has a total of seven design decision variables including forebody position, position, X ramp 1, forebody angle, fb, combustor length, L comb X fb, cowl position, X cowl, ramp 1, nozzle expansion angle, nozz, 41

58 Figure 2.14 Propulsion subsystem components and airframe subsystem (Brewer, 2006). Table 2.10 Mission segment details (Brewer, 2006). Segment Description Accelerate and climb from Mach 6 (at 23.2 km) to Mach 1 8 (at 26.9 km), t = 90 sec 2 Mach 8 cruise for 600 sec Accelerate and climb from Mach 8 (at 26.9 km) to Mach 3 10 (at 30.0 km), t = 90 sec 4 Mach 10 cruise for 600 sec 5 Perform a 180, 2g sustained turn at Mach 10 6 Descend and decelerate to Mach 6 Figure 2.15 Total scramjet vehicle mission (Brewer, 2006). and percent nozzle length, % nozz. The single operational decision variable is the angle of attack of the vehicle, 0. The design and operational variables for the hypersonic vehicle are given in Table

59 The first optimization objective used is maximizing thrust efficiency. The overall mission thrust efficiency is found from a weighted average of each mission segments thrust efficiency, or 5 m f ti i i i 5 (2.15) m t i0 fi i Table 2.11 Mission design and operational decision variables for the inlet, nozzle, combustor, and airframe (Brewer, 2006). where m fi is the fuel mass flow rate, t i is the segment time, and i is the mission segment thrust efficiency. The second optimization objective function used is minimize fuel mass burned which is defined as 5 W g m t. (2.16) fuel io fi i where g is the gravitational constant. The final optimization objective function is that of minimizing the exergy destruction plus exergy lost from unburned fuel in the combustor and is defined as Ex Des Ex loss 5 T S Ex t i0 0, i (2.17) irr, total, i loss, i where S irr, total, i is the total rate of entropy generated by the vehicle for the mission segment, i, while is the rate of exergy loss due to unburned fuel in segment i. The irreversibilities E x loss, i i 43

60 included in the total rate of entropy generated are those for the inlet, combustor, nozzle, and airframe. The vehicle was optimized using a genetic algorithm (GA) Queuing Multi-Objective Optimizer (QMOO) (Leyland, 2002). This type of algorithm is more suited to a hypersonic vehicle optimization problem than a gradient-based optimizer due to the nature of the problem: highly constrained, mixed-integer variables, and non-linear spaces in the solution space would cause a gradient based method to often get stuck in local optima. There is also the difficulty of finding one or more initial points in such a highly constrained problem with a gradient based approach. QMOO was developed by Leyland and Molyneaux and Laboratoire d Energetique Industrielle (Laboratory of Industrial Energy Systems, LENI) at the Ecole Polytechnique Federale de Lausanne (EPFL). Both the hypersonic vehicle and QMOO are developed in MATLAB enabling a straightforward coupling of the GA and the model. Due to the random nature of the GAs, multiple optimizations are needed to establish that the algorithm has indeed found the global optimum instead of a local optimum. This problem, however, is not exclusive to GA optimizations since many gradient-based methods require multiple runs as well to verify that a true optimum (whether it be the global or best local optimum) has been found. The model validation is performed by comparing against previously published results (Riggins, 1996; Bowcutt, 1992; and Starkey, 2004). The model results of Brewer (2006) are all well within reasonable tolerances of those reported in these references. Note also, that at the time of publication, there was no known documentation of mission-level optimization results for a hypersonic aircraft in the literature and very little for single-segment hypersonic optimizations. After validation, the optimizations for the three different objective functions are run. The hypersonic mission proves to be a very difficult problem to solve, as the optimizations took up to two months to complete, running constantly from start to finish on personal computers with processors ranging in speeds from Pentium III equivalent clock speeds of 1.5 to 3.05 Ghz with 512 to 1024 MB of internal memory. The computational cost is due to the sparse optimal solution space for the hypersonic mission. This is in contrast to an easier mission to fly or a system that has a dense optimal solution space. A random sample of the solutions for a hypersonic aircraft mission versus a morphing-wing supersonic aircraft mission (Butt, 2005) is shown in Table The sparse optimal solution space only shows two feasible solutions in the 44

61 random sample, while the dense solution set shows feasible solutions in every sample space shown. The average number of feasible solutions in the 125-member population for the hypersonic optimization is just 10%, while the average number of feasible solutions in the 70- member population for the morphing-wing optimization is nearly 100%. The sparse optimal solution space requires the constraints of the variables shown in Table 2.11 be restricted further to allow the optimum solution to be found in a more timely manner (1 2 weeks). The initial forebody angle is found by the GA (QMOO), while the rest of the angles Table 2.12 Samples of results populations for sparse and dense optimal solution spaces. Note that the very large numbers (i.e. E+15) represent infeasible solutions (Brewer, 2006). Sparse Optimal Solution Space Min. Exergy Destruction Plus Fuel Exergy Loss Population Sample [GJ] (Brewer, 2006) E E E E E E E E E E E E E E E E E E E E E+15 Dense Optimal Solution Space Min Fuel Usage Population Sample [kg] (Butt, 2005) E E E E E E E E E E E E E E E E E E E E E+02 on the hypersonic body are found by iteratively trying different ramp angles until a feasible solution allows progression to the next angle on the forebody. This method, although very computationally expensive, is still faster than simply allowing the GA to find a solution by setting all the ramp angles to optimization decision variables as was attempted by Muñoz. Brewer discovered that one of the weaknesses exhibited by QMOO was its inability to suppress significant digits assigned to decision variables. A large amount of computational time is wasted due to the fact that for each decision variable, QMOO would only vary the last 12 to 15 significant digits of the decision variable from one generation to the next (unless the decision variable is flagged as an integer, of course). Any decision variable that is not defined as an integer automatically has 15 significant digits and Brewer could not find a way to modify this easily within the MATLAB QMOO code. Despite the large amount of initial work and required tailoring of the design/operational decision variable constraints, Brewer is able to attain results for all three objective functions. These results are given in Table As can be seen from the table, the second and third 45

62 Table 2.13 Optimal objective function results (Brewer, 2006). Objective Min. Exergy Destroyed Max. Thrust Efficiency Min. Fuel Mass Function + Fuel Exergy Lost Run Thrust Efficiency [%] Fuel Mass [kg] Exergy Destruction [GJ] objective functions, minimize fuel mass and minimize exergy destroyed plus fuel exergy lost, respectively, yielded very similar optima. The explanation for the similarities can be found in one simple fact: the main source of exergy, or available energy, on the vehicle is found in the fuel. If the fuel mass required to fly the mission is minimized this is, in reality, minimizing the largest source of exergy destruction: fuel combustion. The thrust efficiency objective function, which is purely a first law of thermodynamics performance metric, yields a significantly worse performing vehicle in terms of fuel mass burned and exergy destruction than the other second law of thermodynamics based objective functions, despite having a higher thrust efficiency than the other two optimal vehicles. 46

63 Chapter 3 Model Description and Synthesis/Design Problem Description This chapter discusses the synthesis/design problem as well as the subsystem models used in this thesis work. 3.1 Problem Definition Solving the air-to-air fighter (AAF) synthesis/design optimization problem starts with developing appropriate subsystem models for the aircraft. Next, the mission is examined in detail to enable mission segments (or logical pieces of the mission) to be defined. Each subsystem is then prepared to fly the mission by defining the interactions with the other subsystems. Individual subsystem model convergence is checked by running optimizations on the decoupled or non-interacting subsystems before they are integrated. Finally, the subsystems are integrated and the coupling functions and optimization synthesis / design and operational decision variables are defined. Multiple optimizations for each different objective function and AFS-A configuration (i.e. morphing / fixed-wing) are then run to verify that a global optimum has been reached. The subsystems modeled in this thesis include the following: Airframe subsystem Aerodynamics (AFS-A); Propulsion Subsystem (PS); Environmental Controls Subsystem (ECS); Fuel Loop Subsystem (FLS); Vapor Compression / Polyalphaolefin Subsystem (VC/PAOS); Oil Loop Subsystem (OLS); Electrical Subsystem (ES); Central Hydraulic Subsystem (CHS); Flight Controls Subsystem (FCS). The fighter aircraft mission flown is shown conceptually in Figure 3.1. The details of the mission are derived from a request for proposal (RFP) found in Mattingly, Heiser, and Pratt (2002) and the mission segments are given in Table

64 Figure 3.1 Supersonic fighter aircraft mission from the RFP found in Mattingly, Heiser, and Pratt (2002). 3 Mission Segment 1 Table 3.1 Air-to-air fighter (AAF) mission segments and details. Description Warm-up and take off, 2000 ft altitude, 1 min for military power warm up, takeoff ground roll + 3 s rotation distance < 1500 ft, V 1.2 V TO 2 Accelerate to best subsonic climb Mach (BCLM) 3 Minimum time to climb to 20,000 ft at military power 4 Continue minimum time to climb to best cruise Mach (BCM1) and best cruise altitude (BCA1) 5 Subsonic cruise until total distance for climb/cruise is 150 nmi 6 Perform combat air patrol loiter for 20 min at 30,000 ft and best mach for endurance (BCM2) 7 Accelerate to Mach Accelerate to Mach Accelerate to Mach 1.5, total time for acceleration t < 50 s 10 STALL Supersonic Penetration at Mach 1.5 until total distance for accel + supersonic penetration is 100 nmi, supercruise if possible (no afterburning) 11 Combat segment: perform 360 degree, 5 g sustained turn at 30,000 ft, M = Combat segment: perform two 360 degree, 5 g sustained turns at 30,000 ft, M = Combat Segment: Accelerate from Mach 0.8 to Mach 1.0 in max power 14 Combat Segment: Accelerate from Mach 1.0 to Mach 1.2 in max power 15 Combat Segment: Accelerate from Mach 1.2 to Mach 1.6 in max power 16 Combat segment: drop payload of 2 AIM-9L s and 250 rds of 25mm ammunition (1309 lb) 17 Escape dash at M = 1.5 and 30,000 ft for 25 nmi, supercruise if possible 18 Climb/decelerate to BCM and BCA at military power, no distance credit 19 Subsonic cruise at best cruise mach (BCM3) and best cruise altitude (BCA2) until total distance is 150 nmi from escape dash 20 Loiter for 20 minutes at 10,000 ft and best mach for endurance (BCM4) 21 Descend and land 3 AIAA, reprinted with permission. 48

65 Note that each best cruise Mach or best cruise altitude for a given mission segment is unique to the other best cruise Mach or best cruise altitudes for the aircraft in different mission segments (e.g. BCM 1 BCM 2 and BCA1 BCA2 ). The subsequent sections in Chapter 3 detail the previously bulleted subsystems that comprise the AAF, starting with the AFS-A. 3.2 Airframe Subsystem The airframe subsystem-aerodynamics (AFS-A) is the largest subsystem in the AAF and serves not only as the structure required to house the other subsystems but also as the subsystem that produces lift and the aerodynamics required to fly the mission. The airframe houses all the subsystems mentioned in Chapter 3 as well as the rest of the items not detailed in the following sections. The items not detailed are accounted for simply by a fixed weight value. The AFS-A is developed from Raymer (2000) as well as Rancruel (2003) and is based on drag-polar relationships. The free-body diagram of the aircraft is shown in Figure 3.2. Analysis of Figure 3.2 and an energy balance will yield the master flight equation which is given in Table Figure 3.2 Free-body diagram of the aircraft (Rancruel, 2003) The clean drag term of the master flight equation, D, is the drag due to AFS-A aerodynamics. Additional drag from other subsystems (such as the FLS ram/air heat exchanger) is accounted for in the term, R. The thrust, T, is the installed thrust of the propulsion subsystem (PS) which is detailed in Section 3.3. The lift, L, and drag, D, terms are determined using the lift-drag polar relationship from Mattingly, Heiser and Daly (1987) and Raymer (2000) which is detailed in section The analysis of the mission follows the lift-drag discussion. The AFS-A weight equations are 49

66 presented in Section and, finally, the calculation of the takeoff weight completes the discussion of the AFS-A. Table 3.2 Master flight equation and governing flight equations. Component Variable Description Model Equations T Thrust D Clean Drag Master Flight 2 dh d V R Additional Drag T D RV W W Equation dt dt 2g V Aircraft Velocity W Aircraft instantaneous weight T Installed Thrust Governing 2 V T D R 1 d V i Velocity of aircraft at segment, i i Flight h W W V dt i i Equation h 2g i Altitude of aircraft at segment, i g Gravitational acceleration Lift and Drag The total drag on the aircraft is a combination of parasitic drag or zero-lift drag as well as drag due to lift or lift-induced drag. This section details the lift and drag relationships and equations that were used to develop the aerodynamics model of the AFS-A. In short, the drag model used is the component buildup method called the Parasite-Drag Buildup Method detailed in Raymer (2000). The supersonic wave drag is calculated using the Sears-Haack supersonic wave drag correlation. A summary of the general equations for the lift and drag relationships is given in Table 3.3. Notice that the total drag on the aircraft is a combination of the drag due directly to the performance characteristics of the aircraft as well as the additional drag from other subsystems shown in the master flight equation in Table 3.2. The wave drag is calculated using the Sears- Table 3.3 Lift and drag equations for the AFS-A. Component Variable Description Model Equations D Clean Drag q Dynamic Pressure D qsc D Clean Drag Wing S Reference where Area 2 Parasitic Drag C D K C Do Total Drag coefficient Lift factor Coefficient of parasitic drag C C D Do C C C Do KC Do _ wing Do _ misc 50 L C C Do _ fuselage Do _ wave C C Do _ tail Do _ canopy

67 Component Variable Description Model Equations D LE Ewd M 0.57 q wave 100 Coefficient Wave Drag C Do _ wave of shock wave drag D q C wave Do wave S 0.77 D q SearsHaack Lift-induced Drag C L Coefficient of lift Weight fraction of aircraft C L WTO qs Haack wave drag correlation shown in Table 3.3. The wave drag constant, E wd, is set to 2 in this thesis work in order to correspond to currently built fighter aircraft. The clean drag is the only drag accounted to the AFS-A in the exergy destruction calculations (see Section 3.2.5); however, other subsystem drag must be included in the master flight equation to accurately predict AFS-A performance Mission Analysis The mission consists of many dissimilar mission segments as well as a payload drop. Some important equations for the mission segments are given in Table 3.4 with a subsequent discussion of the individual mission segments. The first mission segment includes a 60 second warm-up at military power (full throttle in the main burner, no afterburning), then takeoff/acceleration, and finally, a ground roll for takeoff. The total distance for acceleration and ground roll is constrained to not exceed 1500 ft. The final aircraft speed at the end of the ground roll is set to 1.2 times the estimated stall speed, V stall, which is initially estimated in order to calculate the drag and lift characteristics. The actual stall speed is recalculated, and the actual drag and lift parameters are iterated until convergence to increase accuracy. The aircraft velocity at the end of segment 1 is 1.2 V stall. The second mission segment is a horizontal acceleration segment with a specified time of 30 seconds. The aircraft accelerates from 1.2 V stall to the best climb Mach (BCLM). The best 51

68 climb Mach is a degree of freedom (DOF) for the second mission segment in the AFS-A optimization problem. Table 3.4 Mission segment model equations. Component Variable Description Model Equations Warm-up weight W A f W A fraction A 1 SFC 60 Wi WTO V Stall speed Takeoff model stall C L max Takeoff lift coefficient V Velocity at end of B acceleration Takeoff acceleration B weight fraction V Velocity at end of C ground roll WTO Vstall 2g A CL max S V 1. 1 B V stall B exp g SFC 2 VB D R/ T 1 Final weight fraction for takeoff V 1. 2 c V stall C exp SFC V 50 2 C V 2 B D R/ T 2g W 1 Weight of aircraft after mission segment 1 W 1 1 W TO A B C G Glide ratio (vertical velocity/horizontal velocity) T W G 2 CDo Ae AR Aspect Ratio where for 30 LE Climb model e C Do b S Weissinger span efficiency Coefficient of parasitic drag Wing span Wing reference area e for 30 LE e AR b AR AR cos S LE Sustained turn model d x n Time to turn Number of turns Load factor for turn d g L D T W 2Vx 2 n 1 1 C Do nw S q n W S qae n L D 52

69 The third and fourth mission segments are sequential, constant Mach, minimum time-toclimb segments in military power from 2000 ft to the best cruise altitude (BCA1). The climb angle is determined by the excess thrust available from the PS. The horizontal distance traveled for the third and fourth mission segments is tracked because the total distance for segments three, four, and five is 150 nmi. Mission segment five is a cruise segment at best cruise Mach (BCM1) and altitude (BCA1). The BCM1 and BCA1 are AFS-A DOF. Mission segment six is a 20 minute loiter segment at 30,000 ft at best Mach for endurance (BCM2), which is also an AFS-A DOF. Mission segments seven, eight, and nine split up the acceleration from BCM2 to Mach 1.5: BCM2 to Mach 1.0, Mach 1.0 to Mach 1.2, and, finally, Mach 1.2 to Mach 1.5. Each of the three segments is time constrained to 15 seconds for a total acceleration time of 45 seconds. Mission segment ten is a supersonic penetration mission segment at Mach 1.5 that continues until the total distance traveled for segments seven, eight, nine, and ten is 100 nautical miles. Segment ten specifies to supercruise (supersonic cruise without afterburning) if possible. The combat simulation consists of a single 5-g sustained turn at Mach 1.6 and two 5-g sustained turns at Mach 0.9, acceleration from Mach 0.8 to Mach 1.6 at maximum power (full throttle for the main burner and afterburner), and a payload drop of 1309 lbs which is calculated from firing two AIM-9L s and 250 rounds of 25 mm ammunition. The combat simulation ends with the aircraft at Mach 1.6, and a supercruise (if possible) escape dash at 30,000 ft from 25 nautical miles follows the combat simulation. Mission segment eighteen is a deceleration / climb at military power to best cruise Mach, BCM3, and best cruise altitude, BCA2. This mission segment usually does not include a weight drop as the energy equation for this segment involves a trade of kinetic energy for potential energy. Mission segment nineteen is simply a 150 nautical mile cruise at BCM3 and BCA2. The final mission segment before landing is a 20 minute loiter / observation mission segment at best cruise mach and best cruise altitude until the total distance since the escape dash is 150 nautical miles. The descent and landing mission segment has a 6 minute time allowance with no weight drop for the descent. The final weight fraction to account for the landing, ground roll, braking and taxi is set to W f W i The weight fraction calculations are presented in the following section. 53

70 3.2.3 Weight Fraction Model The gross weight of the AAF decreases after each mission segment due to the burning of fuel in the PS. The gross weight decreases are tracked in each mission segment using weight fractions, or initial to final mission segment weight ratios that are calculated based on the aircraft energy change over the mission segment, specific fuel consumption (SFC), and flight conditions/aerodynamics. The mission segments are first categorized into mission segment type: for example, loiter, cruise, horizontal acceleration, constant speed climb, climb / acceleration, etc. The master energy equation for the weight fraction calculation is shown in Table 3.5 as well as the main derivatives of the master equation with a discussion following. Table 3.5 Weight fraction model equations. Component Variable Description Model Equations W f W i Final/initial weight ratio W 2 f W Master i sfc V sfc Specific fuel consumption i exp h weight W W i i1 V (1 u) 2g V Aircraft velocity fraction where equation g Gravitational acceleration u D R/ T i Weight ratio for segment i Climb h Altitude Wf sfc D Clean drag exp h Wi V (1 u) R Additional drag Acceleration T Installed thrust W 2 f sfc V exp W V u i (1 ) 2g Cruise S Range for cruise Wf sfc C exp D S W V C i L Loiter t Time for loiter Wf sfc exp 4CDK1 t W i V The weight fraction of the aircraft starts out at 1 and decreases throughout the mission until the aircraft lands. The aircraft weight is the product of each mission segment weight fraction, i, multiplied by the gross aircraft takeoff weight, W TO. Wing loading, W S, is a strong function of weight fraction, and the average weight of the aircraft is used to calculate the wing loading for each mission segment. The average weight fraction is iterated to increase accuracy when a large weight drop occurs in a mission segment. The takeoff weight calculations are detailed in the following section. 54

71 3.2.4 Calculation of W TO The gross takeoff weight, W TO, of this model is primarily developed from component weight estimates in Raymer (2000). The method of estimating fuel weight and empty weight from Mattingly, Heiser, and Pratt (2002) was initially used. However, this method proved to be somewhat slow and sometimes non-convergent for this application. The method used for this thesis is to allow gross takeoff weight to be a design decision variable and participate in the optimization. Using this method requires an initial fuel weight to be estimated to conceptually size some of the components. For this work, the initial fuel weight was estimated to be 40% of W T0. Each subsystem and component weight is then estimated to find the empty weight of the aircraft, W EMPTY. The actual fuel weight is found by subtracting W T0 from W EMPTY. If the aircraft runs out of fuel during the mission, the chosen W TO is simply thrown out and a new W TO is chosen along with a new set of decision variable values. The conceptual sizing method in Raymer (2000) details an iterative process to decrease the amount of fuel at take-off to the amount required to fly the mission to avoid taking off with excess fuel. This iteration is not used with the following justification: a lower W TO can be found by simply picking a better AFS-A design. This avoids spending computational time trying to make a likely worse solution improve by finding the actual amount of fuel the aircraft needs to fly the mission. Allowing the WT 0 to be a DOF forces the optimizer to find the takeoff weight and corresponding aircraft design and operational decision variable values that minimize the difference between fuel used for the mission and fuel at takeoff. Table 3.6 outlines the main subsystem weight calculations and the final W T 0 equation. Note that the Nomenclature section defines the variables that are used but not defined in Table 3.6. Table 3.6 Main subsystem weight calculations and W T0. Component Variable Description Model Equations W Wing Weight wing (lb) 0.5 W Horizontal Tail Wwing KdwKvs( Wdg Nz ) Sw h _ tail Weight (lb) W 2.0 AFS A F Wdg Nz W _ w W Main Landing main _ gear h tail Gear Weight (lb) B h 1000 W Nose Landing nose _ gear Gear Weight (lb) A S ht ( t / c) 0.4 root

72 Component Variable Description Model Equations W Fuselage weight fuselage Wmain _ gear KcbKtpg ( Wl Nl ) Lm (lb) W Inlet duct weight Wnoze _ gear ( Wl Nl ) Ln Nnw air _ induction (lb) W K W N L D W W PS W PS W FLS W FLS W ECS W ECS W VC / PAO W VC / PAO W CHS W CHS W OLS W OLS W FCS W FCS W ES Rated W TO W Engine mount engine_ mounts weight (lb) Gross takeoff weight Engine weight W fuselage air _ induction dwf 13.29K vg dg L Ls / Ld De z d K d W engine _ mounts N en TSL N z W 0.063( T ) M exp 0.81 (lb) PS SL Fuel Loop subsystem weight (lb) Environmental controls subsystem weight (lb) Vapor compresson/pao loop subsystem weight Central hydraulic subsystem weight Oil loop subsystem weight Flight controls subsystem weight System generating capability (kw, or kva) W W W W W FLS HX Fuel _ Tank ECS VCPAOS W HX W W HX Mechanical W W W Mechanical WCHS KvshNu (kg) W OLS Nen Scs s N en Mechanical Ducting WFCS M N Nc WES 172.2K mc Rated N c La N gen W Ducting WTO WAFS A WPS WECS WVCPAOS WFLS WECS WCHS WES WFCS W fuel Morphing-wing Considerations The AFS-A must account for morphing-wing technology, not only with respect to aerodynamic performance but also with respect to takeoff weight and in-flight weight fractions. In order to account for aerodynamics performance, five wing geometry parameters are allowed to vary over the mission: Aspect ratio, A ; 56

73 Wing length, L ; Wing sweep angle, LE ; Thickness to chord ratio, t c Taper ratio,. ; and The AFS-A morphing is treated in a quasi-stationary sense in that the wing geometry may change for each of the 21 mission segments, but the change is considered an instantaneous change (no time allowance for wing geometry change is considered). In other words, each of the mission segments has a unique wing configuration for the morphing AFS-A. Note that the payload drop (mission segment sixteen) has no time allowance, thus, the wing configuration for that segment has no effect on the aircraft aerodynamics or performance. The morphing-wing AFS-A has a total of 20 unique wing configurations for the 21 mission segments flown. The fixed-wing AFS-A has a single configuration for the entire mission and does not change wing geometry during the mission. The actuation cost and additional weight of adding morphing wings to the aircraft must also be taken into account in the morphing-wing aircraft. This is done by using wing weight and fuel penalties to account for the additional actuators required for morphing wings and the energy required to morph the wings, respectively. The wing weight penalties are factors that are multiplied by the equivalent fixed-wing weight and are added to the aircraft takeoff weight. For example, if an aircraft wing with a specified geometry weighs 2,000 lb and the morphing wing weight penalty is 2, the actual wing weight will be 4,000 lb. A difficulty in calculating the wing weight of a morphing-wing is picking the geometry at which to calculate the equivalent wing weight before multiplying by the wing weight penalty. Analysis of the wing weight equation in Table 3.6 indicates that the largest wing weight is yielded by using the following values of the morphing-wing geometry: Lowest aspect ratio, A ; Largest wing length, L ; Largest sweep angle, LE ; Smallest thickness to chord ratio, t c ; and 57

74 Largest taper ratio,. Using this method to establish the wing weight puts the morphing-wing aircraft at the largest disadvantage with respect to gross takeoff weight. Note that the work of Butt (2005) used a sweep angle of zero to establish the equivalent fixed-wing weight. Thus, the sensitivity study presented at the end of Butt (2005) should have larger wing penalties showing a greater advantage over the fixed-wing weight than in this thesis work. Consequently, the performance metric to compare the fixed-wing performance to the morphing-wing performance in this work will be the gross takeoff weight, W TO. The fuel weight penalties are used to account for the power required to morph the wings. After the aircraft takeoff weight and total fuel weight is established, the fuel weight penalties are multiplied by the total fuel weight to find the morphing fuel weight. The morphing fuel is reserved for morphing the wings and is not available to the PS to fly the mission. For example, if the aircraft has 10,000 lb of total fuel, and the fuel penalty is 25%, then there will be 2,500 lb of morphing fuel, leaving 7,500 lb of fuel available to fly the mission. The morphing fuel is expended over the mission at the same rate as the fuel penalty. For example, if the fuel penalty is 25% and the aircraft burns 1000 lb of fuel on a mission segment, 250 lb of morphing fuel is used as well (in addition to the 1000 lb used to fly the aircraft) to account for the energy required to morph the wings. Note that if excess fuel is sized at takeoff, then excess morphing fuel will also be sized at takeoff which implies that excess fuel will be carried throughout the mission. The aircraft weight fractions are updated at the end of each mission segment to reflect the morphing fuel utilized. The usage of the morphing fuel is not taken into account in the exergy destruction calculations since how exactly it is being used, i.e. the details of the actuators, is unknown at this design stage. The AFS-A exergy model is presented in the following subsection AFS-A Exergy Model The exergy destruction in the AFS-A is composed of two unique parts: the exergy destruction due to parasitic drag and that due to lift-induced drag (or simply induced drag). The parasitic drag and induced drag equations are shown in Section and the exergy destruction rate equations are given in Table

75 The exergy destruction in the AFS-A was tracked for every mission segment. The only mission segment that has zero exergy destruction is the instantaneous payload drop mission segment 16 (see Table 3.1). Table 3.7 AFS-A exergy destruction rate equations. Component Variable Description Model Equations E Exergy Exergy destruction rate due to x des Par parasitic drag T D V destruction Ex des 0 0 D Par 0 Zero lift or parasitic drag T rate due to amb q parasitic drag Dynamic pressure S where Dparasite qsc Wing reference area Do C Coefficient of parasitic drag Exergy destruction rate due to lift-induced drag Do E Exergy destruction rate due to lift x des Ind induced drag D i C L Drag due to lift or lift induced drag Coefficient of lift Lift factor Ex desind where 2 C D i KC L T0 DiV Tamb Di qsc Di Tightly coupled with the AFS-A is the propulsion subsystem which is detailed in the following section. 3.3 Propulsion Subsystem The propulsion subsystem (PS) used in this thesis work is a low bypass afterburning turbofan engine with the on- and off-design models based primarily on Mattingly, Heiser, and Pratt (2002). The following subsections detail the PS model used in this thesis work PS Layout and Station Definitions PS is made up of the following components: fan, high pressure compressor, burner / combustor, high pressure turbine, low pressure turbine, exhaust mixer, afterburner, and nozzle. The PS system layout and station definitions are shown in Figure 3.3. The station definitions must be clear to track properties and operating conditions through the PS. Thus, Figure 3.4 and Table 3.8 give further explanation and formally define the PS nomenclature. Notice in Figure 3.4 that the bleed air consumption is not shown. 59

76 Figure 3.3 Engine system layout (Rancruel, 2002). Bleed air Cooling air #2 1 Inlet 2 Fan 3 3 High pressure Compressor Cooling air #1 3a 4 Burner Coolant mixer 1 4a First rotor Last rotor Coolant mixer 2 4c Low pressure turbine 5 Mixer 6 After Burner Nozzle High pressure spool 5 Low pressure spool Figure 3.4 Engine Station Definitions (Periannan, 2005). The bleed air is used by the environmental control subsystem (ECS), which is detailed in Section 3.4 and Section 4.5. Table 3.8 Low-bypass turbofan engine station definitions. Station Description 0-1 Free stream to diffuser inlet 1-2 Diffuser 2-3 Fan entry to high pressure compressor entry 3 High pressure compressor exit 3 Fan exit to bypass duct 3a Burner entry 4 Burner exit 4a Coolant mixer 1 exit, high pressure turbine entry 4b High pressure turbine exit 4c Coolant mixer 2 exit 5 Low pressure turbine exit, mixer entry 5 Fan bypass duct exit to mixer entry 6 Exhaust mixer entry, afterburner entry 7 Afterburner exit, nozzle entry 60

77 Station Description 8 Exhaust nozzle throat 9 Exhaust nozzle exit The turbofan engine is a much more complicated engine than the turbojet used in Butt (2005) and requires flow and energy balancing through the engine and somewhat involved iterations to find the off-design performance. In short, the PS is modeled with a reference or design engine and each mission segment is considered as being off design at conditions at which the engine was not specifically designed to operate most efficiently. The thermodynamic model of the PS follows PS Thermodynamic Model The thermodynamic models and design equations of the PS components are detailed in this section. The off-design operation of the PS is used for every mission segment, while the design calculations are used to build a reference engine. The off-design simulation requires either an equation solver or iterative process. The latter is used for this thesis model. A comprehensive list and order of calculation of the off-design equations are not repeated here, but may be found in Mattingly, Heiser, and Pratt (2002) Free Stream and Diffuser The engine analysis must start at the free stream conditions. First, the known properties of the air are converted to total or stagnation properties. Temperature and pressure are first converted from standard atmospheric conditions to the actual conditions of the aircraft, which are corrected using tables and interpolation from Heiser and Pratt (1994). The equations and constants for freestream and diffuser entry and exit properties are shown in Table 3.9. Table 3.9 Diffuser and nozzle equations. Component Variable Description Model Equations Isentropic freestream temperature 1 2 r recovery ratio r 1 M 0 2 Isentropic freestream temperature r Freestream recovery ratio T t0 P t0 Total/stagnation temperature Total/stagnation pressure 1 r r Tt 0 T 0 r Pt 0 P 0 r 4 AIAA. The low-bypass turbofan equations are reprinted with permission. See the Mattingly, Heiser, and Pratt (2002) for the comprehensive list of turbofan equations. 61

78 Component Variable Description Model Equations d Total pressure ratio Pt 2 d = d max Rspec P d max Total pressure due to wall friction t0 d max 0.97 d Total temperature ratio Diffuser Tt 2 d =1 Tt 0 Rspec Ram recovery coefficient =1 for M 0 1 Nozzle n n Total pressure ratio Total temperature ratio Fan and High Pressure Compressor Rspec M Rspec for M 0 > 1 P n P T n T The fan and high pressure compressor components are driven by the low and high pressure turbines, respectively, via the low pressure and high pressure spools, respectively. The fan and high pressure compressor are used to compress the incoming air before combustion. The fan runs at a lower pressure ratio than the high pressure compressor, and typically spins slower than the high pressure spool as well. The fan and high pressure compressor calculations are given in Table t9 t7 t 9 Table 3.10 Fan and high pressure compressor equations. Component Variable Description Model Equations P ' T ' t3 t3 c' Total pressure ratio c', c' P T Fan c' c' W c' m c ' C Total temperature ratio Efficiency Power Corrected mass flow rate t2 c' 1 / 1 c' c' c'. c' 1 pc ' c' t7 t2 W m c T = m c m fanc m fan r r 1 T t 2 P t 1 = 1 T t3 Tt 2 Pt 2 d c' pc' d fan fan T t2 c' P t Pt ' c 1 c 1 Compressor ch Total pressure ratio ch 1 / ch ch ch 1 ch Total temperature ratio ch Efficiency P t3 ' ch, P t3' ch ch 1 T T T t3' P t 1 = 1 T t3' Tt 3 Pt 3 t3 t3' 3' 62

79 Component Variable Description Model Equations W ch Power T ' t3 Pt 3 W. ch m ch cpch T m chcpch ch ' P t3 m chc Corrected mass flow rate r d c' ch m chc m ch c Total pressure ratio for the compressor section c c' ch r d c' ch 1 ch ch Main Burner and Afterburner The fuel is added to the airstream in the main burner and afterburner (if being used) and ignited. Note that the main burner only adds fuel to the core air flow, while the afterburner adds fuel to both the core and bypass air as both streams are combined in the mixer before entering the afterburner section. Fixed efficiencies and pressure ratios are used for both the main burner and afterburner. The equations are given in Table Table 3.11 Burner and afterburner calculations. Component Variable Description Model Equations m burn Efficiency 4C pttt 4 m3ac pct3a burn m fabhpr burn Total pressure ratio Pt burn 4 Tt , burn Pt 3a Tt 3a Burner Total temperature burn ratio r ' c ch f f Fuel/air ratio hprb / C pct 0 C pttt 4 Enthalpy ratio C pct0 AB Efficiency m7c pabtt7 m6c pmtt6 AB m fabhpr AB Total pressure ratio Pt AB 7 Tt7 0.97, AB Pt 6 Tt6 Afterburner Total temperature AB ratio 1 AB m th f AB 1 f f 1 hpr AB / AB Fuel/air ratio C pabtt 7 AB AB Enthalpy ratio C pct0 m2 tl M C pm / Cpt C T pc 0 AB 63

80 High and Low Pressure Turbines The high and low pressure turbines allow expansion and work extraction in the core of the engine. The high pressure turbine vanes are cooled by coolant mixers with bleed air extracted from the fan. The equations describing the high and low pressure turbine behavior are given in Table Table 3.12 High and low pressure turbine equations. Component Variable Description Model Equations High Pressure Turbine Low pressure Turbine th Total pressure ratio th Total temperature ratio th Efficiency m thc Corrected mass flow rate th th 1 th, t 1 t 1 1 m thc m th th th r ' ch 1 c 1 1 P P t4b t4 f mh r ' ch c r r 1, tl tl Total pressure ratio tl t 1 t 1 tl tl Total temperature ratio tl Efficiency m tlc Corrected mass flow rate tl 1 d d c' c' ch ch b m1 th tl b 1 a ' 1 r c m1 P P t5 t 4c C TO th / 1 1 f / ml th th r ' ch c CTO PTO / m0c pct ) ( Where, 0 m tlc m tl r r d d c' c' ch ch b m1 th b m1 th mp m2 m2 / / tl tl Coolant Mixers and Exhaust Mixer Turbine cooling is required to avoid exceeding material design temperature limits in the turbines in the PS (the limit imposed is 3200 R). The turbine cooling is performed in the coolant mixers with air bled off of the high pressure compressor. The amount of turbine cooling required depends on the temperature of the burner. After the turbine section, the bypass and core airstreams are combined in the mixer section of the turbofan engine. The mixed streams then enter the afterburner section. The equations for the coolant mixers and exhaust mixer are given in Table

81 65 Table 3.13 Turbine cooling mixer and exhaust mixer equations. Component Variable Description Model Equations Coolant Mixer 1 1, 2 Coolant mixer ratios 000 /16, t T where R T t otherwise cmix1 Total temperature ratio ' / 1 1 f f ch c r cmix cmix1 1 cmix1 Total pressure ratio Coolant Mixer 2 cmix2 Total temperature ratio ' } { 1 1 f f th cmix ch c r cmix cmix2 1 cmix2 Total pressure ratio Exhaust Mixer M Total temperature ratio ),, ( ),, ( ') (1 1 / ' ' ' ' ' ' R M MFP R M MFP A A P P T C T C C C T T f m m M t t M t p t p p p t t M M Total pressure ratio ' Mixer bypass ratio Thrust and Performance Calculations The overall performance equations of the PS are detailed in Table The uninstalled thrust is the engine thrust produced without any losses attributed to the engine cowl or nozzle drag. Table 3.14 Thrust and engine performance calculations. Component Variable Description Model Equations Overall Engine Performance 0 f Total fuel consumption 1 / AB f f f n AB M tl th b ch c d r t P P P P ' AB AB P P M t AB 8 8 / 1) ( P P C C T T t AB pab pc P P V V t r AB T g R M V c c c where c g lbmft/(lbfs 2 ) 9 9 P P t Total/static pressure ratio in nozzle 9 M Nozzle exit Mach 0 9 T T Overall temperature ratio 0 9 V V Overall velocity ratio 0 V Aircraft velocity (ft/s)

82 66 Component Variable Description Model Equations Thrust m 0 F Uninstalled specific thrust (lbf/lbm/s) / m F f S M P P T T V V R R f V V f g V m F c c AB c S Uninstalled specific fuel consumption (1/h) F Uninstalled thrust (lbf) Subsonic inlet drag is estimated by assuming a worst case scenario of massive flow separation at the lip of the inlet and no recovery of additive drag. Additive drag is defined as the positive drag acting on the streamtube which encloses the air that enters the engine inlet (Mattingly, Heiser, and Daley, 1987). Supersonic inlet drag is estimated somewhat conservatively using the idea of the inlet swallowing its projected image. This, in effect, means that the inlet area, 1 A, is larger or equal to the flow capture area, 0 A, and that the inlet must have the ability to vary its geometry. Also, the excess air captured by the inlet must be vented via boundary layer bleed ports. The excess air vented is at a lower velocity than the aircraft, thus, creating drag from the momentum loss. The supersonic inlet loss coefficient reflects this momentum loss drag. The inlet and nozzle drag and installed thrust equations are given in Table Table 3.15 Inlet and nozzle drag and installed thrust equations. Component Variable Description Model Equations Inlet Drag inlet Inlet loss coefficient For M 0 < 1.0: a M m Fg M A A M T T M M c inlet For M 0 > 1.0: a m Fg M M A A c inlet 0 M Freestream Mach 1 M Inlet Mach 0 A Area of engine capture streamtube 1 A Inlet area 0 M Freestream Mach 1 M Inlet Mach 0 T Freestream temperature 1 T Inlet temperature

83 Component Variable Description Model Equations For M 0 < 0.8: Nozzle loss nozzle C A coefficient 10 A M D A0 nozzle Fg c A 10 Nozzle inlet area m0a0 For 0.8 < M 0 < 1.2: Nozzle Drag A 9 Nozzle throat C A 10 A M DP a Freestream A0 0 speed of sound nozzle Fg Convergent c C m0a D nozzle pressure 0 drag coefficient For M 0 > 1.2: Experimental 2 CD M exp M0 C DP pressure drag C D coefficient M 0 1 Installed thrust Installed Thrust T T F 1 nozzle inlet (lb f ) The C D term and C DP terms in Table 3.15 are nozzle pressure drag coefficients derived using the integral mean slope (IMS) method discussed in Mattingly, Heiser, and Pratt (2002) PS Exergy Model The exergy model for the PS was developed on a component by component basis. This accounting structure provides a much more detailed picture of losses and inefficiencies which allows a designer to very quickly see areas needing improvement within the PS. The exergy destruction calculations are shown in Table Notice also that the unburned fuel is taken into account in the exergy destruction equations. Table 3.16 PS exergy model equations. Component Variable Description Model Equations Exergy T ' P ' Fan E x des _ fan destruction rate in the fan 3 3 E xdes _ fan T0m 2 C pc ln Rc ln T2 P2 High Pressure Compressor E x des _ ch Exergy destruction rate in the high pressure compressor E x des _ ch T 0 T 3 m 3 C pc ln T ' 3 P Rc ln P 3 3' 67

84 Component Variable Description Model Equations Burner E x des _ burner Exergy destruction rate in the burner E xdes _ burner 0.35 m fuel _ core h fuel High Pressure Turbine Low Pressure Turbine Mixer Afterburner Unburned E x des _ th E x des _ tl E x des _ mixer E x des _ AB E x loss _ unburn Exergy destruction rate in the high pressure turbine Exergy destruction rate in the low pressure turbine Exergy destruction rate in the mixer Exergy destruction rate in the afterburner Exergy loss rate due to the fuel lost out the back of the PS Ex Ex Ex des _ th des _ tl des _ mixer T m 0 T m 0 4a 4c r i1 C C pt pt m i C T ln T T ln T pm 4b 4a t5 4c P Rt ln P R ' tl T ln R T E xdes _ AB 0.35 m fuel _ AB h fuel Ex loss _ unburn b Ex des _ AB 4b 4 P ln P M t5 4c ' p ln R ln yi p 1 Ex 1 des _ burner AB The PS is not only closely coupled to the AFS-A but also to other aircraft subsystems. One of these is the environmental controls subsystem (ECS), which uses bleed air from the PS as the working fluid to perform its cabin and avionics cooling duties (see Figure 3.5). The ECS is detailed in the following section. 3.4 Environmental Controls Subsystem The environmental controls subsystem (ECS) is responsible for cooling the low-heat generation avionics boxes and keeping the cabin temperature and humidity at comfortable levels for the pilot. The system is a bootstrap system similar to that used in the F-16. The ECS consists of four compact heat exchangers, a water separator, and an air-cycle machine ECS Layout and Definitions The ECS flow rate and input air conditions are determined by two bleed ports located in the PS. The low pressure bleed port is located midway on the high pressure compressor in the PS, while the high pressure bleed port is located immediately after the high pressure compressor. Figure 3.5 shows the bootstrap type ECS along with the PS. The bleed air input pressure to the ECS is controlled via a pressure regulating valve (PRV). 68

85 Figure 3.5 ECS layout and components (Muñoz and von Spakovsky, 2001). Following the path of the bleed air flow from the PS shows that the bleed air is first cooled in the primary heat exchanger via ram air. A portion of the bleed air is then compressed and cooled first through the secondary heat exchanger, then through the bleed air/pao heat exchanger, and finally through the regenerative heat exchanger. The air is then expanded over the turbine that drives the compressor. The water separator removes the water from the air which finally makes it to the cabin and low-heat generating avionics (i.e. air-cooled avionics). The following subsections detail the ECS thermodynamic model, heat exchangers, and ECS exergy model ECS Thermodynamic Model The thermodynamic model of the ECS consists of multiple components. The ECS model is based primarily on the work of Muñoz and von Spakovsky (2001a,b), Periannan (2005), and Rancruel (2003). The station definitions may be seen in Figure 3.6, and the model equations for the ECS components are given in Table

86 Bootstrap Sub-system Preconditioning Sub-system Figure 3.6 ECS station definitions (Rancruel, 2002). Table 3.17 Thermodynamic model of the ECS (Periannan, 2005). Variable Description Model Equation T 9 Load thermodynamic temperature T9 Tload P 9 Load thermodynamic pressure P9 Pload T 8 Water separator inlet temperature T8 T9 P 8 Water separator inlet pressure P 8 P 9 Pws T T 7 6 t Turbine temperature ratio Turbine efficiency T T 1 tb P P Fv T 6 P 6 Turbine velocity factor Regenerative hot-side exit temperature. C min is the smallest of the heat capacities C 5 and C 8. Also, rhx is the heat exchanger effectiveness. Regenerative heat exchanger hot-side exit pressure. A correlation is used for the pressure drop. T N Fv Y 4028 Tin Y 1 1 Y PR 1 6 T 5 rhx C T5 T8' C P P6 1 P5 P 5 min 5 70

87 Variable Description Model Equation T 5 P 5 Bleed air / hot PAOS heat exchanger hot-side exit temperature. C min is the smallest of the heat capacities C 4 and C 14. Bleed air / hot PAOS heat exchanger hot-side exit pressure. A correlation is used for the pressure drop. 5 T4' bleed / pao _ rhx T4' T14 T P5 1 P4 ' P 4' P C C min 4' T 4 ' Secondary regenerative heat exchanger hot-side exit temperature. C min is the smallest of the heat capacities C 4 and C 10. T 4' T 4 sec ond _ hx T T 4 o C C min 4 P 4 ' Secondary regenerative heat exchanger hot-side exit pressure. A correlation is used for the pressure drop. P 4 1 P P 4 P ' 4 T T 4 3 c Compressor temperature ratio Compressor efficiency T T cp P 4 P3 1 1 w shaft Shaft work of the compressor and turbine w shaft h4 h3 h6 h7 T 3 P 3 Primary heat exchanger hot-side exit temperature. C min is the smallest of the heat capacities C 2 and C 12. Primary heat exchanger hot-side exit pressure. A correlation is used for the pressure drop. T 3 T 2 pri _ hx T T 2 P P3 1 P P 2 o 2 C C min 2 k Ram inlet scoop mass flow ratio. i u i ku r f T i Ram inlet scoop pressure recovery factor Ram scoop inlet temperature ( i equals 10 or 12 and T i is the temperature at sonic conditions; see Anderson, 1984). T T i i Po i 2 rf Po 1 2 1M i P i P oi Ram scoop inlet pressure ( i equals 10 or 12 and P i is the pressure at sonic conditions; see Bejan (1996). Ram scoop inlet stagnation pressure ( i equals 10 or 12 and P oi is the stagnation pressure at sonic conditions; see Davenport (1983). P P i i P io oi P 1 1 M i 2 ( 1)M M 2 i ( 1) M 2( 1 ) i i 1 71

88 Variable Description Model Equation 4 fl D i Ram scoop inlet augmented friction factor; f assumed equal to 0.01 and L and D are the length and the diameter of the ram air duct, respectively, see Davenport (1983). 4 fl D i 1 M M 2 i 2 i 2 1 ( 1)M i ln 2 2 ( 1) M 2 i D drag Drag due to the presence of the ram air inlet and outlet (e). Pressure drag has been ignored. D m u drag u e P 11or 13 Ram air pressure just ahead of the ram scoop P P 11or 13 1 P 10 or 12 exit (states 11 or 13). P 10 or Heat Exchangers The heat exchanger thermodynamic models used in this thesis work are primarily from Shah (1981) and Kays and London (1984). The models are compact, offset-strip fin type heat exchangers and focus on the liquid/air heat exchange, although the thermodynamic model may be applied to any general heat exchanger with the proper derivation from the general case. The geometric parameters of the offset-strip fin are shown in Figure 3.7. a t s h=b Figure 3.7 Geometric parameters of the offset-strip fin (Muñoz and von Spakovsky, 1999). The weights of the heat exchangers are found using the fin geometry and density. Table 3.18 shows the compact heat exchanger model equations. Table 3.18 Geometric and heat transfer models of the compact heat exchangers. Variable Description Model Equation L h Hot-side length Assigned value L c Cold-side length Assigned value 72

89 Variable Description Model Equation D h Hydraulic diameter A fr Frontal area Afr L Ln D h 2shl sl hl th Ratio of minimum free flow area to frontal area A A O fr Heat transfer area / volume between plates 4 Dh A A f t Fin area / total area A A f t s h 2 s h N p Number of plates N p n p L h n b hb 2a h 2a r V p b V p r Volume between plates, bleed and ram air side V p b V r L p L L ( N 1) h b b L r r N p p h b b A Heat transfer area A Vp A O G Minimum free flow area Mass velocities A O G Dh A 4L m A O R e f j f j Reynolds number Friction and Colburn coefficients (Muñoz and von Spakovsky, 1999) Friction and Colburn coefficients (Muñoz and von Spakovsky, 1999) G D Re For h Re R e (laminar flow) f j R e R e For 1000 e R e R (turbulent flow): f j R e R e A Wall conduction area A L L 2( N ) w R w Wall thermal resistance w b r p 1 a Rw k A b l Fin length l t 2 H Height H b 2a n b b a w w h plates h c 2 73

90 Variable Description Model Equation A f Finned area A fr Frontal area f o n j G Pr h Fin efficiency Outside overall surface efficiency Mixture molar flow rate Colburn factor Maximum mass velocity Prandtl number Heat transfer coefficient 1 2h 2 m, A fr L H kt f tanh( ml) ml A, f 1 1 A m G, A O jgc p h 2 P 3 r c Pr k O f p U Overall heat transfer c h coefficient Cmin minn cc p, n hc p C maxn c, n c C Minimum heat capacity min C Maximum heat capacity max C r NTU Heat capacity ratio Number of transfer units max c c p w OhAb O ha r h h p R, Cr Cmin / Cmax UA NTU Effectiveness 1 exp NTU exp C NTU P The core pressure drop UA C min r 1 Cr 2 P G P 2 2 i K c e i P A i 2 f 1 K e Ao m i e The ECS exergy model is discussed in the following subsection ECS Exergy Model The ECS exergy model losses are primarily due to the heat exchangers and air cycle machine (consisting of a turbine/compressor) losses. No exergy destruction is attributed to the conditioned air exiting the cockpit and avionics after it is used. Any excess bleed air from the PS is considered to be negligible. The exergy destruction rate equations for the ECS heat exchangers are shown in Table

91 Table 3.19 ECS exergy destruction rate equations. Component Variable Description Model Equations Cold Side P Ex DES cold Exergy destruction rate on the cold side of the heat exchanger due to the pressure drop E x m R T ln P / P P DES cold cold cold 0 incold outcold Hot Side P Ex DES hot Exergy destruction rate on the hot side of the heat exchanger due to the pressure drop E x m R T ln P / P P DEShot hot hot 0 inhot outhot Temperature gradient T Ex DES Exergy destruction rate due to a temperature drop T DES Ex Touthot T dt T0 m hot cphot m cold T T1 inhot T outcold incold c Pcold dt T Total Exergy Destruction Rate E x DESHX Total exergy destruction rate in the heat exchangers Ex DES HX Ex T DES Ex P DES cold E x P DES hot The exergy destruction calculations for the compressor and turbine ECS components are found in Table 3.16 as one turbine or compressor is treated the same as another in terms of the exergy destruction. The ECS interacts with the PS via bleed air and the vapor compression / polyalphaolefin subsystem (VC/PAOS), which is detailed in the following section. 3.5 Vapor Compression / PAO Subsystem The vapor compression / PAO subsystem (VC/PAOS) consists of three loops: the cold PAO loop, the hot PAO loop, and the vapor cycle. The VC/PAOS serves as a heat sink for the ECS via a PAO / bleed air heat exchanger and for the high-heat generation avionics via the cold PAO loop. The hot PAO loop rejects heat to the fuel loop subsystem (FLS) and to a ram air / PAO heat exchanger. The motor driving the compressor on the vapor cycle and the pumps on the hot and cold PAO loops are all driven via electrical power from the electrical subsystem (ES). The VC/PAOS layout and station definitions are shown in Figure

92 Figure 3.8 VC/PAOS layout and station definitions (Rancruel, 2002) VC/PAOS Thermodynamic Model The thermodynamic models of the vapor cycle compressor and the PAO / ram air heat exchanger in the VC/PAOS are based on a perfect gas model. The load temperature (or the temperature of the Avionics Box in Figure 3.8) is given by Figliola, Tipton, and Ochterbeck (1997), and the physical model is from Greene (1992). The model equations for the VC/PAOS thermodynamic model are presented in Table Table 3.20 VCPAOS model equations. Component Variable Description Model Equations Initial Condition T load Load temperature of the high-heat generation avionics T10 T load 76

93 Component Variable Description Model Equations Heat transfer rate Q required by the Q Avio Avio T liquid cooled 11 T 10 m PAOC ppao avionics Cold PAO Loop Pump Evaporator Compressor Condenser Expansion Valve m PAO Mass flow rate in h09s h011 T09s T011 p the cold PAO loop h09 h011 T09 T011 W p Pump work rate C Heat capacity of the ppao PAO p Pump efficiency P Pressure rise across p the pump Q load Heating load due to the avionics box and the pump temperature rise Q Heating load on the Evap evaporator Evaporator Evap effectiveness m vap W c COP Vapor mass flow rate in the evaporator Compressor work rate Vapor compression cycle coefficient of performance Q Heat transfer rate in Evap the evaporator Q Heat transfer rate in Comp Cond Q Cond Pr val the compressor Condenser effectiveness Heat transfer rate in the condenser Pressure ratio of the expansion valve 77 W P c p c P Q Q h p load Evap P T09 T011 vp09 P11 vp09 P011 T T vp P m m P vap vap P ( h 9 ) 10 h 09 Evap P 011 Avio_ box P ( h8 h7 ) m pao ( h10 h9 ) C Q Evap min T9 T7 h m m 8 h7 Evap 7 Vap T T evap T T 10 9 P9 10 P10 1 P9 P 9 P7 8 P8 1 P7 P 7 T5 1 P 5 1 T 8 cp P8 W c m vap h 5 h 8 Q load COP W net Q Q Q Q h Cond Cond m Evap vap Comp C C 1 min 9 Vap pipe Evap ( h5 h6 ) m pao ( h12 h4 ) C Q min Cond T5 T4 h m m 6 h5 Evap 5 Vap T T evap T T 12 4 P6 5 P6 1 P P C C min P12 4 P12 1 P4 P 4 h assumes isenthalpic 06 h 07 c v ( T7 T6 ) v( P7 P6 ) Pr val Pr com Evap 0 P P 4 Cond Vap Cond

94 Component Variable Description Model Equations p Pump efficiency h01s h012 T01s T012 p W h01 h012 T01 T p Pump work rate 012 Hot PAO W p ct 01 T012 vp01 P012 Loop Pump vp01 P012 C Heat capacity of the ppao p PAO c T T v P P Bleed Air / Hot PAO Heat Exchanger Fuel / Hot PAO Heat Exchanger Ram Air / PAO Heat Exchanger bleed / pao _ hx fuel / pao_ hx ram / pao_ hx Bleed air / PAO heat exchanger effectiveness Fuel / PAO heat exchanger effectiveness Ram air / PAO heat exchanger effectiveness T 2 T 1 bleed / pao _ hx P P2 1 P1 P 1 T 3 T 2 P P3 1 P P 2 T 4 T 3 fuel / pao _ hx 2 ram / pao _ rhx P P4 1 P P 3 3 T T T T T T bleed _ in fuel _ in ram _ in C C C C C min C 2 1 min min 3 Note that the VC/PAOS ram air inlet has the same thermodynamic model as the inlet ducts in the ECS. The heat transfer model for the condenser and evaporator is from Liu and Kakac (2000). The compact heat exchanger model for effectiveness is from Incropera and DeWitt (1990) and applies to single-pass, cross-flow heat exchangers with unmixed fluids. The thermodynamic properties of the PAO are based on data from the CRC Handbook (1976); Zabransky et al. (1996); and the JANAF Thermochemical Tables (1998) VC/PAOS Exergy Model The exergy component models of the VC/PAOS are effectively the same as those of the ECS. No additional unique components are introduced in the VC/PAOS; and, thus, the exergy destruction equations are not repeated here. The VC/PAOS interfaces with a number of subsystems, including the ECS (cooling loads for the bleed air and liquid-cooled avionics), the ES (to power the pumps and compressor), and the FLS which serves as a heat sink for the hot PAO loop. This latter subsystem is discussed next in the following section. 78

95 3.6 Fuel Loop Subsystem The fuel loop subsystem (FLS) consists of a fuel tank, pumps, fuel lines, and controls required to supply fuel to the PS as well as the hardware necessary to use the fuel as a heat sink by the other subsystems. The FLS thermodynamic model is based on the work of Rancruel (2003) and Periannan (2005). The FLS system schematic is shown in Figure 3.9. Figure 3.9 Schematic of the fuel loop subsystem (Rancruel, 2002). The fuel from the fuel tank is first pressurized by the pump and then is heated by the hot PAO loop via a compact heat exchanger. The fuel is then passed through the fuel / oil heat exchanger where it is further heated by the engine cooling/lubricating oil in the oil loop subsystem (OLS). The fuel is interfaced with the central hydraulic subsystem (CHS) as a heat sink again before finally being burned in the PS. Notice that the fuel loop does not end at the PS but allows for excess fuel to be pumped through the loop if additional cooling capacity is needed 79

96 to meet the heating load requirements of the other subsystems. The additional fuel is cooled in the fuel / ram-air heat exchanger before it is returned to the fuel tank to avoid heating the fuel beyond acceptable limits FLS Thermodynamic Model The thermodynamic model of the FLS is similar to the ECS and VC/PAOS in that compact heat exchangers are also used. The model equations for the heat exchangers can be found in section The fuel tank temperature is not monitored as was done in Rancruel (2003), but rather a constraint is imposed on the allowable temperature of the excess fuel returning to the fuel tank. The FLS thermodynamic model equations are presented in Table Table 3.21 Fuel loop subsystem thermodynamic model equations. Component Variable Description Model Equations Mass flow rate m fuel _ out of fuel leaving the tank ( m C p) pao ( Τ pao _ in T pao _ out) m fuel _ out Fuel added to ( T pao _ add T pao _ out) m the fuel fuel _ add Cp T fuel out fuel _ T required by fuel / pao the PS Fuel tank T fuel _ add Temperature of the fuel m fuel _ add m fuel _ out SFC returning to the tank Q fuel / PAO _ HX m fuel C p ( T fuel _ out T fuel _ add ) SFC Specific fuel ( m C fuel / ram p ) min ( T fuel consumption T fuel _ add T fuel _ to _ PS (by the PS) C p m fuel _ Fuel / PAO heat exchanger T fuel _ to _ PS fuel / pao Temperature of the fuel to the PS Fuel / PAO heat exchanger effectiveness fuel/pao Q Q actual max ( m Cp) ( m Cp) pao min (T (T pao_in pao_in T T fuel pao_out fuel_in ) ) pao _ add _ to _ PS add T ram _ in ) Fuel / Oil heat exchanger fuel / oil Fuel / oil heat exchanger effectiveness fuel/oil ( m Cp) ( m Cp) oil min (T (T oil_in oil_in T T oil_out fuel_in ) ) Fuel / Hydraulic Oil heat exchanger Fuel / Ram- Air heat exchanger fuel / hyd fuel / ram Fuel / hydraulic oil heat exchanger effectiveness Fuel / ram air heat exchanger effectiveness fuel/hyd fuel/ram ( m Cp) ( m Cp) Q Q actual max hyd min (T (T hyd_in hyd_in ( m Cp) ( m Cp) ram min T T (T (T hyd_out fuel_in fuel_in ) ) ram_out T T ram_in ) ) ram_in 80

97 The FLS serves as the main thermal management subsystem (TMS) in the aircraft. The FLS exergy model is described by the same equations as for both the VCPAOS and ECS, and thus, the equations are not presented again in this section. The FLS interfaces with a number of subsystems as a heat sink, one of which is the OLS which, as previously mentioned, is the subsystem responsible for cooling and lubricating the engine bearing surfaces. The OLS is discussed in the following section. 3.7 Oil Loop Subsystem The oil loop subsystem (OLS) lubricates and cools the PS bearing surfaces. The OLS may perform secondary functions as well such as cooling the auxiliary power unit (APU) or operating thrust reversers; however, secondary OLS functions are not modeled in this thesis. The oil loop subsystem used in this thesis is similar to the type used in the Pratt and Whitney F100-PW-100 (early F-15 engine) with a few exceptions: a fuel/oil cooler is used here instead of the air / oil cooler as was used in the F100-PW-100. A simplified diagram of the OLS is shown in Figure Aircraft generally have two different configurations for OLSs: either a hot tank or a cold tank configuration. The former configuration (see Figure 3.10) pumps the oil directly from the scavenger pumps to the oil tank. The supply (or pressure) pump moves the hot oil first through the oil cooler and finally to the oil nozzles at the various bearings. The cold tank configuration has the scavenger pumps moving the oil through the oil cooler before heading to the tank, thus, the name cold tank. It is advantageous to have a hot tank design in fighter aircraft because the oil and air separation is more efficient. Multiple scavenger pumps are located throughout the bearing sumps in a jet engine for redundancy as well as for location (a sump is usually needed at each main bearing location). The scavenger pumps have much greater total capacity than the supply pump so the scavenger pumps will inherently pump a quantity of air as well as oil. This requires oil and air separation and an OLS typically has an internal deaerator on the tank. All the pumps on the OLS are driven by a gearbox on the low pressure spool of the PS. 81

98 Figure 3.10 Oil loop subsystem schematic. The supply pump in an OLS is usually an internal/external gear, rotary, positivedisplacement type pump which can effectively pressurize any system over its design limits if enough shaft power is available, and the pump does not encounter mechanical failure. This necessitates placing relief valves throughout the system to avoid over-pressurization OLS Thermodynamic Model The OLS interacts with the PS, FLS, and AFS-A in the aircraft. The interaction with the PS is not only via the heating load, as previously mentioned, but also via the shaft power required to power the supply, booster, and scavenger pumps. The work required by the OLS pumps is calculated using the equations in Table Table 3.22 OLS pump work equations. Component Variable Description Model Equation W Shaft power Q P pump required W pump m Pump m Pump Work efficiency where m Q Volumetric Q flow rate 82 fluid

99 The OLS heating load on the FLS must be established to have a fully integrated model. The development of the OLS model proved to be difficult because of the lack of information on this subsystem. Two statements found in Hudson (1986) were used along with some PS performance/setting correlations to develop the heating load model for the OLS. Hudson (1986) states that the OLS cooling load is 31% of the total aircraft heating load and that the ECS is 21% of the total aircraft heating load during cruise conditions. These two statements are used as the basis of heating load model for the OLS. The ECS heating load is well-defined; thus, a relationship for the OLS was developed relative to the uninstalled cruise thrust and ECS heating load during cruise conditions. Cruise was defined as the flight conditions for mission segment 19. Once the ECS heating load is established for segment 19, the OLS heating load can be estimated using the aforementioned correlation provided by Hudson (1986). This requires the maximum and minimum thrust at the given cruise conditions which are found using the thrust correction for altitude (versus sea level take-off thrust) and a maximum/minimum estimated engine operating speed (which is found using 11,500 RPM for maximum rpm, and 4,200 RPM for minimum-sustainable RPM). Next, a percentage of the total available thrust can be developed for the engine/aircraft pair for cruise conditions from the maximum and minimum thrust for the engine and the required thrust to fly at cruise conditions. For example, the thrust required for cruise is 25% of the maximum thrust of the engine at those operating conditions. This percentage is carried over to the heating load as a correlation between uninstalled thrust and heating load on the OLS. Thus, for any uninstalled thrust setting in the PS, a corresponding heating load on the OLS can be found. The heating load equations and correlations used in the OLS model are given in Table Table 3.23 OLS heating load equations. Component Variable Description Model Equation OLS Heat Exchanger Load Q fuel / OLS _ HXcc Q fuel / ECS _ HXcc Q fuel / OLS _ HX Cooling load on the fuel from the OLS at cruise conditions Cooling load on the fuel from the ECS at cruise conditions Cooling load on the fuel from the OLS at a given sea level, equivalent thrust T max Maximum thrust at altitude T Uninstalled cruise thrust from cruise 83 Q fuel / OLS _ HXcc 31 Q fuel / ECS _ HXcc 21 Tmax T SL

100 Component Variable Description Model Equation the PS Estimated minimum thrust at T min altitude from the RPM relationship RPM cc Estimated engine RPM at cruise Slope of the heating load versus slope QT the installed the thrust curve (assumes a linear relationship) T Uninstalled thrust RPM T Q Q min min P slope ambient P std cc Q T T RPM RPM QT fuel / OLS _ HX cruise max min max RPM T max fuel / OLS _ HXcc Q max fuel / OLS _ HXcc T 1 T Q cruise min Tcruise Tmin slope T Q The heat exchanger in the OLS is the same as the compact heat exchangers detailed in section Note that the weight of the heat exchanger is accounted to the FLS since the heat exchanger is owned by the FLS and not the OLS in this thesis work. A schedule of stainless steel tubing and predicted mass flow rates of the OLS were initially used to model the OLS. The results showed that the pressure drops due to frictional losses were negligible in terms of the exergy destruction and pumping losses. Thus, the final OLS model used in this thesis work does not estimate tube sizing or calculate friction losses OLS Exergy model Exergy destruction is caused in the OLS by pressure losses due to fluid friction or restriction, temperature losses due to heat transfer across finite temperature differences, unrestrained expansions, or mechanical inefficiencies. The exergy equations for the OLS are given in Table Table 3.24 OLS exergy destruction equations. Component Variable Description Model Equation T 0 Ambient temperature Q Volumetric flow rate 1 Ex Pump W actual W ideal Q P 1 m Exergy Mechanical efficiency of m the pump Destroyed Touthot T T dt Ex E Exergy destruction rate due x Pump DES T0 m hot c Phot m cold to pumping irreversibilities T T1 inhot T Exergy destroyed due to T E x DES the heat exchanger temperature gradient QT outcold incold T cruise min c Pcold min dt T 84

101 It is important to note that for this subsystem, there are actually a total of 6 pumps that generate entropy: the main pump, booster pump, and the four scavenger pumps. The total exergy destruction in the OLS is approximated by finding the pumping losses and exergy destruction in the fuel/oil heat exchanger. We now turn to the central hydraulic subsystem (CHS) which is the next subsystem that interfaces with the FLS by means of a heat interaction. The CHS is detailed in the following section. 3.8 Central Hydraulic Subsystem Hydraulics are used to actuate flight control surfaces as well as various other systems in the aircraft (e.g. the landing gear, nose-wheel steering, etc.). Hydraulics are attractive to use in aircraft because they are able to transfer large amounts of power from a central location to where it is needed by means of small diameter hoses. However, despite the advantages of hydraulics, traditional central hydraulic subsystems (CHS) may have a limited future in fighter aircraft. The More Electric Aircraft (MEA) initiative started the fighter aircraft industry moving in the direction of eliminating the CHS by using electric actuation rather than hydraulics. State-of-theart (SOTA) aircraft have replaced the hydraulics on primary flight control surfaces with electric actuators and motors. Ultimately, the CHS will likely be eliminated from future fighter aircraft for environmental reasons, reliability, maintainability, and operations and support (O & S) costs. A traditional CHS is modeled in this thesis work, although MEA considerations are taken into account in the electrical subsystem. The CHS has the following characteristics: accessory gearbox driven pumps, triple redundant hydraulic lines for flight critical loads, and a fuel/hydraulic oil heat exchanger. The two hydraulic pumps are typically interconnected by power take-offs from the accessory drive gearbox. The following hydraulic power consumers on the fighter aircraft have been modeled in the CHS model: ailerons, tail (assume fully moveable), rudder, and landing gear. An example of a CHS layout is shown in Figure CHS Thermodynamic Model The CHS thermodynamic model includes sizing the actuators, pumps, and estimating the flow required by the non-flight critical subsystems that are hydraulically actuated. Sizing the 85

102 Flight Controls Cargo Bay High Lift Devices Landing Gear Valve Electric Pump Braking Gearbox Driven Pumps Hydraulic Reservoirs Figure 3.11 Notional central hydraulics subsystem layout (simplified). hydraulic actuators requires an estimate of the flight control surface and rudder areas as well as the landing gear weight and drag. A high-resolution sizing of the flight control surfaces would require a dynamic analysis of the aircraft as well as roll characteristics and even possibly wind tunnel testing. Obviously, that level of detail is not feasible for a conceptual design study such as the present one. Thus, currently built aircraft were studied for aileron and rudder sizing estimations and corresponding control surface sizing. Main and nose landing gear specifications were also estimated to determine actuator sizing. The flight control actuator sizing must be sized by the most constrained mission segment to ensure that the flight control surfaces can operate properly throughout the mission. In this case, the maximum opposing force to the flight control surfaces is when the aircraft is flying at the highest dynamic pressure. The actuator flow estimation and sizing equations are shown in Table Table 3.25 Actuator flow estimation calculations. Component Variable Description Model Equation F Theoretical maximum available force max from an actuator (losses included) 2 Actuator Flow Fmax D P Estimation D Piston diameter (m) 2 F avail 0.7D P P Pressure at the piston (N/m 2 ) F avail Force available (approximate) 86

103 Component Variable Description Model Equation Q Flow rate into the piston (m 3 /s) V act Velocity of the actuator (m/s) A cyl Area of the cylinder (m 2 ) Q A cyl D Q 4 where V 2 act V Fmax F req act Actuator Design F req Max force on the actuator q max Maximum dynamic pressure encountered by the aircraft A CS Surface area of the control surface C La Coefficient of lift for a flat plate (simple) a Angle of the control surface to the freestream (radians) req La q max F C A 1 qmax MAX V 2 2 CLa a CS 2 Although, as stated above, the actuators must be sized for the highest load they encounter, the pump sizing method is somewhat different. The CHS pumps are sized to the highest flow required at the lowest RPM. This design point usually occurs during landing, when the engine is at a relatively low operating speed and many hydraulic functions are taking place such as extending the landing gear, extending the flaps, deploying the slats, etc. Thus, the main gear and nose gear are sized for the aircraft, and the total flow rates required estimated. The actuator sizing for the flight control surfaces determine the flow rates and little deviation from the design flow rate is observed even at a lower opposing force (lower dynamic pressure). When the flow rate is known for the highest flow rate mission segment, the pump can be sized to that flow rate and pressure requirement. A side effect of traditional hydraulic pump sizing is that the pump is generally oversized in high engine RPM situations which sacrifices pump operating efficiency. However, if an electrically powered pump were used instead of the shaft-driven pump, this disadvantage would be removed. Now, as to the working fluid, the hydraulic oil must be cooled due to heat generated by friction in the actuators and hydraulic lines. The hydraulic oil is cooled via a fuel/hydraulic oil heat exchanger. Similarly to the OLS, the heat exchanger weight is accounted to the FLS and is not included in the CHS. The CHS heating load equations are given in Table Note that the equation units are listed in Table 3.26 because the constant in the heating load equation is not for a general case. 87

104 Table 3.26 Central hydraulic subsystem heating load equations (Majumar, 2003). Component Variable Description Model Equation Heating Load Estimation E CHS Q P Heating load on CHS heat exchanger (kw) Flow rate of oil (gpm) System pressure (psi) CHS System efficiency W out Actuator work required F act Force required in the actuator V act Velocity required from actuator W in Work supplied to actuator 88 E CHS CHS where W W out in Wout FactVact W in Q P 5 PQ1 For the CHS model, the system pressure is set to 4000 psi, while the system efficiency,, varies from 20% to nearly 90% depending on the required actuator power and the power supplied to the actuator. The actuator equations are presented in more detail in Section The exergy model for the CHS is detailed in the following section CHS Exergy Model The CHS destroys exergy due to irreversibilities in the heat exchanger, actuator inefficiencies, frictional losses in the hydraulic lines, and inefficiencies in the hydraulic pumps and motors. The exergy destruction equations are given in Table Note that the frictional Table 3.27 CHS subsystem exergy destruction equations (Bejan, 1996). Component Variable Description Model Equation Exergy Destroyed S gen Entropy generation rate E x CHS Exergy destruction rate Fluid density T 0 Q Dead state temperature Volumetric flow rate S gen mc p T ln T P c p T in E xchs S gen T P m T 1 Mechanical efficiency of Ex Pump W actual W m ideal Q P 1 the pump m E Exergy destruction rate due x Pump to pump irreversibilities losses in the hydraulic lines are the cause of exergy destruction as well. Thus, the hydraulic lines are sized to permit no more than a 25% pressure drop from the pump to the actuator in full-flow conditions. in CHS for:

105 Finally, CHS power takeoff requirement is a small contributor to the total engine power takeoff term, (see Table 3.12). More important in this sense is the electrical subsystem (ES), which is responsible for the majority of the shaft power required from the PS. The ES is, thus, detailed next. 3.9 Electrical Subsystem Aircraft ESs have been a subject of much research recently. The main reason is the More Electric Aircraft (MEA) program started by the Air Force in 1991 (Pearson, 1998; Weimer, 2003; Cloyd, 1997; Moir, 1999). The goal of the program was to transition currently built and future aircraft away from traditional shaft-powered subsystems and towards electrically powered ones. The MEA program required a more reliable, higher output, and more survivable electrical power generation and distribution subsystem. The ES model is based on MEA considerations as well as on a second program focused on the power distribution subsystem called the Power Management and Distribution System for a More Electric Aircraft (MADMEL). Northrup Grumman built a demonstrator of the technologies developed under this program, and the ES components sized in this thesis are based on the MADMEL demonstrator. The model that is used here is patterned after the F-35 in that it employs two integrated 270 VDC starter/generators (IS/Gs) and an integrated power unit (IPU). The main generators are switched reluctance machines (SRMs) with multiple channels, each of which are supported by a channel (non-electrically isolated) SRM (Elbuluk and Kankam, 1996). Switched reluctance machines are chosen due to high power densities and advantages in reliability and fault tolerances compared to synchronous and induction machines. The power requirements are set up to model an aircraft that has electro-hydrostatic or electro-mechanical actuators (EHA or EMA, respectively) since industry seems to be moving in that direction. The 270 VDC generation is claimed to have better efficiency than previously built aircraft generators as well as being required for high powered/high voltage flight control actuators. 5 The power distribution subsystem includes 28 VDC and 115 AC converters. The battery weight and chargers is also included in the ES weight. A simplified schematic of the ES is shown in Figure Raymer (1999) states that an F-16 sized plane with electrically powered flight control actuators requires about 80 kw of additional power generation capacity. 89

106 IS/G #1 270 VDC Integrated Starter/Generators IS/G #2 270 VDC BUS #1 IPU 270 VDC BUS #2 DC/AC Converter DC/DC Converter DC/DC Converter DC/AC Converter Batteries 115 VAC BUS 270 VDC BUS 28 VDC BUS 270 VDC BUS 28 VDC BUS 115 VAC BUS Power Distribution Centers (PDCs) minimum of three Figure phases The Notional IPU electrical is also an subsystem SRM; however, schematic it (simplified). is a three-phase, two- This thesis work involved performing a small survey of currently built aircraft and their power generation capabilities to establish an MEA-based power generation capacity for the aircraft. Notably the F/A-18, F-22, F-16, F-35 and Eurofighter Typhoon were studied to establish generating capacity guidelines. The power generation capacity was correlated to the gross takeoff weight of the aircraft. The fighter aircraft previously built have lower electrical loads than the newer models because traditional aircraft have more mechanically actuated subsystems. For example, the Eurofighter Typhoon has an empty weight of approximately 11,000 kg and has two 30 kva generators supplying 115/200 VAC, 400 HZ, three-phase power. The weight-to-power-generation ratio for the Typhoon is 5.5 VA/kg 6, which is significantly lower than the MEA based F-22 and F-35 which are approximately 9.7 W/kg and 8.3 W/kg, respectively. The guidelines developed for ES generating capacity and component weights are shown in Table The weight equation for the electrical subsystem may be found in Section VA (volt-amperes) is a measure of alternating current (AC) power, while W is a measure of direct current (DC) power. For the purposes of this work, the AC power is considered to have a power factor of near unity which makes the units of VA nearly equivalent to W. This allows a direct comparison between the weight-to-powergeneration ratios with a VA or W rating. 90

107 Non-MEA Aircraft Power Generation Table 3.28 Fighter aircraft power generation/empty weight estimate. Component Variable Description Model Equation Estimated system electrical power R kw _ trad generation capacity for a non-mea R aircraft (kw) kw _ trad 6.2 WE MEA Aircraft Power Generation W E R kw _ mea Empty aircraft weight (kg) Estimated system electrical power generation capacity for an MEA aircraft (kw) RkW _ mea 8. 3 WE ES Thermodynamic Model The ES requires shaft power from the PS and cooling from the VC/PAOS. The power takeoff (or power extraction) from the low pressure spool of the PS is simply the amount of shaft power required by the generators for a given mission segment. The generating efficiency is set to 85%, and the power takeoff equations are shown in Table Table 3.29 Electrical subsystem generator work. Component Variable Description Model Equation P Power takeoff required for electrical TO_ ES Electrical subsystem W out _ gen PTO _ ES Subsystem W Electrical work rate required for a given g out _ gen Work mission segment g Generator efficiency Some of the ES components must be cooled to avoid overheating. The main generators and components are designed to be cooled by the VC/PAO subsystem. Note that the subsystem would be designed to route PAO through various heatsinks and hot areas in the electrical components rather than through a heat exchanger. Table 3.30 gives the heating load equations from the generators and the ES components. The transmission lines generally do not require active cooling because they are oversized to handle current spikes. Table 3.30 ES heating load model equations. Component Variable Description Model Equations Q Cooling load on the PAO loop du T PAO / ES _ gen os T from the ES generators Q 0 k 1 W T S s 0 irr dt Tk Generator W Electrical work rate required for out _ gen a given mission segment m h T0s Heating Power Take-off required for the For steady state, no heat interactions and no Load P TO _ ES ES mass interactions: g Generator efficiency 0 W out gen PTO ES T S 0 irr T 0 Ambient Temperature thus 91

108 Component Variable Description Model Equations T S P W Component Heating Load S irr _ gen W in _ c Entropy generated in the generators Electrical work rate entering a 92 0 irr TO _ ES out _ gen From an entropy balance, the entropy that must be removed from the component is as follows: Q PAO / ES _ gen S irr _ gen T T S component 0 irr _ c W in _ c W out _ c W Electrical work rate leaving a out _ c component S Entropy generated in a irr _ c component Q Cooling load on the PAO loop PAO / ES _ c from the ES components 0 Similarly to the generators, the entropy that must be removed in a heat interaction from the component is as follows: Q PAO / ES _ c S irr _ c T0 Note that the heating load in both the generators and components is the exergy destruction rate, which is the ambient temperature multiplied by the entropy generation rate in that component. This makes sense because the components are converting some form of electrical energy into another with no planned heat interactions. If the components were 100% efficient, the energy balance would indicate no active component cooling is required; however, real components are obviously less than 100% efficient ES Exergy Model The exergy destroyed in the ES is primarily due to power generation and component losses. The three main components are as follows: Generators; Other electrical components (inverters, converters, ELMCs, etc.); and Transmission lines (ohmic heat loss). The equations for the ES exergy destruction are shown in Table We now conclude with the final subsystem, i.e. that for flight control. Table 3.31 ES exergy destruction model equations. Component Variable Description Model Equation S irr Entropy generation rate d(u T0S ) T0 1 Q k (W out _ e dt T Exergy T 0 Ambient temperature k Destroyed in W in _ e ) T S 0 irr E Exergy destruction rate for a x the ES ES _ c component For steady state and no heat interaction: Components 0 W in e W out e T S 0 irr W Electrical work rate into the in _ e component thus

109 Exergy Destroyed in the ES Generators Exergy Destroyed in the Lines Total Exergy Destruction in the ES W Electrical work rate out of the out _ e component c E x ES _ gen Component efficiency Exergy destroyed in the generators P TO _ ES Power takeoff required for the ES W Work rate leaving the generators out _ gen g Generator efficiency E x ES _ trans i R A l Exergy destruction rate due to transmission losses (Elgerd, 1998) Current in the line Total resistance of the line Resistivity of the transmission line Cross sectional area of the line Length of the conductor E x T S W W ES _ c 0 irr in _ e out _ e E xes _ c W lost ( 1 c ) W in _ e W out E x ES _ gen _ gen E 2 xes _ trans i R l R A g PTO _ ES T S 0 gen PTO _ ES W out E x ES Rate of exergy destruction in the ES E xes Ex ES _ c Ex ES _ gen E xes _ trans _ gen 3.10 Flight Controls Subsystem The flight controls subsystem (FCS) consists of the actuators required to operate the flight control surfaces as well as the control hardware associated with the actuators. The two main types of flight control actuators that are discussed here are the traditional hydraulic actuators and electro-mechanical/hydraulic actuators. Having one type of actuator or the other changes many items on an aircraft. For example, traditional hydraulic actuators require a central hydraulic subsystem (CHS) including accessory gearbox driven hydraulic pumps, triple redundant hydraulic lines, emergency hydraulic power subsystems, a hydraulic oil reservoir, etc. Electro-mechanical/hydraulic style actuators are localized, meaning they are installed as a single, easily-replaceable unit that locally houses its own oil (if electro-hydraulic) or gear system (electro-mechanical) and has no other dependence on the rest of the aircraft except for a power connection and control connections. Electrically powered actuation devices (EPAD) have been tested in currently built fighter aircraft and are being used on production aircraft as well. 93

110 FCS Thermodynamic Model The FCS weight must be estimated for this model. The equation in Table 3.32 is based on previously built fighter aircraft and requires the number of hydraulic functions, the number of mechanical functions, the total area of the control surfaces, and the yawing moment of inertia of the aircraft. After the weight of the FCS has been estimated, it must be included in the empty weight of the aircraft. Table 3.32 Flight controls subsystem weight equations (Raymer, 2006). Component Variable Description Model Equation W Weight of flight controls FCS subsystem (kg) Flight M Mach number Controls WFCS 36.28M Scs N s N c Subsystem S cs Total area of control surfaces (ft 2 ) Weight N Number of flight control surfaces s N c Number of crew The actuator sizing was detailed in the CHS model so it will not be repeated here. However, the power required by the actuators must be estimated to determine the CHS system efficiency (see Table 3.26). The actuator power model equations are shown in Table Note that Majumdar (2003) states that a well designed actuator should have a range of losses from 2% to no more than 8%. Also note that the larger diameter piston cylinders generally have lower loss percentages because the break-away force and frictional forces are small compared to the force being applied to the piston. Table 3.33 FCS actuator electrical power and fluid power requirements. Component Variable Description Model Equation P elec Electrical power Required P P act elec act P Actual power required to act F d Actuator P act move the load (no losses) act t Electrical and Fluid Power act Actuator Efficiency act 92-98% P Fluid power supplied to the fluid actuator 15 P fluid P Q Q Flow rate of hydraulic fluid FCS Exergy Model The FCS destroys exergy due to inefficiencies in the actuators alone, since the other losses will be attributed to the AFS-A (drag), ES (if electrically powered actuators), or CHS (if 94

111 fluid powered actuators). The exergy destroyed is, thus, simply the power supplied to the actuator, minus the actual power translated to the flight control surface by the actuator. The exergy destruction equations are given in Table Table 3.34 FCS exergy destruction equations. Component Variable Description Model Equation Exergy Destruction rate in the FCS E x FCS Exergy Destroyed W req Work rate required by actuator 1 Ex FCS W req 1 act The subsystem models included in the AAF have now been discussed. The following chapter details the system synthesis/design, problem definition, and solution approach taken for this thesis work. 95

112 Chapter 4 Large-scale System Synthesis/Design Optimization Problem Definition and Solution Approach This chapter discusses the optimization problem definition, optimization decision variables and limits, subsystem integration, a decomposition approach called iterative local global optimization (ILGO), the solution approach, and the iscript engineering modeling/optimization software as applied to the fighter aircraft system. 4.1 AAF Aircraft System Synthesis/Design Optimization Problem The fighter aircraft system consists of subsystems that are very tightly coupled in that one subsystem can affect the other subsystem operation and/or design significantly by changing a single design or operational decision variable. The nine subsystems modeled all play a role in the optimization. However, some such as the ES, FCS, and CHS do so via a set of system-level degrees of freedom (DOF) called coupling functions, while the remaining subsystems do so visa vie both a set of coupling functions and a set of local (subsystem specific) decision variables. The former subsystems, thus, in effect participate only passively in the optimization since they have no local decision variables but nonetheless still alter their configurations, sizes, and energy consumption since these are dependent on the aircraft geometry (e.g., the CHS and FCS) and aircraft size (e.g., the ES). The interdependence of the subsystems is further illustrated by considering the PS and ECS. The ECS is dependent on the PS bleed port air properties in that any operational or design change in the PS requires a subsequent change in the ECS operational decision variables. Also, any change in the requirement of bleed air in the ECS changes the specific thrust of the PS and, thus, the overall aircraft performance. The optimization problem presented by the fighter aircraft requires large-scale optimization methods to handle the size of the problem. The objectives listed in Chapter 1 are accomplished by performing the following optimizations: Perform an optimization of the entire aircraft consisting of 9 subsystems with a morphing-wing AFS-A and with exergy destruction and exergy fuel loss minimization as the objective; 96

113 Perform an optimization of the PS and fixed-wing AFS-A with fuel burned minimization as the objective; Perform an optimization of the PS and morphing-wing AFS-A with fuel burned minimization as the objective, make a comparison to the fixed-wing results and distinguish this work from that of Butt (2005); Perform a parametric analysis on the morphing AFS-A based on minimum fuel burned; and Perform a parametric analysis on the morphing AFS-A based on minimum exergy destruction and perform analysis between minimum fuel burned results and minimum exergy destruction results. For the purposes of the large-scale optimizations affected in this research, a decomposition strategy is used. It is described in the following sections. However, before giving this description, the system-level optimization problem is defined in the next section System-Level Optimization Problem Definition Three different objective functions are utilized in the definition of the system-level optimization problem in this thesis: i) total exergy destruction plus fuel exergy lost, ii) gross takeoff weight, and iii) fuel burned. Each is minimized with respect to a set of decision variables and equality and inequality constraints. Nine-Subsystem AAF Aircraft System-Level Optimization Problem The first of the objective minimizations is expressed as a minimization of the total exergy destruction, Ex des _ total, plus the fuel exergy loss, fuel loss Ex _, due to unburned fuel lost out the back end of the PS, i.e., Minimize Ex des _ total Ex fuel _ loss Ex des _ AFS Ex des _ PS Ex des _ ECS Ex des _ FLS Ex des _ VCPAOS Ex des _ OLS Ex des _ CHS (4.1) Ex des _ ES Ex des _ FCS Ex fuel _ loss X, w.r.t Y 97

114 h AFS A g AFSA h PS g PS h ECS g ECS h FLS g FLS subject to H h / 0 VC PAOS, G g VC / PAOS 0 (4.2) h OLS gols h CHS g CHS h ES g ES hfcs g FCS where the vectors X and Y represent the synthesis / design and operational decision variables, respectively. The vectors of equality constraints, H, represent the geometric and thermodynamic models for each subsystem. The vectors of inequality constraints, G, represent the physical limits on the independent and dependent variables of the system. Two-Subsystem AAF Aircraft System-Level Optimization Problem The second objective function minimization is that of minimizing the gross takeoff weight and is expressed as follows Minimize W TO W W (4.3) AFS w.r.t.x Y, X, Y subject to AFS A, AFSA PS PS PS 0 h AFSA, g 0 AFSA (4.4) 0 h PS, g 0 PS (4.5) Where the vectors of equality constraints, hafsa and h PS, represent the geometric and thermodynamic models, and the vectors of inequality constraints, g AFSA and g PS, represent the physical limits imposed on the independent and dependent variables. The third objective function is minimized in a similar fashion and is given by 98

115 Minimize W n Fuel burned W Fuel _ burned WFuel _ (4.6) i1 where n is the total number of mission segments, and i is the mission segment. Equation (4.6) is minimized with respect to the same set of decision variables and subject to the same set of constraints as for the previous objective. Furthermore, it can be shown that this last objective is, in fact, equivalent to the previous one since the amount of fuel burned is directly proportional to the gross takeoff weight of the aircraft. For this reason, only objectives one and three are used to generate the results presented in Chapter 5. Now having defined the system-level objective of the optimization, a discussion of the need for decomposition in large-scale optimization is presented next Need for Decomposition Large-scale engineering optimization problems may require decomposition techniques or strategies in order to make the manageable or even solvable. Such techniques reduce the problem into sub-problems. The aircraft optimization problem in this thesis work is much too large to handle as a single problem; and thus, both physical and time decomposition strategies are used. Physical decomposition results in a set of aircraft subsystems and boundaries, which for the aircraft of this thesis are detailed in Chapter 3. The particular physical decomposition technique used here is called iterative local-global optimization (ILGO) developed by Muñoz and von Spakovsky (2001a,b, 2003) and applied to a 5-subsystem fighter by Rancruel and von Spakovsky (2003, 2004). Time decomposition is used to split the mission, represented by a timeframe from mission start to mission completion, into the stationary mission segments of a quasi-stationary description. The mission segments are defined based on the aircraft flight characteristics (e.g. climb, cruise, loiter, accelerate, etc.) or requirements (e.g. drop a payload, fly a distance, loiter an amount of time). Individual mission segments are modeled as steady state and transient behavior is approximated via the quasi-stationary description. The resulting mission segments are given in Table 3.1 of Chapter 3. Initially, every mission segment was split into 5-7 smaller segments to increase fuel burn/gross weight calculation accuracy. However, it was found that in mission segments with little-changing flight characteristics (e.g., loiter, cruise), a single time step gave sufficient accuracy. Note that decreasing the number of time steps in each segment significantly 99 i lossi

116 decreases the computational burden as well, since the aerodynamics, weight fractions, etc. are updated for each time step within the mission segment. Now, before applying these decomposition strategies to the system-level optimization problems, a description of ILGO is given in the following sections. 4.2 Iterative Local-Global Optimization (ILGO) Approach The purpose of this section is not to give a detailed discussion of ILGO since it was not developed in this thesis work, but rather to give a top-level overview of the ILGO decomposition strategy for large-scale optimization. The reader is referred to Muñoz and von Spakovsky (2000a,b,c,d; 2001a,b) for details on this strategy. A basic discussion of local-global optimization (LGO), however, must precede the discussion of ILGO since the former is the basis for most if not all of the physical decomposition strategies found in the literature. In the process, the differences between LGO and ILGO are highlighted and the uniqueness of the ILGO approach revealed Local-Global Optimization (LGO) Section 2.3 of Chapter 2 discusses the existence / requirement of the ORS (see Figure 2.7) in the LGO approach and also the computational burden required to explicitly or implicitly generate the ORS for a problem of any magnitude. For further explanation, consider the following system problem that has been decomposed into two subsystems or units as shown in Figure 4.1. The vectors Z 1 and Z 2 are the so-called local-level decision (i.e., independent) variables for each unit and the arrows between the two units are represented by vectors of the functions, u 12 and u 21, that couple the two units. R 1 2 u12 Z 1 u 21 R Z 2 Figure 4.1 Physical decomposition of a 2-unit system. 100

117 The system-level objective function, C, is expressed as a sum of the unit-contributions, i.e., C1 C2. Each of the unit contributions to the system-level objective function has a set of terms that define the unit contribution. For example, for the first unit, one might have that C k R C (4.7) capital1 In equation (4.7), R1 is an external resource used by unit 1 (e.g., fuel or power) and 101 C capital is a 1 function related to the size of the unit (e.g., weight/volume or cost) while k 1 is a conversion factor. For each of the units, the coupling functions bring resources from the other unit depending on the needs of the receiving unit. The amount of these resources and the external ones depend on the values of the decision variables of the coupled units. So, looking only at the contribution of the first unit to the system-level objective function, C, the following expression is written: C1 k1r1 Z1, u12 Z1, Z 2, Z sys, u21z1, Z 2, Z sys Ccapital Z1 (4.8) 1 where C capital is, for example, the capital cost in a thermoeconomic problem and Z 1 sys the set of system-level decision (i.e., independent) variables which cannot be assigned strictly locally, i.e., to one unit only. In thermodynamic problems, this term is either ignored as is done in a stationary system problem or is converted into a physical term such as weight for a nonstationary system problem. The reader is referred to the Evans-El-Sayed formalism for more information about thermoeconomics in the context of decomposition (Evans and El-Sayed, 1970). Considering both units with the system-level optimization problem is as follows: Minimize C k1r1 Z1, u12 k R Z, u 2 w.r.t. Z 1 Z1, Z 2, Z sys, u21z1, Z2, Z sys Ccapital Z1 1 Z, Z, Z, u Z, Z, Z C Z 2 2, Z2, 12 Z sys 1 2 sys sys capital2 subject to the primary constraints h 1 H 0 (4.10) h 2 2 (4.9)

118 g 1 G 0 (4.11) g 2 and to the secondary constraints 1, 12 2, u Z Z Z sys 0 (4.12) 1, 21 2, u Z Z Z sys 0 (4.13) where the vectors of equality constraints, H, represent the physical and thermodynamic models for each subsystem. The vectors of inequality constraints, G, represent the physical limits on the independent and dependent variables associated with each unit in the system. Equations 4.12 and 4.13 indicate that the coupling functions take on the values and constrained by the following two expressions: u 12 u min 12 max (4.14) u 21 u (4.15) min 21 max Problem (4.9) can be physically and mathematically decomposed into a set of two subproblems which must be repeatedly solved for different values of and constrained within the limits set by expressions (4.14) and (4.15). The two subproblems are, thus, expressed as Subproblem 1: Minimize C1 k1r1 Z1,, Ccapital Z1 (4.16) 1 w.r.t. Z 1 subject to h 1 0 (4.17) g 0 (4.18) 1 Subproblem 2: Minimize C2 k2r2 Z 2,, Ccapital Z 2 (4.19) 2 w.r.t. Z 2 subject to h 2 0 (4.20) 2 0 g (4.21) Note that for a subproblem, either is fixed for a given optimization and is calculated (i.e., a result of the subproblem optimization) or vise versa. For example, in the case of subproblem 2, if is fixed for subproblem 1, then it must be calculated in subproblem 2; conversely, if is 102

119 fixed for subproblem 1, then it must be calculated in subproblem 2. Also, note that the repeated optimizations required of subproblems 1 and 2 in effect result in a set of nested optimizations for which the inner part of the nest is comprised of the set of subproblem optimizations and the outer part of the nest of a single system-level optimization with the and as decision variables. Thus, the LGO decomposition results in a multi-level optimization which reduces an overall system-level problem into a set of smaller subsystem problems which in theory should be easier to solve. In fact, for large-scale system optimization problems involving many degrees of freedom and non-linearities, such a multi-level approach may be the only way to arrive at a solution. However, it introduces an additional computational burden due to the nesting. This may become so large that it renders the problem unsolvable at least from a practical standpoint. The multi-level optimization resulting from the application of LGO to the two-unit system of Figure 4.1 is illustrated in Figure 4.2. System-level optimization Unit-level optimizations Unitsystem interactions Unit 1 Unit 2 Simulations System - optimizer Figure 4.2 Multi-level optimization resulting in a set of nested optimizations. Note that in this figure, there are a total of three optimization loops shown: one each for units 1 and 2 as well as a third at the system-level. Multi-level optimization requires a new unit-level optimization for each iteration (i.e., completion) of the system-level optimization loop. As already mentioned, such nesting can become very computationally burdensome. ILGO was 103

120 specifically developed to address this difficulty. The following section gives an overview of this approach ILGO Approach Iterative local-global optimization (ILGO) is a decomposition strategy for large-scale optimization developed by Muñoz and von Spakovsky (2000a,b,c,d; 2001a,b) which eliminates the need for nested optimizations found with LGO. It not only reduces the computational burden of multi-level optimization but also allows the optimal decentralized development of aircraft subsystems in the context of an overall system-level optimization. ILGO eliminates the nesting of LGO by embedding at the unit-level the system-level information found in the outer loop of the LGO optimization. In doing so, the outer optimization loop is no longer needed since it is implicitly present at the unit-level. The embeddiong is accomplished via a set of coupling functions and a set of associated shadow prices. The coupling functions are simply the unit-level or subsystem-level interactions required as inputs or outputs by each physical unit. These functions, thus, couple or integrate the unit-level problems with each other. Figure 4.3 shows an example of a simple three unit system with u 12,u 21 u 13,u 31 Unit 1 Unit 2 u 23,u 32 Unit 3 u 12,u 21 u 13,u 31 u 23,u 32 unit 2 and unit 1 interactions unit 1 and unit 3 interactions unit 2 and unit 3 interactions Figure 4.3 An example of three subsystems and their associated coupling functions. coupling functions. The functions associated shadow prices measure changes in the optimal values of the local (unit-level) functions with respect to changes in the coupling functions. The shadow prices allow the decomposed optimizations to progress along the system-level ORS in the direction of the system-level optimal solution. Furthermore, the unit-level objective functions used by LGO are morphed into system-level unit-base objective functions in ILGO. For example, the unit-level objective functions (equations (4.16) and (4.19)) of the two-unit 104

121 system depicted in Figure 4.1 are transformed into system-level unit-base objective functions as follows: C C 2 (1) 2 (1) C u (2) 1 (2) C u (4.22) 1' C1 u (4.23) 2 ' C u21 where the last here terms in each of these equations are 1 st order Taylor Series expansions about some reference point o on the ORS relative to changes of the coupling functions u 12 and u 21 with respect to unit 1 (equation (4.22)) or unit 2 (equation (4.24)), respectively. The ' s in these equations are the partial derivatives associated with these expansions. They are, in fact, the shadow prices. Note that for purposes of simplification, the vectors u 12 and u 21in equations (4.16) and (4.19) have been assumed to contain only a single coupling function each (i.e. u 12 or u 21 ). As should be evident, optimizing equations (4.22) or (4.23) in effect optimizes not only the local objective (equation (4.16) or (4.19)) of each unit but each unit s system-level effects via the additional terms appearing in equations (4.22) and (4.23). Thus, each local optimization not only optimizes the unit but also optimizes the system as a whole. For more details of the ILGO formalism, the reader is referred to Muñoz and von Spakovsky (2000a,b,c,d; 2001a,b), Rancruel (2002, 2005), and Rancruel and von Spakovsky (2005, 2006). To understand how the process of optimization with ILGO proceeds, a notional flow diagram is given in Figure 4.5. As with Figure 4.2, there are two units participating in the optimization. The key differences with ILGO are no system-level simulations required and the outer loop optimization has been eliminated. Each system-level, unit-based optimization results in a new set of shadow price and coupling function values which are then used in a new ILGO iteration unless convergence has been reached. Based on past experience, the number of iterations required for convergence for even complex systems is less than six or seven. In the following sections, the ILGO decomposition is applied to the AAF aircraft optimization problem which is the subject of this thesis. 105

122 ILGO Start Decomposition (define subsystem boundaries, coupling) - optimizer System-level information is sent to the system-level unit-base subproblems (shadow prices, coupling function changes) Unit-level optimizations New ILGO Iteration Unit 1 Unit 2 System-level, unit-based optimization results Convergence Figure 4.4 Notional flow diagram of the application of the ILGO decomposition strategy to the two-unit system of Figure System-Level, Unit-Based Synthesis/Design Optimization Problem Definitions Section 4.1 defines the system-level optimization problem. The following sections define the system-level, unit-based synthesis/design optimization problems for each subsystem starting with the AFS-A. However, the subsystem integration and coupling functions for the AAF fighter are defined first in the next section Subsystem Integration and Coupling Functions Subsystem integration is necessary for the overall system optimum synthesis/design to be found. Subsystem integration, in a programming sense, requires subsystem interactions to be defined explicitly between the subsystem models. In the context of the fighter aircraft system, subsystem integration is a very involved process due to the number of subsystem interactions present. An additional difficulty is added to the integration process when a subsystem has a two way interaction (e.g., a heating interaction between two subsystems) since this may involve a pinch temperature difference constraint that affects both subsystems. 106

123 While a subsystem interaction is simply an operating parameter (e.g., bleed air temperature) or a design parameter (e.g., subsystem weight) required by one subsystem from another, applying the ILGO decomposition to a system requires that the subsystem interaction not be treated as a real-time interaction as it would in an overall system simulation, but rather as an independent variable or fixed parameter during a local or unit-level optimization. The interactions are updated after each ILGO iteration to effectively maintain the subsystem interactions despite the de-coupled local optimizations. The subsystem interactions are, thus, called coupling functions within the context of ILGO as they are not to be confused with the subsystem interactions as found in the LGO decomposition which are maintained in real time and result in the multi-level optimizations of the LGO approach. The reader is referred to the work of Muñoz and von Spakovsky (2000a,b,c,d; 2001a,b), Rancruel (2002, 2005), and Rancruel and von Spakovsky (2005, 2006) for details on coupling functions and the updating methods. The AAF aircraft system of this thesis has numerous subsystem interactions and each subsystem is highly dependent on the other subsystems for both design and operational considerations. The fighter aircraft subsystem interactions are shown in Figure 4.5 with an explanation of the main interactions following the figure. Figure 4.5 does not give a comprehensive list of all of the subsystem interactions; however, Figure 4.5 does clearly illustrate the interdependence and tight coupling of the aircraft subsystems. A comprehensive list of all coupling functions is not given here because of the very large number involved, i.e However, the number associated with each subsystem is given in Table 4.1 where for the Table 4.1 Number of coupling functions associated with each subsystem. Subsystem Input (fixed within subsystem) coupling functions Output (variable within subsystem) coupling functions AFS-A PS ECS VC/PAOS FLS ES 6 21 FCS 6 1 CHS 6 41 OLS Totals

124 application of ILGO used here the input coupling function and associated shadow price values are assumed fixed in any given ILGO iteration while the output coupling function and shadow price values are determined by each subsystem optimization in a given ILGO iteration. u 12,u21 1. OLS 2. PS u 23,u ECS X H OLS OLS, Y, G OLS OLS u 15,u CHS H, CHS G CHS u 78,u FLS X FLS, Y H, G FLS u FLS 57,u FCS H, FLS FCS G FCS u u 24,u 42 u 45,u 54 58,u 85 X H PS PS, Y, G PS PS u 25,u AFS X AFS, Y H, G AFS AFS AFS u 62 u 56,u 65 u 59,u ES H, ES G ES u 35,u 53 X H ECS ECS, Y, G 9. VC/PAOS X VCPAOS, Y H, G VCPAOS u ECS ECS 39,u 93 VCPAOS VCPAOS u 12,u 21 u 23, u 32 u 24,u 42 u 62 u 15,u 51 u 45,u 54 u 25,u 52 u 56,u 65 u 35,u 53 u 39,u 93 u 57,u 75 u 78,u 87 u 58,u 85 u 59,u 95 Interaction Uninstalled thrust, OLS power takeoff Bleed air properties, bleed air mass flow required Fuel flow rate required, FLS power takeoff ES power takeoff OLS weight, mission segment time FLS weight, drag, ram air properties, mission segment time PS weight, drag, AFS-A thrust required, specific fuel consumption, mission segment time ES weight, W, mission segment time TO ECS weight, drag, ram air properties, mission segment time Bleed air/pao heat exchanger properties CHS weight, mission segment time Control surface sizing, hydraulic oil flow rate required FCS weight, aircraft wing and tail geometry, mission segment time, flight conditions VC/PAOS weight, drag, ram air properties, mission segment time Figure 4.5 Aircraft subsystem interactions and coupling functions. Note that each of the subsystem connections in Figure 4.1 has a two way arrow which indicates that there is a coupling function vector for each direction of the interaction. The one exception is the interaction between the ES and PS, u 62, which represents the shaft power takeoff (i.e. power extraction from the low pressure turbine), P TO, required from the PS to generate power for the aircraft. There is no interaction required by the PS from the ES. 108

125 The subsystem interactions are now briefly discussed starting with the AFS-A. Every aircraft subsystem has a weight associated with it, and each subsystem weight must be defined as a coupling function to the AFS-A. In addition, the AFS-A requires a certain amount of thrust to fly the aircraft as well as specific fuel consumption, sfc, rates for the corresponding thrust. Thus, both the thrust required and the corresponding sfc are interactions with the PS. Also, the subsystems that are associated with inlet ducts (i.e., the ECS, FLS, and VC/PAOS) have momentum drag associated with the inlet / exit ducts and these drag values are an interaction with the AFS-A as well. An additional AFS-A subsystem interaction results from the wing geometry of the AFS-A and the FCS. The wing geometry is used as a basis for estimating control surface sizing in the FCS. Note that the morphing-wing AFS-A wing geometry that is used to size the FCS is the same geometry as used for establishing the wing weight (see subsection 3.2.5). The PS is the next most highly integrated and interdependent subsystem. The PS requires power takeoff (or power extraction from the low-pressure spool), PTO, values from the OLS, ES, FLS, CHS, and VCPAOS (the CHS and VCPAOS power takeoff interactions with the PS are not shown in Figure 4.1). The ECS requires bleed air temperature and pressure from both the high and low pressure bleed ports of the PS. In addition, the ECS specifies the bleed air mass flow rate required from the PS, which affects the total performance of the PS depending on the mass flow rate required and which bleed port is selected by the ECS. The OLS requires uninstalled thrust values from the PS to estimate the heating load on the FLS/OLS heat exchanger. The PS also outputs the sfc for each mission segment to the FLS (and to the previously mentioned AFS- A). The VC/PAOS interfaces with the ECS via the bleed air / polyalphaolefin (PAO) heat exchanger. The PAO temperature and heat exchanger physical geometry are all coupling functions between the ECS and VC/PAOS. The model and ILGO decomposition was initially set up to only have a heat interaction as a coupling function, but the optimization then created a thermodynamically impossible heat exchanger (i.e., the exit temperature of the bleed air side or hot side of the heat exchanger was significantly lower than the PAO side or cold side exit temperature). The thermodynamic characteristics of the heat exchanger, thus, had to be owned by one of the unit-level optimizations to avoid this scenario. Consequently, the bleed air/pao 109

126 heat exchanger geometry is sized in the ECS code, while the heat exchanger weight and exergy destruction is accounted to the VC/PAOS. The interdependence of the subsystems can be seen very easily when one observes the effects on the rest of the aircraft of changing one subsystem parameter. For example, if the weight of a subsystem increases 100 kg, the aircraft requires more lift during cruise, more thrust during acceleration, and subsequently more fuel. Also, if the increased weight is due to a subsystem requiring additional cooling (e.g., additional power generating capacity in the ES increases the subsystem weight and cooling requirements), then the VC/PAOS is affected and perhaps even the ECS ram air inlet duct sizing, etc. Needless to say, if one item on an aircraft is changed significantly, it generally requires a synthesis / design or operational change in many other subsystems on the aircraft due to the high level of integration in such systems. This effect is even more pronounced if the aircraft is being optimized to find the best synthesis/design and if over designing a subsystem is not an acceptable solution. Having outlined and discussed the coupling functions for the AAF aircraft system and its subsystems, the optimization decision variables and their associated variable constraints are discussed next AFS-A System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The AFS-A synthesis/design and operational decision variables are presented in Section 4.4 as are the variable inequality constraints, which are based on typical fighter aircraft configurations (Raymer, 2002) and previous work done in fighter aircraft optimization (Periannan, 2004; Rancruel, 2005; Butt, 2005; Smith et al. 2007). The system-level objectives that used for the AFS-A in this thesis work are minimizing the exergy destruction plus exergy loss due to unburned fuel exiting the rear of the PS and minimizing the fuel burned. Both are determined with respect to the entire mission. Now, it is only for the former system level objective, which is used for the ninesubsystem AAF aircraft system optimization, that the ILGO decomposition strategy is used. Thus, a set of system-level unit-base optimization problems for the nine-subsystems must be determined as described in Section 4.2. That for the AFS-A is as follows: 110

127 AFS-A System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' desafs A w.r.t. X Ex Ex Ex 6 j1 i1 AFSA des AFS A 5 desvc / PAOS des FCS j1 3 5 t M, Alt ti uijvar ious, Y i AFSA u Ex 8 desps Ex W j j1 i1 ij var ious W Ex desols j n sfc sfc T i1 Ex M, Alt desecs 4 j1i1 i deses n Ex i Ex D ji desfls D deschs j i Treq i reqi (4.24) subject to h 0 AFSA (4.25) g 0 (4.26) AFSA where the vector of equality constraints, hafsa, represents the aerodynamic and geometric model of the AFS-A and the vector of inequality constraints, g, the physical limits AFSA imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.24) represents the system-level minimization resulting from varying the AFS- A (or local) decision variables only. The local or unit-based objective function is denoted by Ex desafs A and the expression for the system-level, unit-based objective function, Ex', des AFS A includes products of paired shadow prices and associated coupling functions. As to the shadow prices given in problem (4.24), the first summation represents the subsystem weights, the first double summation term represents the drag of the inlet ducts (one term for each mission segment for the ECS, FLS, and VC/PAOS), the double summation term represents the time for the mission segments that are functions of the mission decision variables (i.e., the mission segment 111

128 time is determined by BCM 1, BCM 2, BCLM, etc.), the next double summation is for the three subsystems that require the mission decision variable values from the AFS-A (i.e., the PS, ECS, and FLS), the next coupling function represents the miscellaneous AFS-A coupling functions not listed here, and the last summation couples the AFS-A and the PS with the thrust required and fuel consumption rate. All told, the AFS-A has 88 input and 88 output shadow prices (see Section 4.3.1). The shadow prices for the coupling functions given in Problem (4.24) are defined as 8 j1 Ex des j W j (4.27) W j 3 n j1i Ex des j D (4.28) ji D j i Ex des j (4.29) ti j1 i1 ti (4.30) 3 5 Ex des j M, Alti j1 i1 M, Alti des j Ex uij (4.31) various u ijvarious and the final two shadow prices represent a total of 40 unique shadow prices, one for each of the mission segment legs excluding the payload drop mission segment desps AFS A Ex sfc (4.31) i sfc i desps AFS A Ex T (4.32) req i Treq i Note that there are both input and output shadow prices for every subsystem. For example, an input shadow price for the AFS-A is equation (4.24) since the specific fuel consumption, sfc, is an interaction with the PS and is defined in the PS. Thus, the value of the shadow price is constant within the AFS-A system-level, unit-based optimization. In contrast, an output shadow 112

129 price example is equation (4.32) as the thrust required to fly the aircraft is determined in the AFS-A and is output to the PS. Were all the shadow prices for the AFS-A explicitly stated here, 88 input and 88 output shadow prices would be defined for the AFS-A for a total of 176 shadow prices. Equations (4.27) through (4.32) represent the effect of the marginal changes in the optimum value of the system-level, unit-based objective function for the AFS-A due to changes in the coupling functions. Problem (4.24) has sixteen additional terms, W j,,, M, Alt, u, sfc, ijvarious and are defined as 8 j1 3 W n j1i j1 i1 6 5 j1 i1 j D t W ji ji j D t W j i M, Alt ji j i i j t D j i and j i M, Alt j i T req i u ij u var ij u ious various ijvarious D j i that represent the variations in the coupling functions M, Alt j i (4.33) (4.34) (4.35) (4.36) (4.37) n i1 n i1 sfc T i reqi sfc T i req i sfc i (4.38) T req i (4.39) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Similarly to the shadow prices above, equations (4.33) to (4.39) are not comprehensive; the AFS-A has 176 total equations, one for each shadow price, but the examples provided here are adequate for generating the other equations. The PS system-level, unit-based synthesis/design optimization problem definition is next. t i 113

130 4.3.3 PS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The PS system-level, unit-based optimization problem is defined as follows: PS System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' desps Ex' w.r.t. X PS, Y PS des fuel _ loss Ex Ex Ex n i1 desps deschs Ex desvc / PAOS des fuel _ loss Ex WPS W sfci sfci T Tbleed _ l, h i T inst inst T i i deses PTOi PS Ex P Ex 5 i1 TOi bleed _ l, hi desafsa desols t ti i Ex Ex mbleedi m Pbleed _ l, h i desecs deses uivarious bleedi P u i various bleed _ l, hi (4.40) subject to h PS 0 (4.41) g 0 (4.42) PS where the vector of equality constraints, h PS, represents the thermodynamic and geometric model of the PS and the vector of inequality constraints, g PS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.40) represents the system-level minimization resulting from varying the PS decision variables only. The local or unit-based minimization objective function is denoted by Ex plus an additional term, Ex, to account for the unburned fuel lost out the back des PS des fuel _ loss of the PS expressed. The expression for the unit-based, system-level objective function, Ex' Ex', includes products for paired shadow prices and associated coupling desps des fuel _ loss functions. The first represents the weight of the PS, the second represents the time coupling with 114

131 the AFS-A, the third represents the various other coupling functions, and the six within the summation are the PS interactions. The shadow prices are defined as desafsa Ex (4.43) WPS W PS 5 i1 ti Ex t desps i (4.44) des j Ex uij (4.45) various u ijvarious And the next six shadow prices which represent a total of 240 independent shadow prices are defined as des AFS A, FLS, VC / PAOS Ex (4.46) sfci sfc i desfls, ES, VC / PAOS, CHS Ex (4.47) PTOi P des ECS bleed i TO i Ex m bleed (4.48) i m desecs Ex T bleed (4.49) _ l, hi T bleed _ l, hi desecs Ex P bleed (4.50) _ l, hi P bleed _ l, h i desafs A, OLS Ex (4.51) Tinsti T inst i where equations (4.44), (4.45), (4.46), (4.47), (4.48), (4.49), (4.50), and (4.51) represent a total of 5, 11, 60, 80, 20, 40, 40, and 20 unique shadow prices, respectively. All told, the PS has 115 input, or fixed, shadow prices and 162 output (or variable i.e. internally calculated) shadow prices for a total of 277 shadow prices for the PS. Equations (4.43) through (4.51) represent the effect of marginal changes in the optimum value of the system-level, unit-based objective function for the PS due to changes in the coupling 115

132 functions. Problem (4.40) has nine additional terms, W PS, t i, u, sfc,, m bleedi, T bleed _ l, h, P bleed _ l, h, and listed above and are defined as W 5 i1 PS t i W t i PS i W t i 0 PS u ij u var ij u ious various ijvarious i T inst i 116 ivarious i P TOi that represent variations in the coupling functions (4.52) (4.53) (4.54) sfc i sfc i sfc i P m T P T TOi bleedi P TOi m bleed _ l, hi bleed _ l, hi insti T TOi P bleedi T P insti m bleed _ l, hi bleed _ l, hi T insti bleed i T bleed _ l, h i P bleed _ l, h i (4.55) (4.56) (4.57) (4.58) (4.59) (4.60) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Similarly to the AFS-A, equations (4.52) to (4.60) are not comprehensive; the PS has 277 total equations, one for each shadow price, but the examples provided here are adequate for generating the other equations. The ECS system-level, unit-based synthesis/design optimization problem definition is next ECS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The ECS system-level, unit-based optimization problem is defined as follows: ECS System-Level, Unit-Based Exergy Destruction Optimization Problem

133 Minimize Ex' des ECS w.r.t. X ECS, Y ECS Ex Ex i1 n des ECS t ti i DECS i Ex des VC / PAOS Wbleed / PAO _ HX 5 D des AFS A W uivarious ECSi W ECS T u Ex W ECS i various mbleed i des PS m A P i1 Tbleed _ l, h bleed _ l, hi Pbleed l h bleed _ l, h i _, i i bleed / PAO _ HX bleedi Ainlet inlet (4.61) subject to h ECS 0 (4.62) g 0 (4.63) ECS where the vector of equality constraints, h ECS, represents the thermodynamic and geometric model of the ECS and the vector of inequality constraints, g ECS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.61) represents the system-level minimization resulting from varying the ECS decision variables only. The local or unit-based minimization objective function is denoted by ExdesECS. The expression for the unit-based, system-level objective function, Ex' des ECS, includes products for paired shadow prices and associated coupling functions. The first shadow price represents the weight of the ECS, the second represents the weight of the bleed air / PAO heat exchanger, the third represents the area of the inlet for the ram air / PAO heat exchanger, the fourth represents the time coupling with the AFS-A, the fifth represents the various other coupling functions, and the four within the summation are the ECS interactions. The shadow prices are defined as 117

134 desafsa ECS Ex W (4.64) ECS W ECS desvc / PAOS Ex W bleed (4.65) / PAO _ HX W desvc / PAOS ECS inlet bleed / PAO _ HX Ex A (4.66) inlet A 5 Ex desecs i 1 ti t i (4.67) Exdes j uij (4.68) various u ijvarious and the next four shadow prices which represent 144 independent shadow prices are defined as des AFS A Ex D ECS (4.69) i D ECS i des PS, VC / PAOS Ex (4.70) mbleedi m bleed i desps, VC / PAOS Ex (4.71) Tbleed _ l, hi T bleed _ l, h i desps, VC / PAOS Ex (4.72) Pbleed _ l, hi P bleed _ l, h i where equations (4.67), (4.68), (4.69), (4.70), (4.71), and (4.72) represent a total of 5, 13, 28, 28, 48, and 48 unique shadow prices, respectively. The ECS has a total of 93 input and 72 output shadow prices for a total of 165 shadow prices. Equations (4.64) through (4.72) represent the effect of marginal changes in the optimum value of the system-level, unit-based objective function for the ECS due to changes in the coupling functions. Problem (4.61) has nine additional terms, W ECS, W bleed / PAO _ HX, A inlet, t i, u,,,, and that represent variations in the ivarious D ECSi m bleedi T bleed _ l, hi coupling functions listed above and are defined as ECS ECS ECS 118 P bleed _ l, h i W W W (4.73)

135 bleed / PAO _ HX Wbleed / PAO _ HX Wbleed / PAO _ HX inlet Ainlet Ainlet W (4.74) A (4.75) 5 i1 t i t i t i 0 u ij u var ij u ious various ijvarious (4.76) (4.77) ECSi ECSi ECSi bleed i D D D (4.78) m T P bleedi m bleed _ l, hi bleed _ l, hi bleedi T P m bleed _ l, hi bleed _ l, hi T bleed _ l, h i P bleed _ l, h i (4.79) (4.80) (4.81) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Equations (4.73) to (4.81) are not comprehensive but rather represent 165 total equations for the ECS, one for each shadow price, but the examples provided here are adequate for generating the other equations. The VC/PAOS system-level, unit-based synthesis/design optimization problem definition is next VC/PAOS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The VC/PAOS system-level, unit-based optimization problem is defined as follows: VC/PAOS System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize 119

136 Ex' desvc / PAOS w.r.t. Ex Ex Wbleed / PAO _ HX 5 i1 n i1 4 desvc / PAOS t DVC / PAOSi Ex desecsvc / PAOS ti i m fuel FLS mbleed i m W D m desafsavc / PAOS Ex uivarious bleedi P bleed / PAO _ HX u VC / PAOSi i various i1 P bleed _ l, bleed _ l, h hi i X VC PAOS YVC / PAOS /, desflsvc / PAOS fuel FLS Tbleed _ l, h i T Ex Ainlet WVC / PAOS A Q fuel /' PAO _ HXi despsvc / PAOS bleed _ l, hi inlet Q W VC / PAOS fuel / PAO _ HXi (4.82) subject to h 0 (4.83) VC / PAOS g 0 (4.84) VC / PAOS where the vector of equality constraints, h VC / PAOS, represents the thermodynamic and geometric model of the VC/PAOS and the vector of inequality constraints,, the physical limits g VC/ PAOS imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.82) represents the system-level minimization resulting from varying the VC/PAOS decision variables only. The local or unit-based minimization objective function is denoted by Ex' des VC / PAOS Ex desvc / PAOS while the expression for the system-level, unit-based objective function,, includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of changes in the weight of the VC/PAOS, while the second pair represents the weight of the bleed air / PAO heat exchanger, the third represents the area of the inlet for the ram air / PAO heat exchanger which is sized in the ECS. The fourth pair represents the time coupling with the AFS-A, the fifth represents the various other coupling functions. The pairs within the summation from i to n account for the effects of changes in the 120

137 VC/PAOS inlet drag, the heat interaction with the FLS via the fuel / PAO heat exchanger, and the mass flow rate of the fuel in the FLS while the three shadow price / coupling function pairs within the final summation account for the effects of changes in the bleed air / PAO heat exchanger working fluid properties. The formulation of the bleed air / PAO heat exchanger pair is unique to the VC/PAOS and ECS. The heat exchanger is owned by the ECS and the influence of this heat exchanger on both the ECS and VC/PAOS is represented by more than just a heat interaction coupling function. The properties of both working fluids (bleed air and PAO) are defined as coupling functions so the optimization does not find a thermodynamically impossible heat exchanger such as one having the hot-side exit temperature lower than the cold-side inlet temperature. Thus, additional coupling functions are defined for the PAO working fluid properties at the bleed air/pao heat exchanger so that the optimization can size the heat exchanger properly. Furthermore, the heat exchanger weight is passed from the ECS to the VC/PAOS as a coupling function as the bleed air / PAO heat exchanger weight is accounted in the VC/PAOS. The shadow prices for the VC/PAOS are defined as desafsa Ex (4.85) WVC / PAOS W VC / PAOS desafsa Ex W bleed (4.86) / PAO _ HX W desecs inlet bleed / PAO _ HX Ex (4.87) Ainlet A 5 i1 ti Ex desvc / PAOS t i (4.88) des j Ex uij (4.89) various u ijvarious and the next six shadow prices, which represent 84 independent shadow prices, are defined as desafsa Ex D VC (4.90) / PAOSi D VC / PAOS i 121

138 ExdesFLS Q fuel / PAO _ HX i Q fuel / PAO _ HXi (4.91) des FLS Ex m fuelfls (4.92) i m fuel FLS i des ECS Ex m bleed (4.93) i m bleed i desecs Ex T bleed (4.94) _ l, hi T bleed _ l, h i desecs Ex P bleed (4.95) _ l, hi P bleed _ l, hi where equations (4.88), (4.89), (4.90), (4.91), (4.92), (4.93), (4.94), and (4.95) represent a total of 5, 4, 20, 20, 20, 8, 8, and 8 unique shadow prices, respectively. The VC/PAOS has a total of 50 input shadow prices and 48 output shadow prices for a total of 96 shadow prices. Equations (4.85) through (4.95) represent the effect of the marginal change in the optimum value of the coupling function on the system-level, unit-based objective function for the VC/PAOS due to changes in the coupling functions. Problem (4.82) has seven additional terms, W VC/ PAOS,,, t i, u i,, Q, m, W bleed / PAO _ HX A inlet various 122 D VC / PAOSi fuel / PAO _ HXi fuelfls m bleedi, T bleed _ l, h, and P i bleed _ l, h i that represent the effect that the variation in the VC/PAOS decision variables has on the coupling functions listed above, equations (4.85) through (4.95), and are defined as W W W (4.96) VC / PAOS VC / PAOS VC / PAOS bleed / PAO _ HX Wbleed / PAO _ HX Wbleed / PAO _ HX inlet Ainlet Ainlet W (4.97) A (4.98) t 5 0 t i ti i i1 u ij u various ij u various ijvarious (4.99) (4.100)

139 VC / PAOS D i VC / PAOS D i VC / PAOSi fuel / PAO _ Q fuel / PAO _ Q HXi HXi fuel / PAO _ HXi bleed mbleed m i i bleed i D (4.101) Q (4.102) m T P bleed _ l, hi bleed _ l, hi T P bleed _ l, hi bleed _ l, hi T bleed _ l, h i P bleed _ l, h i (4.103) (4.104) (4.105) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Similarly to the ECS, equations (4.96) to (4.105) are not comprehensive but rather represent 96 total equations for the VC/PAOS, one for each unique shadow price. The FLS system-level, unit-based synthesis/design optimization problem definition is next FLS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The FLS system-level, unit-based optimization problem is defined as follows: FLS System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' desfls Ex Ex Ex n desfls desecs desols w.r.t. X FLS, Y FLS subject to DFLSi Ex Ex D sfc desafsa desvc / PAOS 5 WFLS ti ui (4.106) WFLS FLSi Ex Ex i1 PTOFLS i desps deschs ti P TOFLSi uivarious Q various i1 sfc i i QFLSCHS, OLS, VC / FLSCHS, OLS, VC / PAOS PAOSi i h 0 FLS (4.107) 123

140 g 0 (4.108) FLS where the vector of equality constraints, h FLS, represents the thermodynamic and geometric model of the FLS and the vector of inequality constraints, g FLS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.106) represents the system-level minimization resulting from varying the FLS decision variables only. The local or unit-based minimization objective function is denoted by ExdesFLS. The expression for the unit-based, system-level objective function, Ex' des FLS, includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of variations in the FLS weight, the second pair for the mission segments with variable time, the third pair for various other interactions, while the pairs in the summation reflect the effect of variations in the FLS coupling functions. The FLS shadow prices are defined as desafsa Ex (4.109) WFLS W FLS 5 i1 ti Ex t desfls i (4.110) des j Ex uij (4.111) various u ijvarious And the next four shadow prices, which represent 120 independent shadow prices, are defined as desafsa Ex D FLS (4.112) i D FLS i des PS Ex P TOFLS (4.113) i P desps i TO FLS i Ex (4.114) sfci sfc 124

141 des CHS, OLS, VC / PAOS Ex Q (4.115) FLS CHS, OLS, VC / PAOSi Q FLS CHS, OLS, VC / PAOSi where equations (4.110), (4.111), (4.112), (4.113), (4.114), and (4.115) represent a total of 5, 9, 20, 20, 20, and 60 unique shadow prices, respectively. The FLS has a total of 94 input shadow prices and 41 output shadow prices for a total of 135 shadow prices. Equations (4.109) through (4.115) represent the effect of the marginal change in the optimum value of the coupling function on the system-level, unit-based objective function for the FLS due to changes in the coupling functions. Problem (4.106) has seven additional terms, W FLS, t i, u i various, D FLSi, P TOFLS, sfc, i i and Q FLS CHS, OLS, VC / that represent PAOSi variations in the coupling functions listed above and are defined as FLS FLS FLS W W W (4.116) t 5 0 t i ti i i1 u ij u various ij u various ijvarious D FLS DFLS D i i FLS P TO P FLS TO PTO i FLSi FLSi sfc i sfci sfci Q FLS CHS, OLS, VC / PAOS i QFLS CHS, OLS, VC / PAOS i QFLS CHS, OLS, VC / PAOS i 125 (4.117) (4.118) (4.119) (4.120) (4.121) (4.122) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Similarly to the ECS, equations (4.116) to (4.122) are not comprehensive but rather represent 135 total equations for the FLS. The OLS system-level, unit-based synthesis/design optimization problem definition is next OLS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The OLS system-level, unit-based optimization problem is defined as follows: OLS System-Level, Unit-Based Exergy Destruction Optimization Problem

142 Minimize Ex' des OLS w.r.t. X OLS, Y OLS subject to Ex Ex n i1 des OLS desfls T uninst i Ex T des AFS A 5 W t u (4.123) WOLS uninst i Ex OLS des PS i1 ti QFLSOLS i i Q uivarious FLSOLS i i various h OLS 0 (4.124) g 0 (4.125) OLS where the vector of equality constraints, h OLS, represents the thermodynamic and geometric model of the OLS and the vector of inequality constraints, g OLS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.123) represents the system-level minimization resulting from varying the OLS decision variables only. The local or unit-based minimization objective function is denoted by Exdes OLS while the expression for the system-level, unit-based objective function, Ex' des OLS, includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of changes in the weight of the OLS, the second pair represents the time coupling with the AFS-A. The pairs within the summation from i to n account for the effects of changes in the uninstalled thrust from the PS and the heat interaction with the FLS via the fuel / oil heat exchanger. The OLS shadow prices are defined as desafsa WOLS Ex (4.126) WOLS 5 i1 ti Ex desvc / PAOS t i (4.127) 126

143 des j Ex uij (4.128) various u ijvarious and the next two shadow prices, which represent 40 independent shadow prices, are defined as des AFS A Ex T uninst (4.129) i T uninst i desfls Ex Q FLS (4.130) OLS i Q FLSOLSi where equations (4.128), (4.129) and (4.130) represent 5, 20, and 20 unique shadow prices, respectively. The OLS has a total of 30 input shadow prices and 21 output shadow prices for a total of 51 shadow prices. Equations (4.126) through (4.130) represent the effect of the marginal change in the optimum value of the coupling function on the system-level, unit-based objective function for the FLS due to changes in the coupling functions. Problem (4.123) has five additional terms, W OLS, t i, u i,, and various T uninsti Q that represent the effect that the variation in FLS OLS i the OLS decision variables has on the coupling functions listed above and are defined as OLS OLS OLS W W W (4.131) t 5 0 t i ti i i1 u ij u various ij u various ijvarious T uninst Tuninst T i i uninst Q FLSOLS QFLS OLS Q i i FLSOLS i (4.132) (4.133) (4.134) (4.135) where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Again, equations (4.131) to (4.135) are not comprehensive but rather represent 51 total equations for the OLS. The CHS system-level, unit-based synthesis/design optimization problem definition is next. 127

144 4.3.8 CHS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The CHS system-level, unit-based optimization problem is defined as follows: CHS System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' deschs i1 desafs A Ex Ex Ex Ex deschs desps desfls W u (4.136) n WCHS CHS P TO CHSi P uivarious TO CHS i i various QFLSCHSi Q FLSCHS i subject to h CHS 0 (4.137) g 0 (4.138) CHS where the vector of equality constraints, h CHS, represents the thermodynamic and geometric model of the CHS and the vector of inequality constraints, g CHS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Note that the CHS has no design decision variables and/or operational decision variables, thus there are no vectors of design decision or operational decision variables associated with problem (4.136) and only the shadow prices and coupling function pairs vary during the local optimization of the CHS. Problem (4.136) represents the system-level minimization resulting from varying the CHS interactions only as the CHS has no decision variables. The local or unit-based minimization objective function is denoted by while the expression for the systemlevel, unit-based objective function, Ex' des CHS 128 Exdes CHS, includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of changes in the weight of the CHS, the second pair accounts for the effects of changes in various coupling functions in the CHS, and the pairs within the summation from i to n account for the effects of changes in the

145 uninstalled thrust from the PS and the heat interaction with the FLS via the fuel / hydraulic oil heat exchanger. Notice that problem (4.136) has no shadow price and coupling function pair representing the AFS-A mission segment times since the CHS is calculated within the OLS unitlevel optimization and the time variables are directly assigned. The CHS shadow prices are defined as desafsa Ex (4.139) WCHS W CHS des j Ex uij (4.140) various u ijvarious des PS Ex P TOCHS (4.141) i P TO CHS i desfls Ex Q (4.142) FLS CHSi Q FLSCHS i where equations (4.140), (4.141), and (4.142) each represent 6, 20, and 20 unique shadow prices, respectively. The CHS has a total of 6 input shadow prices and 41 output shadow prices for a total of 47 shadow prices. Equations (4.139) through (4.142) represent the effect of the marginal change in the optimum value of the coupling function on the system-level, unit-based objective function for the CHS due to changes in the coupling functions. Problem (4.136) has four additional terms, W CHS, u i,, and various P TOCHS i Q that represent the effect that the variation in the FLS CHS i CHS degrees of freedom has on the coupling functions listed above and are defined as CHS CHS CHS u ij u var ij u ious various ijvarious P TO P CHS TO PTO i CHSi CHSi Q FLSCHS QFLS CHS Q i i FLSCHS i W W W (4.143) (4.144) (4.145) (4.146) 129

146 where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Again, equations (4.143) to (4.146) are not comprehensive but rather represent 47 total equations for the CHS. The ES system-level, unit-based synthesis/design optimization problem definition is next ES System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The ES system-level, unit-based optimization problem is defined as follows: ES System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' des ES Ex i1 Ex des AFS A W t W (4.147) n des ES WES ES P TO ESi 5 i1 P ti TO ES i Ex i des PS WEmpty Empty subject to h ES 0 (4.148) g 0 (4.149) ES where the vector of equality constraints, h ES, represents the thermodynamic and geometric model of the ES and the vector of inequality constraints, g ES, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Problem (4.147) represents the system-level minimization resulting from varying the ES decision variables only. The local or unit-based minimization objective function is denoted by Exdes ES while the expression for the system-level, unit-based objective function, 130 Ex' des ES, includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of changes in the weight of the ES, the second pair is the AFS-A mission

147 segment time, the third pair represents the AAF empty weight, and the pair within the summation accounts for the effects of changes in the shaft power extracted from the PS. Note that the ES has no design decision variables and/or operational decision variables, thus there are no vectors of design decision or operational decision variables associated with problem (4.147) and only the shadow prices and coupling function pairs vary during the local optimization of the ES. The ES shadow prices are defined as des AFS A Ex (4.150) WES W ES 5 i1 ti Ex t deses i (4.151) desafsa Ex W Empty (4.152) W TO ESi Empty desps Ex P TOES (4.153) i P where equations (4.153) represents 20 unique shadow prices. The ES has a total of 6 input shadow prices and 21 output shadow prices for a total of 27 shadow prices. Equations (4.150) to (4.153) represent the effect of the marginal change in the optimum value of the coupling function on the unit-based, system-level objective function for the ES. Problem (4.147) has four additional terms, W ES, t i,, 131 W Empty and P TO that represent the ESi effect that the variation in the ES degrees of freedom has on the coupling functions listed above and are defined as ES ES ES W W W (4.154) 5 i1 t i Empty t i t i 0 Empty Empty (4.155) W W W (4.156) n i1 P TOESi P TOESi TOESi P (4.157)

148 where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Equations (4.154) through (4.157) are comprehensive for the ES in this thesis work and represent 27 total equations for the ES. The FCS system-level, unit-based synthesis/design optimization problem definition is next FCS System-Level, Unit-Based Synthesis/Design Optimization Problem Definition The FCS system-level, unit-based optimization problem is defined as follows: FCS System-Level, Unit-Based Exergy Destruction Optimization Problem Minimize Ex' des FCS Ex C t des FCS C t As _ Ws Ex A s C r des AFS A C _ W s r b W FCS W L FCS b L A _ W A _ W (4.158) subject to h ES 0 (4.159) g 0 (4.160) ES where the vector of equality constraints, h FCS, represents the thermodynamic and geometric model of the ES and the vector of inequality constraints, g FCS, the physical limits imposed on the subsystem. The superscript indicates the optimum value from the previous ILGO iteration of a given subsystem objective function. Note that the FCS has no design decision variables and/or operational decision variables, thus there are no vectors of design decision or operational decision variables associated with problem (4.155) and only the shadow prices, representing the subsystem interactions, vary during the local optimization of the FCS. Thus, problem (4.158) represents the system-level minimization resulting from variation in the FCS interactions only. The local or unit-based minimization objective function is denoted by Exdes FCS while the expression for the system-level, unit-based objective function, Ex' des FCS, 132

149 includes products for paired shadow prices and associated coupling functions. The first pair accounts for the effects of changes in the weight of the FCS. The subsequent paired shadow prices and coupling functions account for the effects of changes in the wing tip and root chord lengths, wing span, wing length, aileron chord / wing chord, and aileron span / wing span. The FCS shadow prices are defined as des AFS A Ex (4.161) WFCS W t FCS des AFS A Ex (4.162) Ct C des AFS A Ex (4.163) Cr C r Exdes AFS A b (4.164) b Exdes AFS A L (4.165) L Exdes AFS A A _ W (4.166) A _ W des AFS A Ex As _ (4.167) Ws A _ W s s The ES has a total of 6 input shadow prices and 1 output shadow prices for a total of 7 shadow prices. The ES is calculated within the OLS, thus no coupling functions related to mission decision variables are required. Equations (4.161) to (4.167) represent the effect of the marginal change in the optimum value of the coupling function on the system-level, unit-based objective function for the FCS. Problem (4.158) has seven additional terms, W FCS, C t,, b, L, A _W, and A s _ Ws that represent the effect that the variation in the FCS degrees of freedom has on the coupling functions listed above and are defined as FCS FCS FCS W W W (4.168) C r 133

150 C C C (4.169) t r t C C C (4.170) r b t r b b (4.171) L L L (4.172) A _ W A _ W A _ W (4.173) s A _ W A _ W A _ W (4.174) s s s s s where the superscript " " indicates the optimum coupling function value from the previous ILGO iteration. Again, the FCS has no design decision or operational decision variables that participate in the optimization. This concludes the discussion of the system-level, unit-based synthesis/design optimization problem definitions for the AAF model developed in this thesis work. 4.4 Optimization Decision Variables and Variable Constraints This section details the optimization synthesis / design, operational, and mission decision variables and constraints for each of the subsystems included in this thesis work. The first subsystem to be detailed is the AFS-A. The fixed-wing AFS-A decision variables and variable constraints are given in Table 4.2 Table 4.2 AFS-A fixed-wing design and operational decision variables and inequality constraints. Component Design Decision Variables Constraints Fixed-wing AFS-A W Gross takeoff weight (lb) 10,000 W 60, 000 TO L Wing length (ft) 30 L 65 wing TO wing LE Leading edge sweep angle 15 LE 60 AR Aspect ratio 2 AR 10 Taper ratio 0 1 t c Thickness-to-chord ratio 0.06 t c AR Tail aspect ratio 3.5 AR 6. 5 tail t c tail Tail thickness-to-chord ratio 0.06 t tail c tail 134

151 The fixed-wing AFS-A has only a single configuration for the mission and, therefore, has a total of 8 design decision variables. Note that the gross takeoff weight, W TO, participates in the optimization as a means to establish the aircraft takeoff weight and fuel weight. The morphing-wing AFS-A design and operational decision variables and variable constraints are given in Table 4.3. Note that the design decision variables are listed first and only take one value for the entire mission. The operational decision variables each have 20 (one for each mission segment) unique values, thus, the morphing-wing AFS-A has 20 unique wing configurations which brings the total number of optimization decision variables for the AFS-A to 103. Table 4.3 AFS-A morphing-wing design and operational decision variables and inequality constraints. Component Design Decision Variables Constraints W Gross takeoff weight (lb) 10,000W 60, 000 Morphing-wing AFS-A TO AR Tail aspect ratio 3.5 AR 6. 5 tail t Tail thickness-to-chord ratio 0.06 t c tail TO tail c tail Operational Decision Variables Constraints L Wing length (ft) 30 L 65 LE Leading edge sweep angle AR Aspect ratio 2 AR 10 Taper ratio 0 1 t c Thickness-to-chord ratio 0.06 t c The AFS-A also has 6 mission decision variables that participate in the optimization. This set of variables is given in Table 4.4. The variable names correspond to the mission segments detailed Table 4.4 AFS-A mission decision variables and inequality constraints. Component Mission Decision Variables Constraints BCM 1 Best cruise Mach #1 0.6 BCM BCM 2 Best cruise Mach #2 0.6 BCM AFS-A Mission BCM 3 Best cruise Mach #3 0.6 BCM BCA 1 Best cruise altitude 1 (ft) 30,000 BCA 1 55, 000 BCA 2 Best cruise altitude 2 (ft) 30,000 BCA 2 55, 000 BCLM Best climb Mach 0.5 BCLM 1. 0 in Table 3.1. The mission variables could have been included in any of the other subsystems; but since the mission decision variables affect the AFS-A performance and weight fractions significantly, it makes sense to have them participate in the AFS-A optimization. Note that each 135

152 of these mission decision variables affects the time required to fly the mission which is an integration concern for all the subsystems. Also note that the takeoff weight, W TO, is a decision variable for both the fixed-wing and morphing-wing AFS-A optimizations. The next set of subsystem decision variables and inequality constraints to be discussed are those for the PS. The model of the PS has a reference or design engine so there are decision variables associated with the reference conditions of the engine as well. The decision variables and corresponding variable constraints for the PS are given in Table 4.5. Note that the mission defined in Table 3.1 requires military or maximum power for some of the mission segments which removes the decision variables for the corresponding mission segment. Thus, the PS has a total of 11 design decision variables and 27 operational decision variables. Table 4.5 PS design and operational decision variables and inequality constraints. Component Design Decision Variables Constraints Propulsion Subsystem M Design Mach number 0.8 M Alt Design altitude 30,000 Alt 50, 000 m Design total mass flow rate (lb/s) c 0 Design total compressor pressure ratio c' Design fan pressure ratio 2.0 c ' 6. 0 Design bypass ratio T Design burner temperature (R) 1000 T t4 t7 t T Design afterburner temperature (R) 1000 T t M Design mixer Mach number for core 0.4 M P Design power takeoff (kw) 0.0 P TO Design bleed air ratio Operational Decision Variables Constraints T Burner temperature (R) 1000 T t4 t7 c TO t T Afterburner temperature (R) 1000 T t Next comes the ECS. The decision variables and the corresponding constraints are shown in Table 4.6. The ECS has a total of 23 synthesis / design decision variables. Note, however, that there are two possible configurations for the regenerative heat exchanger. This requires that another independent set of regenerative heat exchanger variables must be used as well bringing the total to 28 synthesis / design decision variables. Table 4.6 ECS optimization synthesis / design and operational decision variables and inequality constraints. Component Design Decision Variables Constraints 136

153 Primary Heat Exchanger Secondary Heat Exchanger Regenerative Heat Exchanger Fin Bleed-air fin type 11 Fin 20 b Fin Ram-air fin type 11 Fin 20 b r L Bleed-air side length, m 0.5 L 0. 9 L Ram-air side length, m 0.06 L 0. 9 r L Non-flow length, m 0.5 L 0. 9 n Fin Bleed-air fin type 11 Fin 20 b Fin Ram-air fin type 11 Fin 20 b r L Bleed-air side length, m 0.5 L 0. 9 L Ram-air side length, m 0.06 L 0. 9 r L Non-flow length, m 0.5 L 0. 9 n Fin Hot-side (bleed air) fin type 11 Fin 20 h Fin c h Cold-side (from the water separator) fin type b n b n b r r b r r h 11 Fin 20 L Hot-side length, m 0.3 L 0. 5 b c L Cold-side length, m 0.15 L 0. 3 c L Non-flow length, m 0.3 L 0. 5 n Fin Bleed-air fin type 11 Fin 20 b n c b Bleed Air / PAO Heat Exchanger Fin PAO fin type 11 Fin 20 b r L Bleed-air-side length, m 0.5 L 0. 9 L PAO-side length, m 0.06 L 0. 9 r b r r Compressor L Non-flow length, m 0.5 L 0. 9 n PR des Design compressor pressure ratio n 1.8 PR 3.0 Inlet Duct 1 A 1 Inlet duct 1 area, cm 2 8 A Inlet Duct 2 A 2 Inlet duct 2 area, m 2 8 A Component Operational Decision Variables Constraints Pressure Regulating Valve PR Pressure setting 1.0 PR 6. 0 Bleed Port Air Cycle Machine Splitter Regenerative Heat Exchanger vv Bleed port selection (high or Bleed Bleed (0,1) low pressure bleed port) v Hot-air bypass 0.0 v 1. 0 hot bypass vv des hot v Compressor bypass 0.0 v 1. 0 m Regenerative heat exchanger reg mass flow rate, kg/s bypass 0.0 m 0.2 reg 137

154 Because of certain difficulties encountered during the optimization process in this thesis work, the ECS flies only 4 of the 21 mission segments to establish the ECS subsystem synthesis / design and subsystem interactions. The mission segments flown are warmup and takeoff (segment 1), subsonic climb (segment 4), supersonic turn (segment 11), and subsonic cruise (segment 19). This brings the total number of operational variables to 20, though if the entire mission were used, 100 operational decision variables would be required for the ECS. (In the work of Rancruel (2003), the subsonic cruise mission segment was used as the design segment.) The next subsystem to be discussed is the FLS. The synthesis / design decision variables and constraint limits for the FLS are given in Table 4.7. The FLS optimization only has 8 Ram Air / Fuel Heat Exchanger Table 4.7 FLS optimization decision variables and inequality constraints. Component Design Decision Variables Constraints Fuel/Oil Heat Exchanger Fin Ram air fin type 11 Fin 20 r Fin Fuel side fin type 11 Fin 20 r f L Ram air side length, m 0.05 L 0. 9 L Fuel side length, m 0.05 L 0. 9 f L Non-flow length, m 0.05 L 0. 9 n L c Cold-side length (m) Lc 0. 9 L h Hot-side length (m) 0. 1 Ln 0. 9 L n Non-flow length (m) 0. 1 Ln 0. 9 synthesis / design decision variables since the heat exchangers are sized in other subsystems. Note that the fuel added to the PS-required fuel mass flow rate, m, is an operational fuel _ add variable. However, the fuel added to the flow is iterated within simulation and does not participate as a decision variable in the optimization (see section 3.6.1). Thus, the FLS has no operational decision variables. The synthesis / design and operational decision variables and inequality constraints for the VC/PAOS are shown in Table 4.8. The VC/PAOS flies only a single mission segment to establish its synthesis / design conditions and performance. The mission segment chosen is the most constrained mission segment of the four flown by the ECS which is the segment with the highest cooling load required from the cold PAO loop. If the entire mission were being optimized, a total of 60 operational decision variables would participate in the optimization; r f r f n 138

155 however, for this thesis work, the VC/PAOS has 20 synthesis / design decision variables and only 3 operational decision variables. Table 4.8 VC/PAOS optimization synthesis / design and operational decision variables and inequality constraints. Condenser Evaporator Component Synthesis / Design Decision Variables Constraints Fin Vapor-side fin type 11 Fin 20 Ram Air / PAO Heat Exchanger Fuel / PAO Heat Exchanger v Fin Liquid-side fin type 11 Fin 20 v p L Vapor-side length, m 0.5 L 0. 9 L Liquid-side length, m 0.06 L 0. 9 p L Non-flow length, m 0.5 L 0. 9 n v v n Fin Vapor-side fin type 11 Fin 20 Fin Liquid-side fin type 11 Fin 20 v p L Vapor-side length, m 0.5 L 0. 9 L Liquid-side length, m 0.06 L 0. 9 p L Non-flow length, m 0.5 L 0. 9 n Fin Hot-side (bleed air) fin type 11 Fin 20 h Fin Ram-air fin type 11 Fin 20 h r L Hot-side length, m 0.3 L 0. 5 L Ram-air side length, m 0.05 L 0. 9 r L Non-flow length, m 0.3 L 0. 5 n Fin Bleed-air fin type 11 Fin 20 p Fin Fuel-side fin type 11 Fin 20 p f L Bleed-air side length, m 0.5 L 0. 9 L Fuel-side length, m 0.05 L 0. 9 f L Non-flow length, m 0.5 L 0. 9 n Component Operational Decision Variables Constraints m Vapor mass flow rate, kg/s 0.2 m 2. 2 Mass Flow Rates vapor pao _ hot v n h n p n v p p v p p h r r f p f vapor m Hot PAO loop mass flow rate, kg/s 0.2 m _ 3. 5 m Cold PAO loop mass flow rate, kg/s 0.2 m _ 3. 5 pao _ cold The next subsystem is the OLS. Its operational decision variables and variable constraints are given in Table 4.9. The OLS heat exchangers are sized in the FLS. The OLS has a total of 20 operational decision variables. Table 4.9 OLS optimization operational decision variables and inequality constraints. Component Operational Decision Variables Constraints OLS operational variables m Mass flow rate of oil, kg/s 0 m 4. 0 oil pao pao oil hot cold 139

156 Finally, the remaining subsystems, the ES, CHS, and FCS are passive subsystems in terms of the optimization as they have no local design or operational decision variables associated with them in this thesis work. They do, however, participate via system-level degrees of freedom, i.e. via coupling functions. Thus, they must be considered in the subsystem integration as their operation has a direct effect on the overall system-level objective function and on system performance. 4.5 Solution Approach Both physical and time decomposition are applied in this thesis work to the AAF aircraft which is modeled and optimized based on 9 subsystems separated by physical or thermodynamic boundaries and flown over a mission separated into 21 time segments. Even though the AFS-A, PS, ECS, FLS, VC/PAOS, CHS, OLS, ES, and FCS are modeled and optimized separately, coupling functions update the subsystem interactions between ILGO iterations. Each of the subsystem optimizations, thus, represent a unit (or local) component of the overall system optimization problem. Figure 4.6 shows the notional flow of the ILGO approach for the AAF aircraft system. Starting from the individual subsystem optimizations in Figure 4.6, the ILGO process is as follows: Assign shadow prices to the subsystems and pass these and their updated coupling functions to the subsystems Individual Subsystem Optimizations AFS-A PS ECS FLS OLS VCPAOS Simulate Dependent Subsystems CHS ES FCS Calculate the systemlevel objective function Update the coupling functions ILGO Iteration Calculate the new shadow prices Figure 4.6 Diagram of optimization problem solution approach using ILGO. 140

157 1. The initial subsystem optimizations start with arbitrary values for the coupling functions and shadow prices, but within specified limits. 2. After the AFS-A, PS, ECS, FLS, OLS, and VC/PAOS are optimized, the CHS, ES, and FCS are simulated based on outputs from the optimized subsystems. 3. The overall system-level objective value is calculated. 4. New shadow prices are calculated based on the initial result. 5. The ILGO iteration count is incremented. 6. The coupling function values are updated based on the individual subsystem optimization results. 7. Finally, new shadow prices are assigned to the appropriate unit-level subproblems and these and their associated updated coupling function values are passed to the subsystems so that the next set of individual subsystem optimizations can proceed. The explanation of ILGO provided here is merely a top-level overview. The reader is once again directed to Muñoz and von Spakovsky (2000a,b,c,d; 2001a,b), Rancruel (2002, 2005), and Rancruel and von Spakovsky (2005, 2006) for a detailed explanation of ILGO and the other facets of the ILGO approach mentioned here. The software package used in this thesis work is briefly described in the following subsection. 4.6 iscript Scripting Language and Optimization The software used for the optimization of this thesis work is called iscript which is in development by TTC Technologies, Inc. iscript was developed under an Air Force Small Business Innovative Research (SBIR) Phase II project. The reasons for using iscript include the following attributes that make it attractive for this thesis work: Quick learning curve for the programming syntax; Built-in optimization tools; Component-based programming structure; Automated ILGO; and Automated parallelization for the optimization. 141

158 Another large contributing factor to the decision to use iscript is the fact that the subsystems were already mostly written (or being written) in iscript as a demonstration to the Air Force and industry. The optimization in iscript is performed on two levels. The unit or subsystem level is optimized using a genetic algorithm (GA) developed by researchers at the Laboratoir d energetique industrielle at the Ecole Polytechnique Federale de Lansanne in Lausanne, Switzerland (Leyland, 2002). The parameter set in the original implementation was condensed to five parameters by TTC including the population size, initial evaluations, convergence criteria, mutation frequency, and maximum number of generations to simplify the user interaction with the method. To arrive at a system-level optimum, the ILGO decomposition strategy is implemented in iscript. GAs are based on Darwin s theory of natural selection or survival of the fittest. The initial population or set of optimization decision variable values is generated by stepping through the range of values possible for each optimization variable. After the initial population is generated, the algorithm reorders the optimization variables based on the values of the objective functions they yield, i.e., the solutions are ordered from the best to the worst solution based on the value of the objective. The GA then selects the better sets of optimization decision variable values, called parents, mutates them slightly, then repopulates the bottom half of the population with the mutated optimization decision variable values or offspring of the parents. The new population members or sets of decision variable values are then evaluated in the model and the entire population is again reordered from the best to the worst solution and the selectionmutation-repopulation process continues until the convergence criteria is met or until the maximum number of generations specified is reached. The implementation of ILGO in iscript uses a gradient-based method to converget the decomposed system to a system-level optimum. In this approach, gradients are computed using the shadow prices of the coupling functions between the subsystems. The gradients are used along with the coupling functions to search the system-level ORS for the overall system optimum. The shadow prices represent the effect of the unit-level coupling functions on the system-level objective function. Such an approach allows multiple subsystem optimizations to take place at the same time since each subsystem is effectively isolated from the others while at the same time assuring a system-level optimum by periodically updating the subsystem interactions between ILGO steps via coupling functions. 142

159 Chapter 5 Results and Discussion This chapter discusses the results obtained during this thesis work. The first section starts with a set of results from the first phase of this thesis work that is an extension of the work of Butt (2005) from a paper presented at the 2007 AIAA Thermophysics Conference held in Miami, FL (Smith, et. al, 2007). The next section presents results and analysis of a partial optimization of the 9 subsystem AAF that is written in iscript along with predicted optimal results and analysis. The final section is a parametric study of the morphing AFS-A behavior. 5.1 Two-Subsystem Optimization Results The work of Smith, et al. (2007) studies the benefits of using morphing wing technology in an AAF. Two subsystems are modeled: the AFS-A, both morphing- and fixed-wing models, and a turbofan PS as described in Chapter 3. The AFS-A models were developed and used by Butt (2005). The morphing-wing AFS-A is physically similar to that described in Chapter 3 with a few differences which are as follows: root- and tip-chord lengths are operational decision variables while the AFS-A described in Chapter 3 uses the taper ratio as the operational decision variable to establish the root- and tip-chord lengths; morphing AFS-A wing-weights are established based on setting the wing sweep to zero. This makes the wing-weight estimate lower than if the wing had some amount of sweep, thus, the metric to compare fixed-wing and morphing-wing results in Smith et al. (2007) is gross takeoff weight; and wing-weight penalties and fuel-weight penalties are established in the manner described in Section ; however, the excess fuel that is carried to power the actuators that morph the wings is not expelled in Smith et al. (2007) in order to match what was done in Butt (2005). The mission flown in Smith et al. (2007) is similar to that described in Chapter 3 but with some differences. The AAF model does not fly the entire mission but rather flies a subset of the 143

160 mission and no DOF are associated with the mission (i.e. the mission segments are fixed and do not participate in the optimization) as was done in the work of Butt (2005). The optimization is handled in MATLAB using a genetic algorithm developed by Leyland (2002). The fixed-wing AFS-A has 4 synthesis/design decision variables (wing length, wing sweep, and root- and tip- chord lengths) while the morphing-wing AFS-A has the same number of synthesis / design decision variables and a total of 72 operational decision variables (18 sets of wing length, wing sweep, and root- and tip- chord lengths). The turbofan PS has 9 synthesis/design decision variables (Mach number, mass flow rate, altitude, compressor pressure ratio, fan pressure ratio, bypass ratio, main burner temperature, afterburner temperature, and mixer Mach number) and 28 operational decision variables (nineteen main burner temperatures and nine afterburner temperature settings). The optimization objective function used is that of minimizing the fuel burned over the mission. Figure 5.1 shows the sensitivity of the total fuel consumed over the mission with respect to the wing- weight and fuel- weight penalties for the morphing-wing AAF. Note that the fixedwing result is shown in Figure 5.1 with the red horizontal line. The shaded area below the red line indicates the fuel savings region (i.e. any combination of wing-weight and fuel-weight Figure 5.1 Sensitivity analysis of morphing-wing effectiveness for different wing- and fuel-weight penalties (Smith et al., 2007). 144