Optimum Propeller Design for Electric UAVs by David Lee Wall A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama August 4, 2012 Keywords: Propeller, Design, Optimization Copyright 2012 by David Lee Wall Approved by Gilbert Crouse, Associate Professor of Aerospace Engineering Roy Hart eld, Professor of Aerospace Engineering Brian Thurow, Associate Professor of Aerospace Engineering George Flowers, Dean of the Graduate School, Professor of Mechanical Engineering Abstract A propeller behaves as a rotating wing producing lift in the direction of the axis of rotation. Many previous propeller optimization methods have been developed, but usually focus on piston or turboprop applications. This study discusses the more fundamental pro- peller theories and uses a hybrid blade element momentum theory to model the propellers. A brushless motor model is developed and coupled with the propeller theory in an opti- mizer. Two single point optimizations are made, one for a climb condition and the other for a cruise condition. A third optimization is presented with optimization at climb and cruise conditions. The optimizations are conducted with a hybrid pattern/search particle swarm optimizer. The airfoils for the propellers are optimized with the same optimizer and a simplex method. Multiple objective functions are evaluated for each of the conditions. One having non-dimensional values and another with dimensional values. Dimensional val- ues prove to provide better results for all of the conditions. The optimized cruise propellers display smaller chords, higher pitches, and larger diameters while the optimized climb pro- pellers have larger chords, lower pitches and smaller diameters. The multipoint optimization yields higher pitches with chords and diameters between the single point optimizations. All optimized propellers show improvement over comparable baseline propellers. ii Acknowledgments I would like to thank my parents, Larry and Patsy Wall, for their loving support over the years. I would like to thank Dr. Gilbert Crouse for his guidance with this thesis and studies. I would also like to thank Dr. Brian Thurow and Dr. Roy Hart eld for their help throughout my stay at Auburn University. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Airfoil Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Parameterization of Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Bezier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Bernstein Polynomials Representation of the Unit Shape Function . . 6 2.3 XFOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Basics of Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.2 Other Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.3 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Propeller Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.1 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 Simple Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . 16 3.2.3 Hybrid Momentum Blade Element Theory . . . . . . . . . . . . . . . 22 4 Electric Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Brushed DC Electric Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 iv 4.2 Brushless DC Electric Motors . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Particle Swarm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Pattern Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 Hybrid Pattern Search/Particle Swarm Method . . . . . . . . . . . . . . . . 36 5.4 Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.1 Airfoil Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.2 Propeller Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.1 Airfoil Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Validation of Propeller Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.2.1 Validation Results for Baseline Cruise Propeller . . . . . . . . . . . . 49 7.2.2 Validation Results for Baseline Climb Propeller . . . . . . . . . . . . 51 7.2.3 Validation Results for Baseline Climb Cruise Propeller . . . . . . . . 54 7.3 Propeller Optimized for Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.4 Propeller Optimized for Climb . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.5 Propeller Optimized for Climb-Cruise . . . . . . . . . . . . . . . . . . . . . . 76 7.6 Cruise, Climb, and Climb-Cruise Comparisons . . . . . . . . . . . . . . . . . 85 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A Xfoil Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 A.1 Example XFOIL Command Inputs . . . . . . . . . . . . . . . . . . . . . . . 95 A.2 Example XFOIL Point File . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 A.3 Example XFOIL Output File . . . . . . . . . . . . . . . . . . . . . . . . . . 97 B Derivation of Adkins-Liebeck Di erential Coe cients . . . . . . . . . . . . . . . 98 v C Airfoil Optimization Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.1 AirfoilMaker() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.2 ParametricAirfoil() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 C.3 ClCdFinder() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 D Propeller Optimization Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 D.1 BrushlessMotor() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 D.2 PropellerPerformance() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 D.3 ClCdFinderAirfoilID() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 E Optimized Airfoil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 F Optimized Cruise Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 G Optimized Climb Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 H Optimized Climb-Cruise Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . 128 vi List of Figures 2.1 Example Airfoil with Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Example of Bezier Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Example of Camber Line, Thickness Line and Corresponding Airfoil . . . . . . . 5 2.4 Example of Upper Surface Using Parameterization . . . . . . . . . . . . . . . . 8 2.5 Example of a Random Airfoil Using Parameterization . . . . . . . . . . . . . . . 8 3.1 Blade Cross Section Velocity Disgram . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Propeller Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Propeller Blade Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Momentum Theory Stream Tube . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Momentum Theory Pressure and Velocity through Propeller Disk . . . . . . . . 14 3.6 Ideal Momentum Theory E ciency and Actual Propeller E ciency Verse Thrust Coe cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.7 Example of a Blade Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.8 Velocity Vector Diagram with Reactions on a Blade Element . . . . . . . . . . . 19 3.9 Simple Blade Element Helix Angle E ciency . . . . . . . . . . . . . . . . . . . 21 vii 3.10 Simple Blade Element Free Stream Velocity E ciency . . . . . . . . . . . . . . 22 3.11 Weick?s In ow Method Velocity Vectors . . . . . . . . . . . . . . . . . . . . . . 23 3.12 Axial and Rotational Interference Factor Blade Element Velocity Vectors . . . . 25 4.1 A Simple Equivalent Circuit of a DC Motor . . . . . . . . . . . . . . . . . . . . 30 4.2 Torque Speed Curve Example for a DC Motor . . . . . . . . . . . . . . . . . . . 31 4.3 Torque Speed Power Curve Example for a DC Motor . . . . . . . . . . . . . . . 32 4.4 Electric Motor and Internal Combustion Engine Comparison . . . . . . . . . . . 33 6.1 Flowchart for Airfoil Optimization Process . . . . . . . . . . . . . . . . . . . . . 39 6.2 Flowchart for Propeller Optimization Process using fminsearch() . . . . . . . . 40 6.3 Flowchart for Propeller Optimization Process using Hybrid Optimizer for Cruise Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.4 Flowchart for Propeller Optimization Process using Hybrid Optimizer for Climb Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.5 Flowchart for Propeller Optimization Process using Hybrid Optimizer for Climb- Cruise Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.1 Optimized Airfoil at 0 Angle of Attack . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Optimized Airfoil Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.3 Thrust Validation for APC 11x10E Cruise Propeller at 8000 rpm . . . . . . . . 49 7.4 Power Validation for APC 11x10E Cruise Propeller at 8000 rpm . . . . . . . . . 50 viii 7.5 E ciency Validation for APC 11x10E Cruise Propeller at 8000 rpm . . . . . . . 51 7.6 Thrust Validation for APC 8x6E Climb Propeller at 8000 rpm . . . . . . . . . . 52 7.7 Power Validation for APC 8x6E Climb Propeller at 8000 rpm . . . . . . . . . . 53 7.8 E ciency Validation for APC 8x6E Climb Propeller at 8000 rpm . . . . . . . . 53 7.9 Thrust Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm . . . . . 54 7.10 Power Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm . . . . . . 55 7.11 E ciency Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm . . . . 56 7.12 Fitness verse Number of Generations Case 1 (Objective Function = Cp) . . . . . 57 7.13 Fitness verse Number of Generations Case 2a (Objective Function =PropellerPower) 58 7.14 Fitness verse Number of Generations Case 2b (Objective Function =SystemPower) 58 7.15 Chord Distribution for Cruise Condition Propellers . . . . . . . . . . . . . . . . 62 7.16 Blade Angle Distribution for Cruise Condition Propellers . . . . . . . . . . . . . 62 7.17 Comparison of Possible Blade Chord Pro les for Case 1 . . . . . . . . . . . . . . 63 7.18 Lift Distribution on Blade Comparing the Number of Elements used to nd Per- formance Parameters for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.19 Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller Thrust Over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . 65 7.20 Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller Power Over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . 66 ix 7.21 Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller E ciency Over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . 66 7.22 Fitness verse Number of Generations Case 3 (Objective Function = 1Ct) . . . . . 69 7.23 Fitness verse Number of Generations Case 4 (Objective Function = 1Ft) . . . . . 69 7.24 Chord Distribution for Climb Condition Propellers . . . . . . . . . . . . . . . . 71 7.25 Blade Angle Distribution for Climb Condition Propellers . . . . . . . . . . . . . 72 7.26 Case 3 Angle of Attack Distribution . . . . . . . . . . . . . . . . . . . . . . . . 73 7.27 Comparison of Case 3 and Case 4 Propellers Thrust over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.28 Comparison of Case 3 and Case 4 Propellers Power over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.29 Comparison of Case 3 and Case 4 Propellers E ciency over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.30 Fitness verse Number of Generations Case 5 (Objective Function = 1 prop motor) 78 7.31 Fitness verse Number of Generations Case 6 (Objective Function = PowerCruiseThrust Climb ) 78 7.32 Chord Distribution for Climb Cruise Condition Propellers . . . . . . . . . . . . 81 7.33 Blade Angle Distribution for Climb Cruise Condition Propellers . . . . . . . . . 81 7.34 Comparison of Case 5 and Case 6 Propellers Thrust over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.35 Comparison of Case 5 and Case 6 Propellers Power over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 x 7.36 Comparison of Case 5 and Case 6 Propellers E ciency over a Range of Free Stream Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.37 Comparison of Blade Angles, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.38 Comparison of Chords, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) 86 7.39 Comparison of Thrust, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) 87 7.40 Comparison of Power, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) 88 7.41 Comparison of E ciency, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xi List of Tables 2.1 6th Order Bernstein Polynomials Parameters . . . . . . . . . . . . . . . . . . . . 6 2.2 Class Function Coe cients and Corresponding Geometric Shapes . . . . . . . . 7 2.3 Binomial Coe cients for Bernstein Polynomial Example . . . . . . . . . . . . . 7 7.1 DC Brushless Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.2 Flight Condition Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3 Objective Functions used for Cruise Condition . . . . . . . . . . . . . . . . . . . 56 7.4 Optimizer Limits for Cruise Condition . . . . . . . . . . . . . . . . . . . . . . . 57 7.5 Propeller Performance Parameters for Cruise Case 1 . . . . . . . . . . . . . . . 59 7.6 Propeller Performance Parameters for Cruise Case 2a and 2b . . . . . . . . . . . 60 7.7 Objective Functions used for Climb Condition . . . . . . . . . . . . . . . . . . . 67 7.8 Optimizer Limits for Climb Condition . . . . . . . . . . . . . . . . . . . . . . . 68 7.9 Propeller Performance Parameters for Climb Case 3 and Case 4 . . . . . . . . . 70 7.10 Objective Functions used for Climb Cruise Condition . . . . . . . . . . . . . . . 76 7.11 Optimizer Limits for Climb-Cruise Condition . . . . . . . . . . . . . . . . . . . 77 7.12 Propeller Performance Parameters for Climb Cruise Case 5 . . . . . . . . . . . . 79 7.13 Propeller Performance Parameters for Climb Cruise Case 6 . . . . . . . . . . . . 80 E.1 Coe cients for Bernstein Polynomial for of Upper Surface of Optimized Airfoils 120 E.2 Coe cients for Bernstein Polynomial for of Lower Surface of Optimized Airfoils 121 E.3 Lift Coe cients, and Drag to Lift Ratios for Optimized Airfoils . . . . . . . . . 122 F.1 Propeller Properties for Cruise Case 1 (Objective Function = Cp) . . . . . . . . 123 xii F.2 Propeller Properties for Cruise Case 2a (Objective Function = PropellerPower) 124 F.3 Propeller Properties for Cruise Case 2a (Objective Function = PropellerPower) 125 G.1 Propeller Properties for Climb Case 3 (Objective Function = 1Ct) . . . . . . . . 126 G.2 Propeller Properties for Climb Case 4 (Objective Function = 1Ft) . . . . . . . . 127 H.1 Propeller Properties for Climb Cruise Case 5 (Objective Function = 1 prop motor)129 H.2 Propeller Properties for Climb Cruise Case 6 (Objective Function = PowerCruiseThrust Climb ) 130 xiii List of Abbreviations Angle of Attack Blade/Helix Angle P Change in Pressure Drag to Lift Ratio E ciency Circulation Reaction Force Angle m Flux Linkage of the Stator Winding Per Phase Flow Angle t Flow Angle at Blade Tip Density Solidity Electric Motor Torque f Parameter in Prandtl Momentum Tip Loss Equation Nondimensional Radius Displacement Velocity Ratio A Area or Disk Area D4 4 xiv a Axial Interference Factor a0 Rotational Interference Factor aair Speed of Sound AR Aspect Ratio B Number of Blades b Chord of a Blade Element, Axial Slipstream Factor c Chord CD Drag Coe cient CL Lift Coe cient Cp Power Coe cient Ct Thrust Coe cient D Diameter, Drag F Prandtl Momentum Tip Loss Factor Ft Thrust I Current I0 Idle Current J Advance Ratio KE Back E.M.F. Constant Kt Motor Torque Constant KV Motor Speed Constant RPMV xv L Lift m Number of Phases n Rotational Speed P Power p Number of Poles in Electric Motor P0 Total Pressure pe E ective Pitch pe Geometric Pitch q Dynamic Pressure R Resultant Force in Blade Element r Radius RI Internal Resistance Tem Electromagnetic Torque u Relative Velocity in Free Stream Direction v0 Vortex Displacement Velocity V0 Initial/Free Stream Velocity Vd Flow Velocity at Disk VS Velocity Downstream/Slipstream Vmotor Voltage VRel Relative Velocity xvi x Empirical In ow Factor z Fitness of Objective Function xvii Chapter 1 Introduction Propellers are one of the fundamental elements of propulsion and aircraft design, acting like a rotating wing to produce lift in the same direction as the axis of rotation. There are several di erent methods used to calculate the performance parameters of a propeller. These include momentum theories [1], blade element theories [2], hybrid blade element momentum theories [3][4], and lifting line theories [5]. Procedures have been developed to optimize propellers that do not require computers. With advances in computers, optimization is becoming more readily available and allows for more design variables to be optimized. One optimization method was developed by Adkins and Liebeck in 1983 [6] and has several limitations, but is easily implemented. This method will only give the optimum blade angles and chords for a particular free stream velocity and will not solve for diameters or multiple design points. Fanjoy and Crossley performed a two dimensional optimization using a genetic algorithm [7]. Their method used a panel method for aerodynamic analysis on the propeller blades and included structural penalty functions to ensure a feasible propeller. This method showed some over prediction in airfoil data which sometimes led to bad results. Miller used a vortex lattice method for a three dimensional optimization of a propeller [8]. One panel was used in this method with no camber. Burger in 2007 [5] developed another method using lifting line theory and a genetic algorithm to optimize propellers for noise reduction over a range of operation. Due to the fact that a propeller acts like a rotating wing its cross section is an airfoil. Consequently one aspect of optimizing a propeller is optimizing the airfoil shapes used along the blade span. Optimizing an an airfoil shape can be challenging if the shape is described using individual points. It can take 50-100 points to e ectively describe an airfoil shape 1 which is to many parameters to optimize e ciently. A common approach to describing the number of parameters used to describe an airfoil is to parameterize the shape. Addressing this obstacle Kulfan has developed a process that uses Bernstein polynomials to represent the points of an airfoil [9]. Her approach was adopted for this e ort. Small UAVs are becoming more popular. Advances in small brushless DC motors and lithium polymer battery technology have created useful drive systems for these UAVs. This leads to a desire to design propellers for UAV systems that are as e cient as possible when used with electric motors. The purpose of this study is to examine propeller optimization with a coupled electric motor. A method will be developed to optimize propellers given a brushless electric motor for single point or multiple point design conditions. A hybrid blade element momentum theory method will be used for propeller performance analysis. Validation of the propeller performance will be included. General trends in propellers at design conditions will be presented along with the results of optimized propellers. 2 Chapter 2 Airfoils 2.1 Airfoil Basics Like a wing the cross section of a propeller blade is an airfoil. Airfoils produce a lifting force by creating a low pressure on the surface in the direction of lift and a higher pressure in the opposite direction of lift. An airfoil has several key geometric features. The leading and trailing edge mark the front and back of the airfoil as well as separate the upper surface from the lower surface. The chord, c, is a straight line drawn from the leading edge to the trailing edge. If the upper and lower surfaces are mirror images of each other the airfoil is said to be symmetric. The line consisting of points halfway between the upper and lower surface is known as the mean camber line [10]. The camber is de ned as the maximum distance perpendicular to chord line and mean camber line. A visual interpretation of the nomenclature for an airfoil is shown in Figure 2.1. 0 0.2 0.4 0.6 0.8 1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Camber Chord Line Mean Camber Line Trailing Edge Upper Surface Lower Surface Leading Edge Figure 2.1: Example Airfoil with Nomenclature 3 2.2 Parameterization of Airfoils Airfoil shapes are potentially hard to optimize if only the coordinates are known due to the high number of points that need to be used to de ne an airfoil accurately. To decrease the number of terms used to de ne an airfoil a process called parameterization is used for upper and lower surfaces. 2.2.1 Bezier Curves Bezier curves are one of the many ways to represent an airfoil. A parametric Bezier curve of degree n is described in Equation 2.1. B(t) = nX i=0 Bi n!i! (n i)!ti (1 t)n i (2.1) Venkataraman [11] used four cubic Bezier curves to describe an airfoil. His method spilt the airfoil into an upper and lower surface and used two curves to de ne each surface. Rogalsky [12] expanded on Venkataramans work by using four cubic Bezier curves to de ne a camber and thickness line. The curves can then be combined to form an airfoil. Using Equation 2.2 an example Bezier curve can be constructed and is shown in Figure 2.2. B(t) = (1 t)3B0 + 3 (1 t)2tB1 + 3 (1 t)t2B2 +t3B3 (2.2) B0; B1; B2; and B3 are coordinate location for each of the control points and t ranges from 0 to 1. Camber and thickness lines can then be made. The rst of the two Beizer curves that make up the camber line is anchored at the origin at one end, and the other end is anchored at the location of maximum camber. The second line is anchored at the location of maximum camber and at one unit in the x-direction. The thickness line is constructed in the same manner with the location of maximum thickness between the inner anchored points. The other points used to construct the curves are placed in such a way to 4 0 0.2 0.4 0.6 0.8 1 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 B 0 B 1 B 2 B 3 Figure 2.2: Example of Bezier Curve provide an appropriate curve. These points along with the locations of the maximum camber and thickness are unknown variables that can be moved to create an airfoil. An example camber line, thickness line, and corresponding airfoil are shown in Figure 2.3 with the Beizer coe cients being displayed in the squares and the solid lines representing the upper and lower surfaces. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01 0 0.01 0.02 0.03 x/c y/c Camber Line 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 x/c y/c Thickness Line 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 x/c y/c Example Airfoil Figure 2.3: Example of Camber Line, Thickness Line and Corresponding Airfoil 5 2.2.2 Bernstein Polynomials Representation of the Unit Shape Function Bernstein polynomials are a special case of Bezier curves. Bernstein polynomials only range from 0-1 on the x-axis. Kulfan discusses a method for parameterizing an airfoil using Bernstein polynomials in Reference [9], and her method will be discussed here. The airfoil shape is divided into an upper and lower surface. The following process will need to be repeated for the lower surface. First an overall shape function for the upper surface will be de ned in Equation 2.3. S( ) = nX i=1 AuiSi ( ) (2.3) Where = xc, Si ( ) is a shape function, and Aui are the unknown coe cients that de ne the contour. A unit shape function is then de ned by the Bernstein polynomials in Equation 2.4. Sr;n (x) = Kr;nxr (1 x)n r (2.4) The Bernstein polynomial is of order n, r = 0;1;2;:::;n and Kr;n are binomial coe cients shown in Equation 2.5. Kr;n (nr) n!r! (n r)! (2.5) An example of Equation 2.4 evaluated for a 6th order Bernstein polynomial is shown in Table 2.1. A class function can now be introduced in Equation 2.6. This class function de nes the i Kr;n Sr;n (x) 0 1 (1 x)6 1 6 6 (1 x)5x 2 15 15 (1 x)4x2 3 20 20 (1 x)3x3 4 15 15 (1 x)2x4 5 6 6 (1 x)x5 6 1 x6 Table 2.1: 6th Order Bernstein Polynomials Parameters 6 leading and trailing edges of the airfoil. CN1N2 ( ) = N1 (1 )N2 (2.6) Table 2.2 provides example values of the N1 and N2 coe cients and the corresponding shape that they yield [13]. A trailing edge o set is de ned by Equation 2.7 where zte is the height N1 N2 Description 0.5 1.0 Round-Nose and Pointed Aft End Airfoil 0.5 0.5 Elliptic or Ellipsoid Body of Revolution 1.0 1.0 Biconvex Airfoil 0.75 0.75 Sears-Haack Body 0.75 0.25 Low-Drag Projectile 1.0 0.001 Cone or Wedge 0.001 0.001 Rectangle, Circular Duct, or Circular Rod Table 2.2: Class Function Coe cients and Corresponding Geometric Shapes of the trailing edge from the x-axis. = ztec (2.7) An equation for the upper surface, upper, can de ned in Equation 2.8 by multiplying Equa- tions 2.4 and 2.6 and adding the o set in Equation 2.7, where = zc. upper = CN1N2 ( )S( ) + upper (2.8) The results of an example of this process is shown in Figure 2.4 where the binomial coe cients can be found in Table 2.3. The coe cients for the class function are N1 = 0:5 and N2 = 1:0, and the leading edge radius is equal to 0.03. i 0 1 2 3 4 5 6 Aupperi 0.2 0.3 0.2 0.2 0.2 0.1 0.1 Aloweri -0.2 -0.2 0.2 - 0.2 0.2 - 0.05 -0.05 Table 2.3: Binomial Coe cients for Bernstein Polynomial Example 7 0 0.0 2 0.0 4 0.0 6 0.0 80.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z / c x/c Figure 2.4: Example of Upper Surface Using Parameterization If this process is repeated for the lower surface di erent binomial coe cients will need to be used to provide a di erent contour. The upper and lower surfaces can then be combined to form an airfoil. An example of a randomly generated airfoil is shown in Figure 2.5 using the binomial coe cients from Table 2.3. Kulfan provides results that indicate that this method -0.1-0.0500.050.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z / c x/c Figure 2.5: Example of a Random Airfoil Using Parameterization is a suitable method for describing an airfoil when the minimum order for the Bernstein polynomials is higher than 5th order [9]. 2.3 XFOIL There are many di erent established computer programs and methods for calculating two dimenisonal lift and drag coe cients of an airfoil. A program developed by Drela in 1986 called XFOIL will be brie y discussed here [14] [15]. XFOIL was originally designed for assisting with the development of airfoils for human powered aircraft and low Reynolds number aircraft. Appendix A contains sample inputs and output les for typical sessions where lift and drag are desired. 8 XFOIL can only provide results for two dimensional airfoils. Its capabilities include both inviscid and viscous solutions. Inviscid solutions are solved using a vortex sheet on the surface of the airfoil and a source sheet the surface of the airfoil and its wake. Once the unknown vortices are found a corresponding pressure distribution and lift coe cient for the airfoil can be found. The viscous solution is much more complicated process. XFOIL?s viscous solution is based on the transonic ISES code with a few improvements. The ISES code solves for the boundary layer and nds separation bubbles using the inviscid solution to solve for the potential ow eld. XFOIL includes the Karman-Tsien compressibility correction which is reliable up to sonic conditions and provides reliable pressure distributions, lift, and drag coe cients at low Reynolds numbers [15]. 9 Chapter 3 Propellers 3.1 Basics of Propellers A propeller is a device used for creating thrust in a uid through rotational means. Figure 3.1 is velocity diagram for a cross section of a propeller blade. This illustrates that both the free stream and rotation velocities that are seen by the propeller. ? ? ? V rot V ? V rel Propeller Cross Section Rotation Axis Figure 3.1: Blade Cross Section Velocity Disgram 3.1.1 Geometry Propellers are very similar to wings. The lifting surface on a propeller is called a blade, and a propeller can have any number of blades. Most propellers have two to four blades. Any given point along a blade the cross section has all the same characteristics as an airfoil: leading and trailing edges, mean camber line, chord line, thickness, etc. Where the blades connect is called the hub which is either directly attached to an engine or to a transmission. The root is the area between the hub and the blade, and the tip is end of the blade opposite 10 the hub. The blade angle, , is the resultant angle between the free stream and rotational velocity components and is shown in the velocity diagram in Figure 3.1. The e ective pitch, pe, is the distance a propeller advances in one rotation. While the geometric pitch, ge, is the theoretical distance an element of a propeller blade would travel in one rotation and may not be constant along the length of blade [16] [17]. Several of these geometric paratmeters can be seen in Figures 3.2 and 3.3. Leading Edge Trailing Edge Tip Rotation Direction Hub Root Figure 3.2: Propeller Geometry 3.1.2 Other Parameters There are many other parameters that are useful in describing propellers. The advance ratio, J, is the ratio between the distance the propeller moves forward through one rotation and the blade diameter. J = VnD (3.1) Where n is in rotations per second. The aspect ratio, AR, is the tip radius divided by the maximum blade width. A spinner is a conical or parabolic shaped fairing that is mounted over the center of the center of the propeller where it is connected to the hub. The blade face is the lower surface of the propeller airfoil and is also known as the thrust or driving face. The blade back is the upper surface of the propeller airfoil. Several of these parameters are shown in 3.3. The rake or tilt of a propeller is the mean angle between a line drawn through the center of area of each section of a blade and a plane perpendicular to the rotation axis. Some of these parameters are shown in Figure 3.3. 11 Back Face Leading Edge Trailing Edge ? Rotation Direction Figure 3.3: Propeller Blade Cross Sections 3.1.3 Types Propellers are either tractor or pusher propellers. A tractor propeller is placed in a con guration where the engine is downstream of the propeller and pulls the aircraft. While a pusher propeller is placed where the engine is upstream of propeller and pushes the aircraft. Propellers can also be classed as either xed or variable pitched propellers. A xed pitch propeller?s blades are rigidly connected to the hub. A variable pitch propeller?s blades can be adjusted either on the ground or during ight to allow the propeller to operate at maximum performance throughout its operation range. 3.2 Propeller Theories There are several methods for solving for propeller performance factors. The following discusses a few fundamental methods which are computationally friendly and provide ac- curate results. Before these methods are explained, nondimensional expressions for thrust, power, and e ciency will be given. These expressions are similar to the lift and drag coe - cients used to characterize airfoils and show how the performance of a propeller changes with scale or rotation speed. The thrust coe cient, Ct, power coe cient, Cp, and the e ciency, 12 , are shown in Equations 3.2 - 3.4. Ct = Ft n2D4 (3.2) Cp = Q n3D5 (3.3) = CtJC p (3.4) where n is the rotation speed, D is the Diameter of the propeller, is the density of air, and J is the advance ratio. 3.2.1 Momentum Theory Momentum theory is most the fundamental of all of the propeller theories. The following explanation for the momentum theory was taken from Nelson in Reference [16]. This theory assumes the propeller is a disk that creates a uniform thrust through a pressure di erential between the front and back of the propeller. The theory does not take in account com- pressibility or viscous e ects. Figure 3.4 is a reproduction from Nelson?s work in Reference [16] and shows continuous stream lines that form a stream tube. The pressure and velocity V 0 V S V d V 0 Propeller Disk P 0 P 0 P ' 0 P ' 0 +?P Boundary Figure 3.4: Momentum Theory Stream Tube 13 before and after the disk are shown in Figure 3.5. The far upstream pressure, P0, is shown to change by P at the propeller disk then return to P0 far downstream. It should also be noted that the pressure drops by P0 P00 at the beginning of the disk then quickly rises by P before asymptotically returning to P0.The velocity is shown to start at V0 upstream and slowly rise to a nal value of Vs. With this information thrust from a propeller can be P 0 P 0 P ' 0 P ' 0 +?P Propeller Disk Location Pressure Velocity V 0 V d V S Figure 3.5: Momentum Theory Pressure and Velocity through Propeller Disk calculated using classic momentum theory. Ft = A Vd (VS V0) (3.5) where A Vd is the mass per unit time through the disk and (Vs V0) is the velocity increase from far upstream to far downstream. The pressure change across the propeller disk, velocities upstream and downstream, and the area of the propeller can be used to 14 calculate the thrust using Bernoulli?s equation. Ft = A P P = P0 + 12 V2s P0 + 12 V20 P = 12 V2s V20 Ft = A 2 V2S V20 (3.6) Combining Equations 3.5 and 3.6 the velocity at the disk, Vd, can be found. A Vd (VS V0) = A 2 V2S V20 Vd = A 2 (V 2 S V 2 0 ) A (VS V0) Vd = Vs +V02 (3.7) It can then be seen that half of the downstream velocity, VS, is added before the propeller disk. E ciency is de ned as the work output divided by the input work. Kinetic energy can be used to describe the input work, and thrust times velocity de nes the work output. The following process shows the e ciency in terms of the free stream velocity and the downstream velocity. = TV0K:E: = TV0 1 2A Vd (V2 S V 2 S) 15 plug in Vd from Equation 3.7, and FT from Equation 3.6; = A 2 (V 2 S V 2 0 )V0 1 2A Vs+V0 2 (V2 S V 20 ) = 2A V0 (V 2 S V 2 0 ) (A (Vs +V0)) (V2S V20 ) = 2V0V S +V0 = 21 + V S V0 (3.8) Ft 2 V20 = 1 2 (3.9) Equations 3.8 and 3.6 can be combined to get a theoretical e ciency in terms of density, free stream velocity, disk area, and thrust. Equation 3.9 can be used to observe how thrust, free stream velocity, and disk area a ect the e ciency of a propeller. Figure 3.6 shows e ciency verses the thrust coe cient, Ct. Where Ct is a dimensionless thrust found in Equation 3.10. Ct = Ft1 2 V 20 A (3.10) This shows that increasing thrust a propeller produces, slowing the free stream velocity, or decreasing the propeller disk area decreases the e ciency. The most e cient propeller would then produce no thrust, have a high free stream velocity, and be in nitely large. 3.2.2 Simple Blade Element Theory Blade element theory is the next fundamental propeller theory. This theory separates each blade of a propeller into elements and calculates the lift and torque generated by each of the element. The thrust and torque are then summed up to nd the total thrust and torque. The following process is a summarized explanation of Dommasch, Nelson, and Weick?s work in References [2], [16], and [17]. An example blade element is shown in Figure 3.7 which a 16 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 ? C t APC 10x7E Data Ideal Momentum Theory Figure 3.6: Ideal Momentum Theory E ciency and Actual Propeller E ciency Verse Thrust Coe cient reproduction from Dommasch?s work in Reference [2]. First consider an individual element. The lift and drag on the element can be calculated using a di erential form of the classic lift and drag calculations. dL = qCLbdr (3.11) dD = qCDbdr (3.12) Where the dynamic pressure is de ned in Equation 3.13. q = 12 V2Rel (3.13) VRel is the relative ow velocity which is shown in Figure 3.8 which is a slightly modi ed version of Figure 3.1 discussed in Section 3.1. 17 r R dr b Rotation Direction Figure 3.7: Example of a Blade Element Using the reaction force diagram in Figure 3.8 the di erential thrust can be calculated by adding the lift and drag components. dFt = dLcos( ) dDsin( ) (3.14) Substitute Equations 3.11, 3.12, and 3.13 into Equation 3.14. dFt = 12V2Relbdr(CLcos( ) CDsin( )) (3.15) A new angle , which describes the reaction force of the lift and drag components can now be used to simplify the di erential thrust equation and is de ned in Equation 3.16. tan( ) = DL = CDC L (3.16) 18 ? ? ? V 0 V rel Rotation Axis V rot = 2?nr dL dD dF t dF dR ? ? Figure 3.8: Velocity Vector Diagram with Reactions on a Blade Element The following process simpli es the di erential thrust equation into its nal form where VRel = V0sin( ). dFt = 12 V2Relbdr(CLcos( ) CDsin( )) = 12 V2Relbdr C L CLCLcos( ) CL CLCDsin( ) = 12 V2RelbdrCL (cos( ) tan( )sin( )) = 12 V2RelbdrCL cos( ) sin( )cos( )sin( ) = 12 V2RelbdrCL cos( )cos( ) sin( )sin( ) cos( ) = 12 V2RelbdrCL cos( + ) cos( ) = 12 V2bdrCL cos( + ) sin2 ( )cos( ) (3.17) 19 The torque can be found in the same manner using dF found in Figure 3.7. dQ = r dF = 12 V2brdrCL sin( + ) cos( )sin2 ( ) (3.18) Equations 3.17 and 3.18 can then be integrated across the radius of the blade to nd the total thrust and torque generated by one blade. That value is multiplied by the number of blades, B, to nd the total thrust and total torque for the propeller. Ft = Z r 0 1 2V 2bdrBCL cos( + ) sin2 ( )cos( ) (3.19) Q = Z r 0 1 2 V 2brdrBCL sin( + ) cos( )sin2 ( ) (3.20) The e ciency can be found with the same process discussed in the momentum theory. = FtV02 nQ (3.21) Nelson in Reference [16] says that if the e ciency is taken at a three quarter radius element that it would characterize the e ciency of the entire blade. He then describes a process to show the e ciency only it terms of the e ective pitch angle, and the reaction angle, . = dTVdQ2 n = dRcos( + )VdRsin( + ) 2 nr = tan( )tan( + ) (3.22) Figure 3.9 is a reproduction of Nelson?s work. It plots multiple curves which each correspond to di erent lift to drag ratio blade elements. It shows that a realistic maximum e ciency a propeller can have is approximately 93%, but using an average lift to drag ratio would be 20 80%. This gure also shows theoretical maximum e ective pitch angle which is shown by 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 ? Effective Pitch Angle, ?(degrees) L/D = 10 L/D = 30 Figure 3.9: Simple Blade Element Helix Angle E ciency the dashed line. If it is assumed that the propeller tip, the fastest location on a propeller, cannot have a relative velocity exceeding the speed of sound, aair, then at the three quarters location it cannot exceed three quarter the speed of sound. An e ective pitch angle of 45 is shown to provide the highest e ciency. tan( ) = V0V Rot = V03 4aair (3.23) If the speed of sound is approximately 1000 feet per second then using Equation 3.23 an e ciency verse free stream velocity curve can be drawn in Figure 3.10.. It should be noted that the free stream velocity corresponds to an e ective pitch angle that provides the max- imum e ciency. The simple blade element theory agrees with the momentum theory that e ciency of a propeller is increased with a higher free stream velocity. The simple blade element theory adds that the blade angle also needs to increase for added e ciency. 21 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 ? V 0 (Miles per Hour) L/D = 10 L/D = 30 Figure 3.10: Simple Blade Element Free Stream Velocity E ciency 3.2.3 Hybrid Momentum Blade Element Theory There are many di erent hybrid momentum blade element theories. Two similar meth- ods will be discussed here. These methods introduce factors to account for radial ow, blade interference, and tip losses. Axial Slipstream Factor Method First will be a method from Weick in Reference [17]. Earlier in the momentum theory section it was shown that half of the velocity increase is in front of the propeller in the slipstream. This leads to a new term known as in ow. In ow is added to the free stream velocity to increase the overall velocity of the ow in the axial direction. Figure 3.11 is a reproduction from Weick?s work showing the additional velocity included in the axial direction. It is shown that the velocity in the free stream direction is given by Equation 3.24 and a new angle of attack needs to be found for the blade element using Equations 3.25 and 22 ? ? ? V 0 V rel Rotation Axis V rot = 2?nr ?' xbV 0 ?' Figure 3.11: Weick?s In ow Method Velocity Vectors 3.26. u = V0 +xbV0 (3.24) tan( 0) = (1 +xb)tan( ) (3.25) 0 = 0 (3.26) The in ow for this model is an average in ow distributed over the whole blade. More robust methods calculate the in ow per blade element. According to Weick?s method in ow can be determined by Equation 3.27. Where b in this case is not the chord but is the axial slipstream factor and x is an empirical factor that ranges from one third to two thirds. Ft = A0 V2b(1 +xb) (3.27) A? is the e ective disk area and is given to be between 0.7 and 0.8. The di erential thrust can then be found using the same process derived in the in the previous section. Where b in 23 Equations 3.28 and 3.29 is the chord for the blade element. dFt = 12 u2bdrCL cos( + 0) sin2 ( 0)cos( ) (3.28) dQ = 12 u2brdrCL sin( + 0) cos( )sin2 ( 0) (3.29) This method must be iterated to nd the value of the axial slipstream factor, b. This is executed by establishing an initial guess for the ow angle . Equation 3.28 is evaluated, and the thrust found is substituted in Equation 3.27. The axial slipstream factor, b, is then solved and now 0 can be found. This 0 is plugged into Equation 3.28 and a new thrust is solved. The new thrust and the old thrust are compared. If they are not within an acceptable tolerance, the new thrust is plugged in Equation 3.27 and a new axial slipstream factor is found. This process is repeated until an acceptable tolerance is achieved. One problem with this method is error introduced by the empirical factor x. This factor varies from propeller to propeller leading to inconsistent results [17]. In ow with Axial and Rotational Interference Factors The second method that will be discussed is an optimum design paper developed by Adkins and Liebeck in 1983 and was reproduced with more detail in 1994 [3]. This is the same procedure discussed by Glauert in Reference [1] with updated nomenclature and a more detailed explanation. This method also uses an axial interference factor, a, and a rotational interference factor, a0. Using momentum theory the axial interference ow factor, a, is the increase in ow velocity in front of the propeller, and the rotational interference ow factor, a0, is the decrease in the relative rotational ow velocity. Figure 3.12 shows these new modi cations to velocity vector diagram for a blade element. Using momentum theory and Figure 3.12 it can easily be shown that the thrust per unit radius is given by Equation 24 ? ? ? V 0 (1+a) V rel Rotation Axis V rot = 2?nr(1-a') Figure 3.12: Axial and Rotational Interference Factor Blade Element Velocity Vectors 3.30, and the torque per unit radius is given by Equation 3.31. dFt = 4 rV2a(1 +a)Fdr (3.30) dQ r = 8 2 nr2Va0(1 +a)Fdr (3.31) With F being the Prandtl momentum tip loss factor. Adkins does not discuss this factor in any detail in his paper, but it is discussed in more detail by Glauert in Reference [4]. It was originally developed by Prandtl and describes the losses due to induced velocities at the tip of the blade. This loss factor would be equal to 1.0 across the radius if the propeller is shrouded or in a duct. If it is not it will start at 1.0 and at a given location will begin to decay to 0.0. The location for the start of the decay is due to geometry. Prandtl?s original method used the in ow angle based on the blade tip which Glauert later revised to using the local in ow angle. Adkins returns to Prandtl?s original method by using the in ow angle at 25 the tip [3] [4]. The Prandtl momentum tip loss factor is expressed in Equation 3.32. F = 2 cos 1 e f (3.32) where, f = B2 (1 )sin( t) (3.33) and the ow angle at the tip, t is de ned by tan( t) = tan( ) (3.34) Adkins includes the addition of circulation into this method. Which a simpli ed version of the circulation is de ned by Equation 3.35. The circulation equation introduces which is the displacement velocity ratio, v0V0 . The vortex displacement velocity, v0, is the axial velocity of the vortex lament in the vortex sheet in the wake of the propeller. = 2 rV0 FB cos( )sin( ) (3.35) This circulation is used to described the lift per unit radius and is given by Equation 3.36. The thrust per unit radius and torque per unit radius can be found using Equation 3.36 and Figure 3.8. With being the drag to lift ratio. dL dr = B Vrel (3.36) dFt = dL drcos( ) (1 tan( )) dr (3.37) dQ r = dL drsin( ) 1 + tan( ) dr (3.38) 26 Cy, Cx, Cl, and Cd replacing dFt, dF, dL, and dD respectively in Figure 3.8. Cy = Clcos( ) Cdsin( ) = Cl (cos( ) + sin( )) (3.39) Cx = Clsin( ) +Cdcos( ) = Cl (sin( ) + cos( )) (3.40) The axial and rotational interference factors are expressed in Equations 3.41 and 3.42. a = KF K (3.41) a0 = K 0 F + K0 (3.42) where, K = Cy4sin2 ( ) (3.43) K0 = Cx4cos( )sin( ) (3.44) The solidity, , is de ned by Equation 3.45. = Bb2 r (3.45) Figure 3.12 shows these new modi cations to velocity vector diagram for a blade element. It is easily shown that the ow angle is Equation 3.46. tan( ) = V2 nr (1 +a)(1 a0) (3.46) The procedure for solving for the thrust, torque, and e ciency for a propeller goes as follows. An initial guess for is found by setting a and a0 equal to zero in Equation 3.46. This then allows for the angle of attack, , for the blade element to be found which also yields the lift and drag coe cients. Equations 3.41 and 3.42 can then be solved for the interference factors. 27 The interference factors are plugged back into Equation 3.46 to solve for a new ow angle, . The new and old values of are compared, and if they are not within a set tolerance of each other they average of the two values is used to repeat the process. These iterations continue until the solution has converged within the set tolerance. It should also be noted that Adkins suggests using a clipping method by Viterna and Janetzke [18] in which a and a0 are limited to a maximum value of 0.7. Once the convergence is met the thrust coe cient and power coe cient for the propeller can be found and is shown in Equations 3.47 and 3.48. These equations can be integrated to solve for the overall thrust and power coe cients. The coe cients can be substituted into Equations 3.2-3.4 for nal thrust, power, and e ciency values. dCt d = 3 4 3F2 Cy [(F + K0)cos( )]2 (3.47) dCp d = dCt d Cx Cy (3.48) Detailed derivations for Equations 3.47 and 3.48 are shown in Appendix B. This deriva- tion presented corrects an error in the di erential thrust equation in Adkins [3]. 28 Chapter 4 Electric Motors Electric motors are being used more frequently in small UAVs due to the high energy density lithium polymer batteries that are becoming more available and the overall decrease in sound production of the electric propulsion system compared to an internal combustion system. Two types of electric motors will be brie y discussed here. First a brushed DC motor will be introduced followed by a brushless DC motor. More details for DC motors can found in References [19], [20], [21], and [22]. 4.1 Brushed DC Electric Motors A brushed DC electric motor as the name implies uses a DC voltage source for power. The motor consists of a rotor, stator, eld system, armature, brushes, and a commutator which are several of main components. The rotor is the rotating part of the motor. The stator is the stationary part of the motor. The eld system provides the magnetic ux used to create the torque on the motor. The armature carries the current that interacts with the eld ux to create torque. For most brushed DC motors the rotor and the armature are one in the same since the rotor will have windings that are used to move the current from the brushes and commutator to the rotor. The brushes connect the armature to the power supply and a motor requires a minimum of one pair of brushes. The commutator distributes the current properly to the armature coils [19]. With these components the motor can be modeled as a resistor or armature resistance and the back e.m.f. A simple equivalent circuit of a DC motor is shown if Figure 4.1. Where the back e.m.f. is the back electromotive force, and KE is the back e.m.f. constant. 29 V supply R a V back e.m.f. = K E ? I a Figure 4.1: A Simple Equivalent Circuit of a DC Motor For the scope of this work it is now assumed that when all of the previous components are combined the motor is a torque generator. The motor generates torque according to Fleming?s left hand rule. A detailed explanation of how a DC motor produces torque can be found in Reference [19]. The torque of a motor can then be found using the torque constant, Kt, and the current draw, I, on the motor. = KtI (4.1) If the constant units are in the SI system KE is equal to Kt and if the imperial system is used 1:352Kt = KE. Another useful term to describe a DC electric is the KV value. This has units of RPMV . This motor speed constant is used to nd the motors speed with no load given a particular input voltage. Equation 4.2 shows the relation between the torque and speed constants where Kt has units of oz inA [21]. Kt = 1000K V 1:352 (4.2) 30 A torque speed curve can be produced using Equations 4.1 and 4.2. The maximum/stall torque is found from the stall current in Equation 4.3 and Equation 4.1 with RI being the internal resistance of the motor. The maximum RPM is found using the motor voltage and the Kv value. A straight line is connected between the maximum RPM and the stall torque to form the torque speed curve. Figure 4.2 shows the torque speed curve has a negative slope, and it is shown that as the voltage is increased the curves remain parallel with a new maximum/stall torque and peak rotational speed. Istall = VmotorR I (4.3) The power required for motor is then found using Equation 4.4. A power curve can be 0 10 20 30 40 50 60 70 80 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 T o r q u e RPM V 2V 3V 4V Figure 4.2: Torque Speed Curve Example for a DC Motor added to the torque speed curve. Figure 4.3 shows an example torque speed power curve for an arbitrary DC motor with Kv = 1000, Kt = 1:5 oz inA , RI = 0:2 , and V = 11:1Volts. P = 2 n60 = VmotorI (4.4) 31 Figure 4.3 is a useful tool to describe a motor. It shows that the maximum power con- 020406080100120140160180 0102030405060708090 0 2000 4000 6000 8000 1000 012 000 P o w e r ( W a t t s ) T o r q u e ( o z - i n ) RPM Torqu e Powe r Figure 4.3: Torque Speed Power Curve Example for a DC Motor sumption is at half the maximum RPM and conversely that little power is used near the maximum and minimum RPM ranges. Figure 4.4 compares the electric motor from Figure 4.3 with a Cox 0.09 2-stroke model aircraft engine running 30% nitro [23]. Even these motor are not operating at the same speeds they produce approximately the same power. This plot shows the main di erence between a internal combustion engine and an electric motor, an electric motor torque starts at a peak and goes to zero with an increase in speed while an internal combustion engine starts at zero reaches a peak then decreases back to zero. An electric motor What Figure 4.3 does not show is the actual limits of a motor. The wasted energy or losses in the motor are due to heat and friction. In most electric motors friction is minimal and can usually be ignored, but the heat produced by the current owing through the motor cannot. The heat that is produced can melt the coils on the armature if too much current is allowed to ow through the motor without su cient cooling. This leads to motors having thermal limits which are identi ed with continuous and maximum/burst current ratings or continuous and maximum/burst power ratings. A continuous rating is 32 020406080100120140160180 0102030405060708090 0 5000 1000 0 1500 0 2000 0 2500 0 P o w e r ( W a t t s ) T o r q u e ( o z - i n ) RPM Elect ric To rque Elect ric Po wer Intern al Co mbus tion T orque Intern al Co mbus tion P ower Figure 4.4: Electric Motor and Internal Combustion Engine Comparison the maximum allowable current or power the motor can experience to run inde nitely. The maximum/burst ratings are the absolute limits on the motor for a certain time span. This allowable time for maximum conditions is set by the manufacturer, and this time ranges from 15-60 seconds. 4.2 Brushless DC Electric Motors Brushless DC electric motor have become more popular over the last several years for small remote controlled aircraft due to there more e cient nature and little to no need for maintenance. Its increase in e ciency over a brushed DC motor comes from the lack of mechanical brushes in the motor. This reduces the friction inside the motor, and removes parts that need to be serviced. For the simplicity friction will be ignored in this motor model. The only advantage of a brushless motor to a brushed motor is the brushless motor has fewer parts to wear out. A brushless DC motor may appear to be an AC motor, but it is not. Besides the type of current being provided to the motor, what separates an AC from DC motor is the brushless 33 DC motor uses sensors to detect the rotor position to control the pulses to the motor [19]. Simply put these pulses create magnetic elds which cause the motor to rotate. Usually the type of sensor used is a Hall E ect sensor, but most hobby grade brushless DC motors which are used with most small electric UAVs have no physical sensors in the motor. The speed controllers used to power the motor read the back e.m.f. from the motor and determine the motors position from that. A brushless DC motor can be modeled in the same manner as brushed DC motors approximately yielding the same type of torque speed power curves. The rotational speed in radians per second of the brushless DC motor under load can be found in Equation 4.5 [22]. !r = Vmotorp m 2 RI m p m2 2 Tem (4.5) Where m is the ux linkage of the stator winding, p is the number of poles in the motor, m is the number of phases, and Tem is the electromagnetic torque de ned by Equation 4.6. A poles is the set of windings in a motor and is an even number. The number of phases is the number of conductors connected to the motor that supply voltage to the motor. All of the motors discussed here will be three phases motor. This leads to the voltage waveforms on each of the phases of the motor will be o set by 60 . Tem = mp2 mI (4.6) The e ciency of the brushless DC motor is found by Equations 4.7-4.9. I0 is the idle current of the motor, this is the current the motor will draw with no load. = PoutP in (4.7) Pout = (Vmotor IRI) (I I0) (4.8) Pin = VmotorI (4.9) 34 Chapter 5 Optimization Methods Optimization methods are used to minimize or maximize a problem with multiple un- knowns. These methods greatly decrease the number of function calls over a brute force method in which every possible combination of unknowns is evaluated. They can also be used to nd local or global minimums. An objective function is evaluated by the optimiza- tion method to determine if the problem is optimized. The optimizers that will be used for this work will be treated as black boxes, and no math will be presented in this section. This chapter will provide a brief overview of a particle swarm method, a pattern search method, and a simplex method. 5.1 Particle Swarm Particle swarm optimization was rst suggested by Kennedy and Eberhart [24] as a stochastic methodology based on crowding behavior and collective intelligence. Similar to genetic algorithms in practice, the particle swarm technique relies on communication and in- teraction among its members of a population to collectively move throughout a design space. Like any stochastic based optimization routine, swarming has the ability to escape the local optima of a problem in search of a better solution. The prime attractor to particle swarm optimization is its simplicity in implementation compared to other stochastic techniques. The particles move through the design space to nd the optimum location with a varying velocity. The particles move in the directions in which the best particles are performing and are swayed by their own best position and the absolute best location of all the particles. Particle swarm methods can move from local optimum if other particles nd an improved solution unlike gradient methods which will get stuck on local optimum points. The objective 35 function used in the particle swarm does not have to di erentiable or smooth. This allows for more problems to be solved. The biggest problem with particle swarms is a particle uses its previous position and velocity to solve where it should go. This is sometimes a problem if a particle is located at the optimum point. This leads to particle swarms are good at nding area of the design space that should yield the optimum global point [24]. 5.2 Pattern Search The pattern search method was originally developed by Hooke and Jeeves and is a direct search technique [25]. This method works by monitoring the changes each of the design parameters have on the objective function. The pattern search starts with an initial design case, and then performs an investigative move on one of the design variables holding the others constant. It then evaluates the objective function and if it is better than a solution it stores it. This process is repeated for the rest of the design variables. A pattern move is then performed which is a changing all of the variables and evaluating the objective function. If this new evaluated objective function is better the design variables are stored, and if it is worse the process is repeated with a decreased initial move by the design variables [25]. The pattern search method is a good technique for nding local optimum locations. Its solution is highly dependent on its initial location. If placed in the correct location in the design space it can provide rapid results, but more design variables used and worst starting location can yield a long computation time [26]. 5.3 Hybrid Pattern Search/Particle Swarm Method The hybrid pattern search and particle swarm method that will be presented was de- veloped by Jenkins and Hart eld [26]. It combines the local optimization features of the pattern search and the global optimization features of the particle swarm methods. This method rst starts by initializing a population with acceptable values of the design parame- ters that ful ll the objective function. The pattern search method is then run through user 36 a de ned number of generations. The particle swarm method is then run went the results of the pattern search. This process is repeated for a de ned number of generations to provide adequate convergence and to ensure a global optimum is found [26]. 5.4 Simplex Method The simplex method that will be presented is a direct search method and does not use gradients. This method can be found in MATLAB 2010a as the fminsearch() function [27]. It uses a method developed by Lagarias et al. in Reference [28]. This function uses only function evaluations like the pattern search and particle swarm methods and does not require derivatives to be solved. This method uses a simplex which is a geometric shape of same number of unknowns in the problem with the number of unknowns plus one points describing the shape. For example if the problem has three unknowns it would be a three dimensional problem with four points describing the simplex which would resemble a pyramid. When the method is executed a new point near the simplex is evaluated and compared to the values of the points of the simplex. If the new point is \better" than one of the simplex points, the \bad" point is replaced. This process is repeated until the diameter of the simplex is within a user de ned tolerance [27] [29]. The simplex method can be used with discontinuous objective functions. It is not guaranteed to provide a global optimum solution for discontinuous function and will usually provide a local optimum minimum [27]. 37 Chapter 6 Implementation The optimization of a propeller is split into two major sections. First an airfoil is optimized for a maximum lift to drag ratio. Second a propeller is optimized using the optimized airfoil. The process was performed in this matter to decrease the run time of the nished optimization. MATLAB 2010a was used to execute the optimizations. 6.1 Airfoil Optimization A set of airfoils for the optimized propellers is found rst. This set of airfoils consists of optimized airfoils for a range of angle of attacks. The process is shown in a ow chart in Figures 6.1 and 6.2. A series of functions, which are found in Appendix C, were constructed using the process discussed in Section 2.2.2 to describe an airfoil. A 6th order Bernstein polynomial was used for the upper and lower surfaces to provide a good representation of an airfoil surface [9]. The rst function will be called AirfoilMaker(). The fourteen unknowns or coe cients as well as a desired angle of attack for the airfoil is sent to this function. It starts by devising a name for the airfoil using the coe cients which will be used as the le name for the airfoil data. The function then calls another function, ParametricAirfoil() to produce x and y coordinates of the airfoil. It also checks so see if the upper and lower surfaces cross, and if they do it returns that the airfoil is a bad airfoil. AirfoilMaker() then calls ClCdFinder() and passes it the x and y values for the airfoil and the airfoil name. ClCdFinder() checks a folder containing already made airfoils and searches for it. If it is not found a le corresponding to the airfoil an instruction le for XFOIL is written. The instruction le looks similar to the example in Appendix A. The function then runs 38 Minimum and Maximum Coe?cients uni0048uni0079uni0062uni0072id uni0050ag425euni0072n uni0053eauni0072cuni0068 uni0050auni0072g415cuni006Ce uni0053uni0077auni0072m uni004Funi0070g415miuni007Aeuni0072 uni004Funi0062uni006Aecg415uni0076e uni0046uncg415on Parametric Airfoil() ClCdFinder() uni004Funi0070g415miuni007Aed uni0041iuni0072uni0066oiuni006C Coe?cients uni0077ituni0068 minimum cd/cuni006C AirfoilMaker() Coe?cients Coe?cients uni0041nuni0067uni006Ce ouni0066 uni0041g425acuni006B Cuni006C and Cd Coe ?cie nts X, Y uni0050oi nts Cuni006C and Cd X, Y uni0050oi nts Xfoil.exe Airfoil uni004Cig332 and Drag Tables uni0046itness Figure 6.1: Flowchart for Airfoil Optimization Process a batch le that runs XFOIL with the points and instruction les. A timeout was added to the batch program in the event that XFOIL could not nish running due to improper convergence. The function then checks to con rm that XFOIL produced an acceptable table of lift and drag coe cients. The table is not acceptable if any of the values for the lift or drag coe cients are NaN (not a number) or if XFOIL failed to produce enough data points to describe the lift and drag curve slopes. If it does fail the process a new instruction le is written increasing the number of panels on the airfoil. This is done until the airfoil successfully passes or reaches a maximum panel size in which case the airfoil is considered to be a bad airfoil. After either creating the new airfoil table or locating a previously made le the function reads in the data from the le. It interpolates for the lift and drag coe cients at the desired angle of attack and returns the values to AirfoilMaker(). 39 uni004Funi0070g415uni006Duni0069uni007Auni0065uni0064uni0020uni0020Auni0069rfouni0069luni0020atuni00200?uni0020 AoAuni0020 uni0048uni0079uni0062runi0069uni0064uni0020uni0050ag425uni0065runi006Euni0020uni0053uni0065aruni0063uni0068uni0020uni0050arg415uni0063luni0065uni0020 uni0053uni0077aruni006Duni0020uni004Funi0070g415uni006Duni0069uni007Auni0065runi0020 uni004Funi0062uni006Auni0065uni0063g415uni0076uni0065uni0020uni0046uni0075uni006Euni0063g415ouni006Euni0020 Parametric Airfoil() ClCdFinder() uni004Funi0070g415uni006Duni0069uni007Auni0065uni0064uni0020Auni0069rfouni0069luni0020atuni0020 1?uni0020AoAuni0020 AirfoilMaker() Couni0065?uni0063uni0069uni0065uni006Etsuni0020 Couni0065?uni0063uni0069uni0065uni006Etsuni0020 Auni006Euni0067luni0065uni0020ofuni0020Ag425auni0063uni006Buni0020 Cluni0020auni006Euni0064uni0020Cuni0064uni0020 Couni0065? uni0063uni0069uni0065uni006Et suni0020 X,uni0020Yuni0020uni0050 ouni0069uni006Ets uni0020 Cluni0020auni006E uni0064uni0020Cuni0064 uni0020 X,uni0020Yuni0020 uni0050ouni0069uni006E tsuni0020 Xfoil.exe Airfoil uni004Cig332 and Drag Tables uni0046uni0069tuni006Euni0065ssuni0020 fminsearch() uni004Funi0062uni006Auni0065uni0063g415uni0076uni0065uni0020uni0046uni0075uni006Euni0063g415ouni006Euni0020 Parametric Airfoil() ClCdFinder() AirfoilMaker() Couni0065?uni0063uni0069uni0065uni006Etsuni0020 Couni0065?uni0063uni0069uni0065uni006Etsuni0020 Auni006Euni0067luni0065uni0020ofuni0020Ag425auni0063uni006Buni0020 Cluni0020auni006Euni0064uni0020Cuni0064uni0020 Couni0065? uni0063uni0069uni0065uni006Et suni0020 X,uni0020Yuni0020uni0050 ouni0069uni006Ets uni0020 Cluni0020auni006E uni0064uni0020Cuni0064 uni0020 X,uni0020Yuni0020 uni0050ouni0069uni006E tsuni0020 Xfoil.exe Airfoil uni004Cig332 and Drag Tables uni0046uni0069tuni006Euni0065ssuni0020 uni004Funi0070g415uni006Duni0069uni007Auni0065uni0064uni0020Auni0069rfouni0069luni0020atuni00202?uni0020AoAuni0020 Tuni0068uni0069suni0020uni0070rouni0063uni0065ssuni0020uni0069suni0020runi0065uni0070uni0065atuni0065uni0064uni0020uni0077uni0069tuni0068uni0020fminsearch() uni0075uni006Eg415luni0020uni006Dauni0078uni0069uni006Duni0075uni006Duni0020AoAuni0020oruni0020uni0069suni0020stuni0065uni0070uni0070uni0065uni0064uni0020uni0064ouni0077uni006Euni0020touni0020 tuni0068uni0065uni0020uni006Duni0069uni006Euni0069uni006Duni0075uni006Duni0020AoAuni0020 Figure 6.2: Flowchart for Propeller Optimization Process using fminsearch() An objective function, AirfoilCostFunction, was developed for the optimizers to call. It called the AirfoilMaker() function with the fourteen coe cients and the angle of attack to be optimized. The objective function returns the tness of airfoil to the optimizer. Since all of the optimization method discussed are minimization methods the optimum airfoil would 40 be found by minimizing the drag to lift ratio. The tness equation is shown if Equation 6.1 where z is the tness. z = cdcl (6.1) If coe cients used in AirfoilMaker() return lift or drag coe cients less than zero the t- ness for that set of coe cients was set to 108. This reiterates to the optimizer that those coe cients produce corrupt results. The optimized airfoils were found be rst using the hybrid pattern search particle swarm method discuss in Chapter 5. An airfoil was optimized at a zero degree angle of attack using this method. Then airfoils at a range of angles of attack were optimized using MATLABs simplex method fminsearch(). It was executed in this matter due to increased run time of the hybrid optimizer verse fminsearch(). The time results from each of the optimization runs will be discussed later in the Chapter 7. The zero degree angle of attack airfoil was used as the initial starting location. As the optimization sweep advanced the previous airfoil was used as the initial starting location for next airfoil. A list of optimized airfoils was saved to be used by the propeller optimizer. 6.2 Propeller Optimization The propeller optimizer code, like the airfoil optimizer, is made up of a series of functions coded in MATLAB 2010a. Flowcharts for the propeller optimization methods, cruise and climb are shown in Figures 6.3 and 6.4. All of the functions used for this optimizer can be found in Appendix D. A brushless DC electric motor model is coupled into the optimizer for the most e cient system. The brushless DC motor model function was named BrushlessDCMotor(). The function has two di erent sets of input and output cases depending on what is being optimized. The Kv value, internal resistance, number of poles, number of phases, the idle current, and which case is being used are inputs that are used for both cases. The rst condition has inputs of 41 uni004Funi0070g415uni006Duni0069uni007Auni0065uni0020uni0066uni006Funi0072uni0020uni0043uni0072uni0075uni0069uni0073uni0065uni003Auni0020 Muni0069nuni0069uni006Duni0075uni006Duni0020anduni0020Maxuni0069uni006Duni0075uni006Duni0020valuni0075uni0065uni0073uni0020 uni006Funi0066uni0020Buni0065tauni0073,uni0020uni0043huni006Funi0072duni0073,uni0020Duni0069auni006Duni0065tuni0065uni0072,uni0020RPMuni0020uni0020uni0020 uni0048uni0079uni0062uni0072uni0069duni0020Pag425uni0065uni0072nuni0020 uni0053uni0065auni0072uni0063huni0020Pauni0072g415uni0063luni0065uni0020 uni0053uni0077auni0072uni006Duni0020uni004Funi0070g415uni006Duni0069uni007Auni0065uni0072uni0020 uni004Funi0062uni006Auni0065uni0063g415vuni0065uni0020 uni0046uni0075nuni0063g415uni006Fnuni0020 ClCdFinderAirfoilID() uni004Funi0070g415uni006Duni0069uni007Auni0065duni0020uni0020Puni0072uni006Funi0070uni0065lluni0065uni0072uni0020 Propeller performance() uni0043uni006Funi0065?uni0063uni0069uni0065ntuni0073uni0020 uni0043uni006Funi0065?uni0063uni0069uni0065ntuni0073uni0020 uni0020 J,uni0020uni0043t,uni0020uni0043uni0070,uni0020anduni0020?uni0020 uni0046uni0069tnuni0065uni0073uni0073uni0020 Auni0069uni0072 uni0066uni006Funi0069l uni0020uni0046uni0069l uni0065na uni006Duni0065 uni0020uni0020 uni0043luni0020a nduni0020 uni0043duni0020 uni0066uni006Funi0072 uni0020 Auni0069uni0072 uni0066uni006Funi0069l uni0020 Brushless motor() Tuni006Funi0072quni0075uni0065uni0020RPMuni0020 Vuni006Fltaguni0065uni0020 uni0043uni0075uni0072uni0072uni0065ntuni0020? uni006Duni006Ftuni006Funi0072uni0020 Figure 6.3: Flowchart for Propeller Optimization Process using Hybrid Optimizer for Cruise Condition RPM and torque and outputs of voltage, current, and motor e ciency. The current is found by dividing the input torque with the torque constant, Kt. The voltage is found using a brute force iterative method evaluating stepping the voltage in Equation 4.5 until the rotational speed matches the input rotational speed. The e ciency is found the same way as the rst case using Equations 4.7 - 4.9 and dividing the output power with the input power. The second condition has inputs of voltage and current and outputs of torque, RPM, and motor e ciency. The rotational speed of the motor is found using Equation 4.5. The torque is found by using the Kv to nd the Kt and then multiplied by the input current. The e ciency is found using the process expressed in Equations 4.7 - 4.9 by dividing the output power with the input power. 42 uni004Funi0070g415uni006Duni0069uni007Auni0065uni0020uni0066uni006Funi0072uni0020uni0043uni006Cuni0069uni006Duni0062uni003Auni0020 Muni0069nuni0069uni006Duuni006Duni0020anduni0020Maxuni0069uni006Duuni006Duni0020vauni006Cuuni0065suni0020 uni006Funi0066uni0020Buni0065tas,uni0020uni0043huni006Funi0072ds,uni0020Duni0069auni006Duni0065tuni0065uni0072,uni0020 uni0043uuni0072uni0072uni0065ntuni0020uni0020 uni0048uni0079uni0062uni0072uni0069duni0020uni0050ag425uni0065uni0072nuni0020 uni0053uni0065auni0072uni0063huni0020uni0050auni0072g415uni0063uni006Cuni0065uni0020 uni0053uni0077auni0072uni006Duni0020uni004Funi0070g415uni006Duni0069uni007Auni0065uni0072uni0020 uni004Funi0062uni006Auni0065uni0063g415vuni0065uni0020 uni0046ununi0063g415uni006Fnuni0020 ClCdFinderAirfoilID() uni004Funi0070g415uni006Duni0069uni007Auni0065duni0020uni0020uni0050uni0072uni006Funi0070uni0065uni006Cuni006Cuni0065uni0072uni0020 Propeller performance() uni0043uni006Funi0065?uni0063uni0069uni0065ntsuni0020 uni0043uni006Funi0065?uni0063uni0069uni0065ntsuni0020 uni0020 J,uni0020uni0043t,uni0020uni0043uni0070,uni0020anduni0020?uni0020 uni0046uni0069tnuni0065ssuni0020 Auni0069uni0072 uni0066uni006Funi0069uni006C uni0020uni0046uni0069uni006C uni0065na uni006Duni0065 uni0020uni0020 uni0043uni006Cuni0020a nduni0020 uni0043duni0020 uni0066uni006Funi0072 uni0020 Auni0069uni0072 uni0066uni006Funi0069uni006C uni0020 Brushless motor() Vuni006Funi006Ctaguni0065uni0020 uni0043uuni0072uni0072uni0065ntuni0020 Tuni006Funi0072quuni0065uni0020Runi0050Muni0020 ?uni006Duni006Ftuni006Funi0072uni0020 Figure 6.4: Flowchart for Propeller Optimization Process using Hybrid Optimizer for Climb Condition The function that calculates the performance parameters of a propeller was named PropellerPerformance(). The functions inputs are the blade angles, , the chord width of the elements, the radial position of the elements, the rotational speed, the number of blades, and the free stream velocity. It uses the method \In ow with Axial and Rotational Interference Factors" discussed in more detail in Section 3.2.3. The function uses the inputs to iterate for the in ow factors, radial factors and the ow angles for each element. The iteration stops when a maximum iteration limit is reached or a tolerance is met. The thrust and power coe cients are found for each element and integrated using the trapezoid method to nd the overall coe cients. The e ciency is then found using Equation 3.4. The function 43 returns the advance ratio of the propeller, the thrust coe cient, the power coe cient, and the e ciency of the propeller. The lift and drag coe cients for each element is found from the function ClCdFinderAir- foilID(). An \Airfoil ID" and the angle of attack is sent to this function. The angle of attack desired from PropellerPerformance() is rounded to the nearest integer value and that is used to identify the which airfoil to use. If the angle of attack is less than or greater than the minimum or maximum optimized airfoil the closest airfoil to that value is used. If the angle of attack is greater than 20 the largest optimized airfoil is used, but the lift and drag coe - cients are estimated using at plate theory [30]. Flat plate theory says Cl = 2sin( )cos( ) and Cd = 2sin( )2. A third condition, optimizing for climb-cruise, is very similar to the rst condition and is a multi-point optimization. The owchart for this method is shown in Figure 6.5. This method has inputs of inputs of RPM and torque of a cruise condition and runs the PropellerPerformance() function and the BrushlessMotor() function. If a \good" propeller is found the process is repeated with the climb condition. The optimizer that was chosen was the hybrid pattern search particle swarm method. This method was chosen over the other methods due to its ability to potentially nd a global optimum better than the other methods. The objective function includes conditional statements to con rm the optimizer has found a viable solution. These include statements to con rm the current draw from the motor is not too high, the propeller does not produce negative thrust, e ciencies are not greater than 100 percent, and several others. The actual objective function used for this optimizer will be presented in the Chapter 7. 44 uni004Funi0070g415uni006Duni0069uni007Auni0065uni0020uni0066uni006Funi0072uni0020uni0043uni006Cuni0069uni006Duni0062-uni0043uni0072uuni0069suni0065:uni0020 Muni0069nuni0069uni006Duuni006Duni0020anduni0020Maxuni0069uni006Duuni006Duni0020vauni006Cuuni0065suni0020 uni006Funi0066uni0020Buni0065tas,uni0020uni0043huni006Funi0072ds,uni0020Duni0069auni006Duni0065tuni0065uni0072,uni0020RPMuni0020uni0020uni0020 uni0048uni0079uni0062uni0072uni0069duni0020Pag425uni0065uni0072nuni0020 uni0053uni0065auni0072uni0063huni0020Pauni0072g415uni0063uni006Cuni0065uni0020 uni0053uni0077auni0072uni006Duni0020uni004Funi0070g415uni006Duni0069uni007Auni0065uni0072uni0020 uni004Funi0062uni006Auni0065uni0063g415vuni0065uni0020 uni0046ununi0063g415uni006Fnuni0020 uni004Funi0070g415uni006Duni0069uni007Auni0065duni0020uni0020 Puni0072uni006Funi0070uni0065uni006Cuni006Cuni0065uni0072uni0020 Propeller performance() and BrushlessMotor() At Cruise uni0043uni006Funi0065?uni0063uni0069uni0065ntsuni0020 uni0043uni006Funi0065?uni0063uni0069uni0065ntsuni0020 uni0020 J,uni0020uni0043t,uni0020uni0043uni0070,uni0020anduni0020?uni0020 uni0046uni0069tnuni0065ssuni0020 Propeller performance() and BrushlessMotor() At Climb Puni0072uni006Fduuni0063uni0065uni0020?Guni006Funi006Fduni0020 Runi0065suuni006Ct?uni0020 Yuni0065suni0020 uni0020 Nuni006Funi0020(?tnuni0065ssuni0020=uni0020108)uni0020 uni0020 Figure 6.5: Flowchart for Propeller Optimization Process using Hybrid Optimizer for Climb- Cruise Condition 45 Chapter 7 Results The propeller optimization code was performed on three particular ight conditions. For each of the conditions a propeller is designed using the same brushless DC motor. The motor is modeled after an E ite Park 450 Outrunner. The parameters for the motor are listed in Table 7.1 [31]. The propeller is designed to be placed on a small aircraft with the Table 7.1: DC Brushless Motor Parameters Parameter: Value: Internal Resistance, RI 0.2 Ohms Kv RPMVolt 890 Idle Current, I0 0.70 Amps Continuous Current 14 Amps Maximum Burst Current 18 Amps for 15s Voltage Range 7.2-12 Volts Weight 2.5 oz requirements listed in Table 7.2. Based on the requirements in Table 7.2 the three optimized Table 7.2: Flight Condition Design Parameters Design Parameter: Value: Stall Speed 15 MPH Cruise Speed 50 MPH Drag at Cruise 7.5 oz propeller optimizations will be for a cruise, a climb, and a multi-point climb cruise case. Two di erent objective functions where used for each of the optimizations to see the e ect an objective function using coe cients would perform verse dimensional values. The number of elements used was 10. This was decided on to give a good representation of the propeller blade and to keep the number of unknowns a small as possible. The element distribution was an even distribution starting at 12% of the radius. The number of blades was xed to 46 2 blades for all of the propellers. In the optimizer the blade angles, , could only decrease from as they traveled outward to the tip, and the chord was allowed to expand until the three quarter radius location then could only decrease to zero at the tip. The diameter was allowed to vary from 8 to 12 inches. 7.1 Airfoil Optimization The set of airfoils used for the propeller were found rst. The rst airfoil found using the hybrid pattern search particle swarm optimizer was implemented at a zero angle of attack. The drag to lift ratio was minimized. It took approximately 22 hours to complete on an Intel i7 930 with 6 gigabytes of ram running 64-bit Windows 7. Xfoil was called and airfoils were made 25,000 times before the optimizer was nished running using a population size of 15 for 30 generations with 2 pattern searches per generation. This computer was used for the remainder of the computations. Figure 7.1 shows the optimized for a zero angle of attack airfoil. At an angle of attack of 0 it has a Cd = 0:01305 , a Cl = 0:8404. This gives a drag to lift ratio of 0:0155. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x/c y/c Figure 7.1: Optimized Airfoil at 0 Angle of Attack 47 The rest of the airfoils, 5 to 15; were found using fminsearch() which is a simplex function in MATLAB. Each of these airfoils took 5-10 minutes to complete. Tables E.1, E.2, and E.3 in Appendix E contain the coe cients for all of the airfoils, the drag to lift ratios, and the lift coe cients at the design point. Figure 7.2 shows a comparison of the di erent optimized airfoils. The axis in the plot are not equal to show the di erences between the airfoils. The 5 airfoil is not show due to the very slight change between the 0 airfoil. It is shown that as the angle of attack increases the upper surface changes, but the lower surface slightly changes. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 x/c y / c Optimized at 0? Optimized at 5? Optimized at 10? Optimized at 15? Figure 7.2: Optimized Airfoil Comparison 7.2 Validation of Propeller Code To con rm the propeller code was providing accurate data three commercial propeller?s were executed and discussed in this section. The three propellers chosen will be the propellers used as a baseline comparison for each of the conditions, cruise, climb, and climb cruise. The 48 propeller for cruise condition comparison was an APC 11x10E, the climb condition baseline propeller was an APC 8x6E, and the propeller for the climb cruise condition was an APC 10X7E. The reasonings for these baseline propellers will be discussed later in this chapter. The geometry was provided from the manufacturer [32] and wind tunnel data for each of these propellers was found in the UIUC Propeller Database [33]. 7.2.1 Validation Results for Baseline Cruise Propeller The cruise propeller chosen as the baseline propeller was an APC 11x10E. The following three gures are a comparison with thrust, power, and e ciency verse free stream velocity. The propeller code and the wind tunnel data was run at a rotational speed of 8000 rpm. The thrust comparison is shown in Figure 7.3. The thrust calculated using the propeller code is shown to slightly over predict, but does follow the same trend as the data. The increase in thrust over the wind tunnel data can be attributed to incorrect lift coe cients from XFOIL or other losses that were not accounted for in the propeller code. 0 5 10 15 20 25 30 35 40 450 2 4 6 8 10 12 14 16 18 20 Free Stream Velocity (MPH) Thru st (o z) APC 8x6E Calculated DataAPC 8x6E UIUC Windtunnel Data Figure 7.3: Thrust Validation for APC 11x10E Cruise Propeller at 8000 rpm 49 The propeller power comparison is shown in Figure 7.4. The propeller code under predicts the power required, but follows the same trend as the propeller code. This under prediction shows that XFOIL?s drag estimation is small. The e ciency plot is shown in Figure 7.5. The calculated e ciency is much higher than the wind tunnel data. This is due to the under estimation in power required and the over estimation of thrust produced. 0 5 10 15 20 25 30 35 400 10 20 30 40 50 60 70 Free Stream Velocity (MPH) Pow er (W atts ) APC 8x6E Calculated DataAPC 8x6E UIUC Windtunnel Data Figure 7.4: Power Validation for APC 11x10E Cruise Propeller at 8000 rpm 50 0 5 10 15 20 25 30 35 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effic ienc y APC 8x6E Calculated DataAPC 8x6E UIUC Windtunnel Data Figure 7.5: E ciency Validation for APC 11x10E Cruise Propeller at 8000 rpm 7.2.2 Validation Results for Baseline Climb Propeller The climb propeller chosen as the baseline propeller was an APC 8x6E. The following three gures are a comparison with thrust, power, and e ciency verse free stream velocity. The propeller code and the wind tunnel data was run at a rotational speed of 8000 rpm. The thrust comparison is shown in Figure 7.6. The thrust calculated using the propeller code is shown to slightly over predict, but does follow the same trend as the data. The increase in thrust over the wind tunnel data can be attributed to incorrect lift coe cients from XFOIL or other losses that were not accounted for in the propeller code. 51 0 10 20 30 40 50 60 700 5 10 15 20 25 30 35 40 45 50 Free Stream Velocity (MPH) Thru st (o z) APC 11x10E Calculated DataAPC 11x10E UIUC Windtunnel Data Figure 7.6: Thrust Validation for APC 8x6E Climb Propeller at 8000 rpm The propeller power comparison is shown in Figure 7.7. The propeller code calculates power required approximately the same for free stream velocities above 30 miles per hour. The propeller code does under predict power required below a free stream velocity of 30 miles per hour. This under prediction shows that XFOIL?s drag estimation could be small. The e ciency plot is shown in Figure 7.5. The calculated e ciency is higher than the wind tunnel data. This is due to the over estimation of thrust produced. 52 0 10 20 30 40 50 60 700 50 100 150 200 250 300 350 Free Stream Velocity (MPH) Pow er (W atts ) APC 11x10E Calculated DataAPC 11x10E UIUC Windtunnel Data Figure 7.7: Power Validation for APC 8x6E Climb Propeller at 8000 rpm 0 10 20 30 40 50 60 700 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effic ienc y APC 11x10E Calculated DataAPC 11x10E UIUC Windtunnel Data Figure 7.8: E ciency Validation for APC 8x6E Climb Propeller at 8000 rpm 53 7.2.3 Validation Results for Baseline Climb Cruise Propeller The climb cruise propeller chosen as the baseline propeller was an APC 10x7E. The following three gures are a comparison with thrust, power, and e ciency verse free stream velocity. The propeller code and the wind tunnel data was run at a rotational speed of 8000 rpm. The thrust comparison is shown in Figure 7.3. The thrust calculated using the propeller code is shown to slightly over predict, but does follow the same trend as the data. The increase in thrust over the wind tunnel data can be attributed to incorrect lift coe cients from XFOIL or other losses that were not accounted for in the propeller code. 0 10 20 30 40 50 600 5 10 15 20 25 30 35 Free Stream Velocity (MPH) Thru st (o z) APC 10x7E Calculated Data APC 10x7E UIUC Windtunnel Data Figure 7.9: Thrust Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm The propeller power comparison is shown in Figure 7.4. Similar to the APC 8X6E pro- peller, the propeller code calculates power required approximately the same for free stream velocities above 30 miles per hour. The propeller code does under predict power required below a free stream velocity of 30 miles per hour. This under prediction shows that XFOIL?s drag estimation is small. The e ciency plot is shown in Figure 7.5. The calculated e ciency 54 is much higher than the wind tunnel data. This is due to the under estimation in power required and the over estimation of thrust produced. 0 10 20 30 40 50 600 20 40 60 80 100 120 140 160 Free Stream Velocity (MPH) Pow er (W atts ) APC 10x7E Calculated DataAPC 10x7E UIUC Windtunnel Data Figure 7.10: Power Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm 55 0 10 20 30 40 50 600 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effic ienc y APC 10x7E Calculated DataAPC 10x7E UIUC Windtunnel Data Figure 7.11: E ciency Validation for APC 10x7E Climb Cruise Propeller at 8000 rpm 7.3 Propeller Optimized for Cruise The rst optimization that was tested was the cruise condition. From Table 7.2 the propeller needs to produce 7.5 ounces of thrust at a free stream velocity of 50 miles per hour. If the thrust is xed for the propeller to be as e cient as possible power for the propeller must be minimized. This leads to the three di erent objective functions used and are shown in Table 7.3 where z is the tness. Case 2a and 2b were chosen to show the e ect on which power is minimized. The range of values used by the optimizer are shown Table 7.3: Objective Functions used for Cruise Condition Case Number Objective Function: 1 z = Cp 2a z=Propeller Power (Watts) 2b z=System Power (Watts) in Table 7.4. The values of could only decrease to the tip, and the chord was allowed to expand to approximately the three quarters radius location then decrease to zero at the 56 tip. The optimizer was run for 100 generations with 5 pattern searches per generation and Table 7.4: Optimizer Limits for Cruise Condition Parameter Minimum Maximum start ( ) 35 65 chordstart(in) 0.5 2.0 Diameter(in) 8 12 RPM 3000 5500 Currrent(Amps) 0 14 Voltage(Volts) 0 11.1 a population size of 15 members. Each case had an execution time of 5 hours. Figures 7.12, 7.13, and 7.14 show the evaluated objective function for each member of each generation. The members that were \bad" propellers are not plotted. As discussed in the previous chapter these \bad" propellers do not meet the requirements and the objective function assigns them a value of 108. Each member is represented by a \x." A dashed line connects the maximum values of each generation as well as the minimum values. A solid line is used to show the best tness found as of that generation. 0 20 40 60 80 100 1200 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Number of Generations Obj ecti ve F unc tion (C p ) Figure 7.12: Fitness verse Number of Generations Case 1 (Objective Function = Cp) 57 0 20 40 60 80 100 12050 55 60 65 70 75 80 85 Number of Generations Obj ecti ve F unc tion (P, (Wa tts)) Figure 7.13: Fitness verse Number of Generations Case 2a (Objective Function = PropellerPower) 0 20 40 60 80 100 12065 70 75 80 85 90 95 100 Number of Generations Obj ecti ve F unc tion (Sy stem Pow er) Figure 7.14: Fitness verse Number of Generations Case 2b (Objective Function = SystemPower) Each case was run for 100 generations to show that the optimizer had converged on a solution. For Case 1 the optimizer found the \best" solution at generation 24 with an 58 objection function value of 0.0218. Case 2a the \best" solution was found on generation 11 with an objection function value of 51.19 Watts. Case 2b the \best" solution was found on generation 40 with an objection function value of 65.35 Watts. It is also shown that Case 2a and 2b nds a wider range of values compared to Case 1 were the values are more localized. The performance the three propellers are shown in Tables 7.5 and 7.6. Detailed properties (i.e. blade angles, chord sizes, pitch, and element position) for the two cases are shown in Tables F.1, F.2, F.3 in Appendix F. Table 7.5: Propeller Performance Parameters for Cruise Case 1 Case 1 (Objective Function = Cp) J 0.8000 Pitch at 3/4 radius 9.83 Ct 0.0244 Diameter (in) 12.0 Cp 0.0218 3=4 ( ) 19.23 prop 89.54% RPM 5500 Motor Power (Watts) 66.20 Motor Voltage (Volts) 7.78 Torque (oz-in) 12.87 Motor Current (Amps) 8.51 Thrust (oz) 7.55 motor 71.70% system 62.20% 59 Table 7.6: Propeller Performance Parameters for Cruise Case 2a and 2b Case 2a (Objective Function = PropellerPower) J 0.8085 Pitch at 3/4 radius 8.67 Ct 0.0248 Diameter (in) 12.0 Cp 0.0220 3=4 ( ) 17.16 prop 91.14% RPM 5442 Motor Power (Watts) 64.62 Motor Voltage (Volts) 7.69 Torque (oz-in) 12.72 Motor Current (Amps) 8.40 Thrust (oz) 7.51 motor 71.63% system 65.29% Case 2b (Objective Function = SystemPower) J 0.8359 Pitch at 3/4 radius 10.20 Ct 0.0283 Diameter (in) 11.6 Cp 0.0261 3=4 ( ) 20.50 prop 90.64% RPM 5433 Motor Power (Watts) 65.35 Motor Voltage (Volts) 7.70 Torque (oz-in) 12.84 Motor Current (Amps) 8.49 Thrust (oz) 7.52 motor 71.53% system 64.83% Table 7.5 shows that both propellers produced slightly higher than the desired thrust of 7.5 ounces. This is due to a tolerance set in the objective function that allowed the optimizer to produce pick propellers that produced more thrust in the early generations and then bring the thrust down as close as possible to the desired thrust. The propeller from Case 1 has an overall system e ciency of 62.20%, Case 2a has a slightly better system e ciency of 65.29%, and Case 2b has a system e ciency of 64.83%.. For this ight condition using the propeller power proved to provide better results. The system power should provide better results but 60 does not. This could be due the selection process from the optimizer. The optimizer was executed a total of three times to try to nd a solution for the system power that exceeded the propeller power case with no success. For these Cases a baseline propeller for comparison was chosen to be an APC 11x10E. This propeller is a commercially available propeller that is 11 inches in diameter with a constant pitch of 10 and is designed for electric motors.This was chosen due to its similar diameter and pitch compared to the optimized propellers. Figures 7.15 and 7.16 show the chord and blade angle distribution across the radius of the blade. The chord in the baseline propeller expands greatly until mid-blade span then decreases to the tip while Case 1 tapers to the tip. Case 2a?s chord expands before it tapers to the tip, and Case 2b maintain until half the blade span then tapers to the tip. Both cases show the blade angle to taper slowly to the tip. All of optimized propellers are shown to have a smaller chord than the baseline propeller. The baselines root angle is approximately 55 and tapers to 18 while Case 1?s root blade angle is 57:75 and tapers to 1:5 at the tip. Case 2a?s root angle is smaller at 40:95 and tapers to 8:7 , and Case 2b?s root angle is larger at 54 and tapers to 4:80 . 61 0 1 2 3 4 5 60 0.2 0.4 0.6 0.8 1 1.2 1.4 Radial Position (in) Cho rd ( in) Cruise 12 inch (Pitch 9.83) Case 1 Cruise 12 inch (Pitch 8.67) Case 2aCruise 11.63 inch (Pitch 10.62) Case 2b Baseline APC 11x10E Figure 7.15: Chord Distribution for Cruise Condition Propellers 0 1 2 3 4 5 60 10 20 30 40 50 60 Radial Position (in) Bla de A ngle ,? ( deg ree s) Cruise 12 inch (Pitch 9.83) Case 1 Cruise 12 inch (Pitch 8.67) Case 2aCruise 11.63 inch (Pitch 10.62) Case 2b Baseline APC 11x10E Figure 7.16: Blade Angle Distribution for Cruise Condition Propellers 62 Figure 7.17 shows the possible blade chord dimensions and optimized blade chord di- mensions for Case 1 only. This plot is for a xed diameter of 12 inches. If the optimizer would have chosen a di erent diameter the plot would be di erent. The solid line is the contour that was picked, and the dashed and dotted lines are the maximum and minimum possible chord contours. -0.3-0.2-0.100.10.20.3 0 0.1 0.2 0.3 0.4 0.5 0.6 c / D r/D Des ign Prof ile Max Cho rd M ax Exp ansi on Min Cho rd M in Exp ansi on Figure 7.17: Comparison of Possible Blade Chord Pro les for Case 1 Di erent number of elements used to nd the performance parameters was used. Orig- inally 10 elements were used to describe each blade, but 100 was also used to see if there was any improvement when lift distribution over the blades was integrated. The 100 ele- ments were found using interpolation between the original 10 optimized elements. For Case 1 Ct = 0:0256 and Cp = 0:0225 with 10 elements and with 100 elements the values changed to Ct = 0:0253 and Cp = 0:0222. Figure 7.18 shows the lift distribution on the blade. The 63 increased elements smooths the distribution, but it does not make a signi cant di erence in the integrated values. 0 1 2 3 4 5 6-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Radial Location (in) Ct / dr 10 Elements 100 Elements Figure 7.18: Lift Distribution on Blade Comparing the Number of Elements used to nd Performance Parameters for Case 1 The performance parameters for each of the propellers were evaluated from a free stream velocity of 10 mph to whenever the propeller fails to produce any thrust. The APC 11x10E baseline propeller was included for comparison. The data for the baseline propeller was calculated using the same propeller code and XFOIL used to calculate the performance parameters of the other propellers. The thrust was calculated assuming an input voltage of 11.1 volts for the same motor used in optimizer, Brushless Park 450. This simulates a full throttle input throughout the free stream velocity range. The rotation speed, rpm, was found through an iteration process between the electric motor model and the propeller performance codes. Figure 7.19 shows the thrust verse free stream with the rpm at 11.1 volts, Figure 7.20 shows the power for the propeller verse free stream and current draw at 11.1 volts, and Figure 7.21 shows the e ciency of the propeller verse free stream. It should 64 be noted that the power plotted in Figure 7.20 is the power required for the propeller. The motor will require more power due to losses in the motor. The optimized locations are shown on the Figures with indicators. 0 10 20 30 40 50 60 70 80 900 5 10 15 20 25 30 35 Free Stream Velocity (MPH) Thr ust (oz) 0 10 20 30 40 50 60 70 80 904000 6000 8000 10000 Free Stream Velocity (MPH) RPM Cruise 12 inch (Pitch 9.83) Case 1Cruise 12 inch (Pitch 8.67) Case 2a Cruise 11.63 inch (Pitch 10.62) Case 2bBaseline APC 11x10E Case 1 Optimal LocationCase 2a Optimal Location Case 2b Optimal Location Figure 7.19: Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller Thrust Over a Range of Free Stream Velocities 65 0 10 20 30 40 50 60 70 80 900 50 100 150 Free Stream Velocity (MPH) Pow er ( Wa tts) 0 10 20 30 40 50 60 70 80 900 10 20 30 Free Stream Velocity (MPH) Cur ren t (A mps ) Cruise 12 inch (Pitch 9.83) Case 1Cruise 12 inch (Pitch 8.67) Case 2a Cruise 11.63 inch (Pitch 10.62) Case 2bBaseline APC 11x10E Case 1 Optimal LocationCase 2a Optimal Location Case 2b Optimal Location Figure 7.20: Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller Power Over a Range of Free Stream Velocities 0 10 20 30 40 50 60 70 80 900 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effi cien cy Cruise 12 inch (Pitch 9.83) Case 1Cruise 12 inch (Pitch 8.67) Case 2a Cruise 11.63 inch (Pitch 10.62) Case 2bBaseline APC 11x10E Case 1 Optimal LocationCase 2a Optimal Location Case 2b Optimal Location Figure 7.21: Comparison of Case 1, Case 2a, and Case 2b Propellers Against Baseline Propeller E ciency Over a Range of Free Stream Velocities 66 All of the optimized propeller are shown to require less power than the baseline propeller with approximately equal thrust. This leads to the baseline propeller be less e cient than the optimized locations. The optimized locations are also shown to have a higher e ciency at the design speed then the propeller at full throttle at the design speed. This is due to the increase in power required and more losses in the motor and propeller. All of the propellers have about the same blade angle distribution, but di erent chord distributions. Since the blades have about the same blade angles then only di erence is the airfoils. The optimized airfoils are shown to out perform the baseline propellers airfoils. Case 2a requires less power and closely matches the desired thrust at the optimized locations it was chosen as the \best" for cruise. 7.4 Propeller Optimized for Climb The second optimization that was tested was the climb condition. From Table 7.2 the propeller needs to produce as much thrust possible at a free stream velocity of 15 miles per hour given the maximum current draw the motor can perform at maximum voltage which was set to 11.1 volts to simulate a 3-cell lithium polymer battery power supply for the motor. If the power is nearly constant for the propeller then to be as e cient as possible thrust needs to be maximized. The optimizer is designed to minimize a problem. This leads to inverse of the thrust is used as the objective function. The two di erent objective functions, one non-dimensional and the other with dimensional values, and are shown in Table 7.7 where z is the tness. The range of values used by the optimizer are shown in Table 7.8. Table 7.7: Objective Functions used for Climb Condition Case Number Objective Function: 3 z = 1Ct 4 z = 1Ft(ounces) The values of could only decrease to the tip, and the chord was allowed to expand to approximately the three quarters radius location then decrease to zero at the tip. For both 67 Table 7.8: Optimizer Limits for Climb Condition Parameter Minimum Maximum start ( ) 20 65 chordstart(in) 0.5 2.0 Diameter(in) 8 12 MotorCurrent(Amps) 1 14 of the objective functions used the optimizer was run for 100 generations with 5 pattern searches per generation and a population size of 30 members. The number of members was increased from the cruise condition due to the inadequate number of members that would nd a viable solution in each generation. Shown in Figures 7.22 and 7.23 a large number the members did not provide a \good" solution. These Figures show the evaluated objective function for each member of each generation. The members that were \bad" propellers are not plotted. As discussed in the previous chapter these "bad" propellers do not meet the requirements and the objective function assigns them a value of 108. Each member is represented by a \x." A dashed line connects the maximum values of each generation as well as the minimum values. A solid line is used to show the best tness found as of that generation. 68 0 20 40 60 80 100 1200 5 10 15 20 25 30 Number of Generations Obj ecti ve F unc tion (1/ C t) Figure 7.22: Fitness verse Number of Generations Case 3 (Objective Function = 1Ct) 0 20 40 60 80 100 1200.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 Number of Generations Obj ecti ve F unc tion (1/ Thr ust) Figure 7.23: Fitness verse Number of Generations Case 4 (Objective Function = 1Ft) 69 Each case was run for 100 generations to show that the optimizer had converged on a solution. For Case 3 the optimizer found the \best" solution at generation 99 with an objection function value of 4.826 and took 15 hours to complete the 100 generations. For Case 4 the \best" solution was found on generation 25 with an objection function value of 0.030 and took 11 hours to complete. Again it is shown that Case 4 nds a wider range of values compared to Case 3 were the values are more localized as the number of generations increase. The performance the two propellers are shown in Table 7.9. Detailed properties (i.e. blade angles, chord sizes, pitch, and element position) for the two cases are shown in Tables G.1 and G.2 in Appendix G. Table 7.9 shows that both propellers produced slightly Table 7.9: Propeller Performance Parameters for Climb Case 3 and Case 4 Case 3 (Objective Function = 1Ct) J 0.2452 Pitch at 3/4 radius 5.25 Ct 0.2072 Diameter (in) 8.12 Cp 0.0982 3=4 ( ) 15.38 prop 51.74% RPM 7956 Motor Power (Watts) 126.19 Motor Voltage (Volts) 11.1 Torque (oz-in) 17.21 Motor Current (Amps) 11.37 Thrust (oz) 28.10 motor 74.62% system 38.61% Case 4 (Objective Function = 1Ft) J 0.2067 Pitch at 3/4 radius 3.29 Ct 0.1054 Diameter (in) 9.73 Cp 0.0415 3=4 ( ) 8.26 prop 52.51% RPM 7876 Motor Power (Watts) 131.10 Motor Voltage (Volts) 11.1 Torque (oz-in) 17.60 Motor Current (Amps) 11.81 Thrust (oz) 28.87 motor 74.05% system 38.88% lower than the maximum allowable current of 18 Amps. This is due to a tolerance set in the objective function that allowed the optimizer to produce pick propellers that used less than the maximum current in the early generations and then bring the current up to drive the thrust as high as possible. The propeller from Case 3 has an overall system e ciency of 70 38.61% and produces 28.10 ounces of thrust at a free stream velocity of 15 miles per hour. Case 4 has a slightly better system e ciency of 38.88% and produces 28.87 ounces of thrust. Case 4 with proved to produce a more e cient propeller with more thrust requiring less power than Case 3. An APC 8x6E propeller was chosen for baseline comparison due to its very similar nature to the optimized climb propellers. The baseline propeller has a diameter of 8 inches with a constant pitch of 6. Figures 7.24 and 7.25 show the chord and blade angle distribution across the radius of the blade. The chord in Case 3 expands before it tapers to the tip while Case 4 remains approximately constant for half of the radius. Both cases show the blade angle to follow an exponential decay to tip. Case 3?s root blade angle is 58:05 and is 4:89 at the tip. Case 4?s root angle is smaller at 42:67 and tapers to 1:46 . It is shown that chords are much larger and the blade angles are approximately when compared to the baseline propeller. 0 1 2 3 4 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Radial Position (in) Cho rd ( in) Climb 8.12 inch (Pitch 5.25) Case 3Climb 9.73 inch (Pitch 3.29) Case 4 Baseline APC 8x6E Figure 7.24: Chord Distribution for Climb Condition Propellers 71 0 1 2 3 4 50 10 20 30 40 50 60 Radial Position (in) Bla de A ngle ,? ( deg ree s) Climb 8.12 inch (Pitch 5.25) Case 3 Climb 9.73 inch (Pitch 3.29) Case 4Baseline APC 8x6E Figure 7.25: Blade Angle Distribution for Climb Condition Propellers To see if any of the propeller blade is stalled at the optimized climb location Figure 7.26 was created. This plot shows that the relative angle of attack on each element for Case 3 which reaches a maximum around 8 . This is not close to the stall range which is important since an average Reynolds number was used for these airfoils, and the Reynolds number dictates stall location. Case 4 is not shown due to its similar blade angles and ight conditions that produces a similar result. 72 0 0.5 1 1.5 2 2.5 3 3.5 40 1 2 3 4 5 6 7 8 Radial Position (in) Ang le o f At tack (de gre es) Element location Figure 7.26: Case 3 Angle of Attack Distribution The performance parameters for each of the propellers were evaluated from a free stream velocity of 10 mph to whenever the propeller fails to produce any thrust. A baseline propeller was included for comparison.This propeller was chosen to be an APC 8x6E due to it similar geometry to the optimized propellers in Case 3 and 4. The data for the baseline propeller was calculated using the same propeller code and XFOIL used to calculate the performance parameters of the other propellers. The thrust was calculated assuming an input voltage of 11.1 volts for the same motor used in optimizer, Brushless Park 450. This simulates a full throttle input throughout the free stream velocity range. The rotation speed, rpm, was found through an iteration process between the electric motor model and the propeller performance codes. Figure 7.27 shows the thrust verse free stream with the rpm at 11.1 volts, Figure 7.28 shows the power for the propeller verse free stream and current draw at 11.1 volts, and Figure 7.29 shows the e ciency of the propeller verse free stream. It should be noted that the power plotted in Figure 7.28 is the power required for the propeller. The 73 motor will require more power due to losses in the motor. The optimized locations are shown on the Figures with indicators. 0 10 20 30 40 50 60 700 5 10 15 20 25 30 35 Free Stream Velocity (MPH) Thru st (o z) 0 10 20 30 40 50 60 707000 8000 9000 10000 Free Stream Velocity (MPH) RPM Climb 8.12 inch (Pitch 5.25) Case 3Climb 9.73 inch (Pitch 3.29) Case 4 Baseline APC 8x6ECase 3 Optimal Location Case 4 Optimal Location Figure 7.27: Comparison of Case 3 and Case 4 Propellers Thrust over a Range of Free Stream Velocities 74 0 10 20 30 40 50 60 700 50 100 150 Free Stream Velocity (MPH) Pow er ( Wa tts) 0 10 20 30 40 50 60 700 5 10 15 Free Stream Velocity (MPH) Cur rent (Am ps) Climb 8.12 inch (Pitch 5.25) Case 3Climb 9.73 inch (Pitch 3.29) Case 4 Baseline APC 8x6ECase 3 Optimal Location Case 4 Optimal Location Figure 7.28: Comparison of Case 3 and Case 4 Propellers Power over a Range of Free Stream Velocities 0 10 20 30 40 50 60 700 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effi cien cy Climb 8.12 inch (Pitch 5.25) Case 3Climb 9.73 inch (Pitch 3.29) Case 4 Baseline APC 8x6ECase 3 Optimal Location Case 4 Optimal Location Figure 7.29: Comparison of Case 3 and Case 4 Propellers E ciency over a Range of Free Stream Velocities 75 The optimized propeller from Case 4 is shown to have a less thrust, require less power, and have a higher e ciency throughout the free stream velocities. Both propellers are shown to have much larger chords than the baseline while the blade angles follow the same trend. The baseline propeller requires more power compared to the optimized propellers. The Case 3 propeller will produce thrust in a wider range in free stream velocities at full throttle. Case 3 and Case 4 were marginally di erent, but Case 4 met the design requirements better than Case 3. 7.5 Propeller Optimized for Climb-Cruise The third optimization that was tested was the multipoint climb cruise condition. From Table 7.2 the propeller needs to produce as much thrust possible at a free stream velocity of 15 miles per hour. The propeller also needs to produce 7.5 ounces of thrust at a free stream velocity of 50 miles per hour. Similar to the previous optimizations two di erent objective functions were used, but the it was changed slightly for the climb cases. An attempt was made combining the methods from the cruise and climb optimizer, but the optimizer could not make an initial population in a reasonable period of time. A minimum desired thrust of 24 ounces was then set for the climb condition. The rst is non-dimensional and is maximizing the e ciency at the two di erent design conditions. The other tries to minimize the power at cruise and the thrust during climb. The two di erent objective functions are shown in Table 7.10 where z is the tness. The range of values used by the optimizer are shown in Table 7.10: Objective Functions used for Climb Cruise Condition Case Number Objective Function: 5 z = 1 cruise climb 6 z = PowerThrust Table 7.11. The values of could only decrease to the tip, and the chord was allowed to expand to approximately the three quarters radius location then decrease to zero at the tip. For both of the objective functions used the optimizer was run for 75 generations with 5 76 Table 7.11: Optimizer Limits for Climb-Cruise Condition Parameter Minimum Maximum start ( ) 45 65 chordstart(in) 0.5 2.0 Diameter(in) 8 12 RPMcruise 3000 5500 RPMclimb 5000 8500 Currrentcruise (Amps) 0 14 Currrentclimb (Amps) 0 18 Voltage(Volts) 0 11.1 pattern searches per generation and a population size of 15 members. As shown in Figures 7.30 and 7.31 a large number the members did not provide a \good" solution. These Figures show the evaluated objective function for each member of each generation. The members that were \bad" propellers are not plotted. As discussed in the previous chapter these \bad" propellers do not meet the requirements and the objective function assigns them a value of 108. Each member is represented by a \x." A dashed line connects the maximum values of each generation as well as the minimum values. A solid line is used to show the best tness found as of that generation. 77 0 10 20 30 40 50 60 70 800.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 Number of Generations Ob ject ive Fun ctio n (1 -? c limb ? cru ise) Figure 7.30: Fitness verse Number of Generations Case 5 (Objective Function = 1 prop motor) 0 10 20 30 40 50 60 70 801.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Number of Generations Ob ject ive Fun ctio n (P owe r/Th rus t) Figure 7.31: Fitness verse Number of Generations Case 6 (Objective Function = PowerCruiseThrust Climb ) 78 Each case was run for 75 generation to show that the optimizer had converged on a solution. For Case 5 the optimizer found the "best" solution at generation 56 with an objection function value of 0.8196 and took 5 hours to complete the 75 generations. For Case 6 the \best" solution was found on generation 30 with an objection function value of 1.587 and took 5.5 hours to complete. Again it is shown that Case 6 nds a wider range of values compared to Case 5 were the values are more localized as the number of generations increase. The performance the two propellers are shown in Tables 7.12 and 7.13. Detailed properties (i.e. blade angles, chord sizes, pitch, and element position) for the two cases are shown in Tables H.1 and H.2 in Appendix H. Table 7.12: Propeller Performance Parameters for Climb Cruise Case 5 Case 5 (Objective Function = 1 cruise climb) Cruise 3=4 ( ) 19.61 Pitch at 3/4 radius 8.92 J 0.9009 Diameter (in) 10.6559 Ct 0.0457 RPM 5500 Cp 0.0470 Motor Power (Watts) 82.20 prop 87.60% Torque (oz-in) 15.33 Thrust (oz) 8.79 Motor Voltage (Volts) 8.11 Motor Current (Amps) 10.14 motor 69.82% system 61.16% Climb J 0.2684 Ct 0.1413 RPM 5538 Cp 0.0739 Motor Power (Watts) 151.45 prop 51.32% Torque (oz-in) 24.43 Thrust (oz) 27.54 Motor Voltage (Volts) 9.37 Motor Current (Amps) 16.16 motor 62.66% system 32.16% Tables 7.12 and 7.13 shows that both propellers produced higher than the minimum thrust at climb conditions. The Case 5 propeller where the objective function was based on the e ciencies at the design points produced a thrust of 24.43 ounces with a system e ciency 79 Table 7.13: Propeller Performance Parameters for Climb Cruise Case 6 Case 6 (Objective Function = PowerThrust) Cruise 3=4 ( ) 19.78 Pitch at 3/4 radius 8.80 J 0.9345 Diameter (in) 10.4566 Ct 0.0441 RPM 5404 Cp 0.0469 Motor Power (Watts) 68.78 prop 87.87% Torque (oz-in) 13.43 Thrust (oz) 7.59 Motor Voltage (Volts) 7.74 Motor Current (Amps) 8.89 motor 70.94% system 62.34% Climb J 0.2230 Ct 0.1244 RPM 6792 Cp 0.0578 Motor Power (Watts) 190.41 prop 48.00% Torque (oz-in) 26.15 Thrust (oz) 33.82 Motor Voltage (Volts) 11.00 Motor Current (Amps) 17.31 motor 65.74% system 31.55% of 32.16%, and Case 6 produced a greater thrust at 33.82 ounces with a system e ciency of 31.55%. At cruise Case 5 produced 8.79 ounces of thrust with a system e ciency of 61.16%,, and Case 6 produced 7.59 ounces of thrust with a system e ciency of 62.34%. Case 5 and Case 6 peak in e ciency at 88%. Case 6 is shown to produce as much thrust as possible and achieve closer to a desired cruise speed compared to Case 5. A baseline propeller was added to the following series of Figures for comparison. The baseline propeller was chosen to be an APC 10x7E which has a diameter of 10 inches and a constant pitch of 7. Figures 7.32 and 7.33 show the chord and blade angle distribution across the radius of the blade. The chord in both cases expands before it tapers to the tip. Both cases show the blade angle to follow an exponential decay to approximately 15 at the tip. Case 5?s root blade angle is 53:74 and is 16:42 at the tip. Case 6?s root angle is smaller at 46:86 and tapers to 13:79 . 80 0 1 2 3 4 5 60 0.2 0.4 0.6 0.8 1 1.2 1.4 Radial Position (in) Cho rd ( in) Climb-Cruise 10.66 inch (Pitch 8.92) Case 5Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Baseline APC 10x7E Figure 7.32: Chord Distribution for Climb Cruise Condition Propellers 0 1 2 3 4 5 610 15 20 25 30 35 40 45 50 55 Radial Position (in) Bla de A ngle ,? ( deg ree s) Climb-Cruise 10.66 inch (Pitch 8.92) Case 5 Climb-Cruise 10.46 inch (Pitch 8.80) Case 6Baseline APC 10x7E Figure 7.33: Blade Angle Distribution for Climb Cruise Condition Propellers 81 The performance parameters for each of the propellers were evaluated from a free stream velocity of 15 mph to whenever the propeller fails to produce any thrust. A baseline propeller was included for comparison. For these cases an APC 10x7 thin electric propeller was chosen because it is similar to Case 5 and Case 6 propellers. This is a 10 inch propeller with a constant blade pitch of 7 inches. The data for the baseline propeller was obtained from the UIUC Propeller Database [33]. Figure 7.34 shows the thrust verse free stream, Figure 7.35 shows the power for the propeller verse free stream, and Figure 7.36 shows the e ciency of the propeller verse free stream. It should be noted that the power plotted in Figure 7.35 is the power required for the propeller. The motor will require more power due to losses in the motor. The Figures are plotted at a xed rotational speed for the propeller which is the optimized rotational speed. The performance parameters for each of the propellers were evaluated from a free stream velocity of 15 mph to whenever the propeller fails to produce any thrust. A baseline propeller was included for comparison.This propeller was chosen to be an APC 10x7E due to it similar geometry to the optimized propellers in Case 5 and 6. The data for the baseline propeller was calculated using the same propeller code and XFOIL used to calculate the performance parameters of the other propellers. The thrust was calculated assuming an input voltage of 11.1 volts for the same motor used in optimizer, Brushless Park 450. This simulates a full throttle input throughout the free stream velocity range. The rotation speed, rpm, was found through an iteration process between the electric motor model and the propeller performance codes. Figure 7.34 shows the thrust verse free stream with the rpm at 11.1 volts, Figure 7.35 shows the power for the propeller verse free stream and current draw at 11.1 volts, and Figure 7.36 shows the e ciency of the propeller verse free stream. It should be noted that the power plotted in Figure 7.35 is the power required for the propeller. The motor will require more power due to losses in the motor. The optimized locations are shown on the Figures with indicators. 82 0 10 20 30 40 50 60 70 800 5 10 15 20 25 30 35 Free Stream Velocity (MPH) Thr ust (oz) 0 10 20 30 40 50 60 70 804000 6000 8000 10000 Free Stream Velocity (MPH) RPM Climb-Cruise 10.66 inch (Pitch 8.92) Case 5Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Baseline APC 10x7ECase 5 Optimal Location Case 6 Optimal Location Figure 7.34: Comparison of Case 5 and Case 6 Propellers Thrust over a Range of Free Stream Velocities 0 10 20 30 40 50 60 70 800 50 100 150 Free Stream Velocity (MPH) Pow er ( Wa tts) 0 10 20 30 40 50 60 70 800 10 20 30 Free Stream Velocity (MPH) Cur ren t (A mps ) Climb-Cruise 10.66 inch (Pitch 8.92) Case 5Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Baseline APC 10x7ECase 5 Optimal Location Case 6 Optimal Location Figure 7.35: Comparison of Case 5 and Case 6 Propellers Power over a Range of Free Stream Velocities 83 0 10 20 30 40 50 60 70 800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effi cien cy Climb-Cruise 10.66 inch (Pitch 8.92) Case 5Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Baseline APC 10x7ECase 5 Optimal Location Case 6 Optimal Location Figure 7.36: Comparison of Case 5 and Case 6 Propellers E ciency over a Range of Free Stream Velocities The optimized propellers from Case 5 and Case 6 is shown to have approximately the same performance parameters at the full throttle setting, but the Case 6 propeller out performs Case 5 at the design condition for climb. Both optimized propellers produce more thrust and require less power when compared to the baseline propeller. The geometry Case 5 and Case 6 are only slightly di erent from the baseline. The chord distributions follow the same trends with approximately the same values, and the blade angles follow the same trends and approximate values as well. The optimized propellers are also shown to operate at a much higher free stream velocity compared to the the baseline propeller. Case 5 does produce higher thrust and require more power compared to the baseline and Case 6. Case 6 does produce higher thrust at the climb design point and more closely matches the thrust desired for cruise. 84 7.6 Cruise, Climb, and Climb-Cruise Comparisons The \best" propeller from each of the optimized conditions was compared. The \best" cases were found to be Case 2a, Case 4, and Case 6. These propellers were shown to out perform the other cases. The blade angles are compared in Figure 7.37. All of the propellers follow the same trend with the Climb-Cruise and Cruise propellers have similar values and the Climb propeller having smaller values. 0 1 2 3 4 5 60 5 10 15 20 25 30 35 40 45 50 Radial Position (in) Bla de A ngle , ? (de gre es) Cruise 12 inch (Pitch 8.67) Case 2a Climb 9.73 inch (Pitch 3.29) Case 4Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Figure 7.37: Comparison of Blade Angles, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) Figure 7.38 shows the chord distribution for each of the \best" propellers. The Cruise propeller is shown to be the largest diameter propeller with the smallest overall chord. The climb propeller has the largest chord and the smallest diameter, and the Climb-Cruise propeller has a diameter and chord that is in between the other conditions. The thrust for these propellers was calculated in the same manner as the previous result sections. The motor model was given 11.1 volts and a rpm was found through iteration to match propeller torque and the motor torque. This simulates a theoretical full throttle 85 0 1 2 3 4 5 60 0.5 1 1.5 Radial Position (in) Cho rd ( in) Cruise 12 inch (Pitch 8.67) Case 2a Climb 9.73 inch (Pitch 3.29) Case 4Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Figure 7.38: Comparison of Chords, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) scenario. The thrust comparison for the \best" propellers can be found in Figure 7.39. The rpm of the propeller/motor is shown in a subplot on the same Figure. The Cruise and Climb-Cruise propellers are shown to produce more thrust than the climb propeller. They are also able to produce thrust longer. This is due to their higher blade angles that allow the relative angle of attack of the propeller to produce positive lift coe cients for each of the elements. The Cruise and Climb-Cruise propellers also have a much lower rotation speed at this full throttle setting due to there increased diameter. The power requirement for the propellers is shown in Figure 7.40. Again the Cruise and Climb-Cruise propellers are shown to be similar. The climb propeller requires less power since it was optimized for this design condition. The current draw is also shown and follows the same trends as the power curves. Once more the Cruise and Climb-Cruise propellers are shown to be very similar in the e ciency plot in Figure 7.41, but the Cruise propeller is slightly better. These propellers peak in e ciency at 90% around 70 miles per hour for the Cruise propeller and 80 miles 86 0 10 20 30 40 50 60 70 800 5 10 15 20 25 30 35 Free Stream Velocity (MPH) Thr ust (oz) 0 10 20 30 40 50 60 70 806000 8000 10000 Free Stream Veloctiy RPM Cruise 12 inch (Pitch 8.67) Case 2aClimb 9.73 inch (Pitch 3.29) Case 4 Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Figure 7.39: Comparison of Thrust, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) per hour for the Climb-Cruise propeller. The Climb propeller matches the Cruise propeller?s e ciency until 45 miles per hour at 81 then rapidly decreases. 87 0 10 20 30 40 50 60 70 800 50 100 150 Free Stream Velocity (MPH) Pow er ( Wa tts) 0 10 20 30 40 50 60 70 800 10 20 30 Free Stream Velocity (MPH) Cur ren t (A mps ) Cruise 12 inch (Pitch 8.67) Case 2aClimb 9.73 inch (Pitch 3.29) Case 4 Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Figure 7.40: Comparison of Power, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) 0 10 20 30 40 50 60 70 800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Free Stream Velocity (MPH) Effi cien cy Cruise 12 inch (Pitch 8.67) Case 2aClimb 9.73 inch (Pitch 3.29) Case 4 Climb-Cruise 10.46 inch (Pitch 8.80) Case 6 Figure 7.41: Comparison of E ciency, , for the \Best" Propellers (Case 2a, Case 4, and Case 6) 88 Chapter 8 Conclusions A method for optimizing an electrically driven propeller for single and multiple con- ditions with a hybrid pattern search particle swarm optimizer was performed. 6th order Bernstein polynomials were used to parameterize airfoils that were used with Xfoil to calcu- late lift and drag coe cients. The hybrid optimizer was used to maximize the lift to drag ratio at zero angle of attack for an airfoil. A simplex optimizer was then used to nd airfoils over a range of angle of attack from 5 15 . A table was generated with the optimum airfoils and was used for the optimized propellers. The propeller analysis was evaluated using a momentum blade element method with axial and rotational in ow factors. The pattern search particle swarm optimizer produced global minimums in the large design spaces of the propellers. This optimizer required a large amount of time compared to the simplex method that was used for the airfoil optimizer beyond 0 airfoil. However the simplex was also shown to be highly dependent on initial location. The objective functions used by the optimizers strongly in uence the design. The objective functions using dimen- sional values proved to produce better results. Several restrictions were also required on the chord and blade angle distributions which if left unrestrained would produce impractical designs. For the Cruise condition the power was minimized. It was shown that an objective function using dimensional values verse non-dimensional values was better. The dimensional value case using the propeller power required had a 3% overall system e ciency improvement over the non-dimensional case. The system power optimization was shown to require slightly more power than the propeller power optimization. The optimized propellers were all shown to have improvements over a commercially available baseline propeller. These optimized 89 propellers were shown to have smaller chords and approximately equal blades when compared to the baseline propeller. For the Climb condition the thrust was maximized. The dimensional value objective function proved marginally better than the non-dimensional. The results were very similar with each case producing 28 ounces of thrust and having an overall system e ciency of 39%. Even though the Case 3 propeller could produce more thrust over a wider range of free stream velocities, Case 4 is the better propeller due to its higher e ciency at the design point. The non-dimensional objective function case did have a run time approximately 40% longer than the dimensional case. When compared to a baseline propeller both cases showed much improvement in power required. The optimized propellers had larger chords than the baseline propeller, and the blade angles were slightly smaller. For the Climb-Cruise condition two di erent methods were used. The rst was non- dimensional objective function maximizing the e ciencies at climb and cruise conditions. The second objection function minimized power at cruise over thrust at climb. Both methods proved to have advantages and disadvantages. The optimizer in the non-dimensional case drove thrust at climb to the minimum desire thrust, but had a higher e ciency across the range of operation compared to the dimensional objective function. The dimensional case showed to have a substantial more amount of thrust at climb, but this caused the e ciency to su er. The optimized propellers were compared to a baseline propeller with marginally di erent chord and blade angle distributions. These results produced two propellers that when compared to a baseline propeller require less power, produce more thrust, and have higher e ciencies. All of the optimized propellers proved to be more e cient than the baseline propellers. The optimized airfoils allowed the propellers to have increased performance and a larger range of operation. Cruise propellers were shown to have larger blade angles, smaller chords, and larger diameters while Climb propellers had smaller blade angles, larger chords, and smaller 90 diameters. Climb-Cruise propellers exhibited blade angles similar to cruise propellers and diameters and chords in between the Climb and Cruise conditions. A method for optimizing an electrical motor driven propeller has been presented and shown to be successful for single and multiple design points. The inclusion of a model for the electric motor assists the optimizer in nding the most e cient rotational speed for the propeller and eliminates the iterative process of manually matching an electric motor and a propeller. Compared to baseline propeller performance, the method produced more e cient propeller designs for single point and multipoint objective functions. As with any method, additional improvements could be made. A better method for calculating induced air ow through the rotor disk could be implemented and would provide more accurate results. Integrating an airfoil optimizer in with the propeller optimizer might provide a signi cant performance increase but would also dramatically increase run time. Improvements in the accuracy at o design conditions where the relative angle of attack on the propeller blades could be in a stalled region would make the analysis more robust. Other optimization techniques such as a genetic algorithm could be evaluated to minimize computational times and possibly improve results. An electric motor optimizer could be developed to investigate if any improvements over commercially available motors are possible. In addition to these method improvements, thoughts for future work include actual prototyping a propeller design optimized by this method and conducting performance tests to further verify and validate the method. Also providing an internal combustion motor model to the method could be considered. This addition would allow comparison between propellers optimized for electric and internal combustion motors and provide insight into the fundamental di erences between propellers tuned to the two types of powerplants. 91 Bibliography [1] Glauert, H., The Elements of Aerofoil and Airscrew Theory, Cambridge, 1947. [2] Dommasch, D. O., Element of Propeller and Helicopter Aerodynamics, Sir Isaac Pitman & Sons, LTD, 1953. [3] Adkins, C. N. and Liebeck, R. H., \Design of Optimum Propellers," Journal of Propul- sion and Power, Vol. 10, No. 5, Sept.-Oct 1994. [4] Glauert, H., Airplane Propellers, Springer Verlag, 1935, Aerodynamic Theory: Volume IV Chapter XI. [5] Burger, C., Propeller Performance Analysis and Multidisciplinary Optimization using a Genetic Algorithm, Ph.D. thesis, Auburn University, 2007. [6] Adkins, C. N. and Liebeck, R. H., \Design of Optimum Propellers," AIAA Paper 83- 0190, AIAA 21st Aerospace Sciences Meeting, 1983. [7] Fanjoy, D. W. and Crossley, W. A., \Aerodynamic Shape Design for Rotor Airfoils via Genetic Algorithm," American Helicopter Society 53rd Annual Forum, 1998. [8] Miller, C. J., Optimally Designed Propellers Constrained by Noise, Ph.D. thesis, Purdue University, 1984. [9] Kulfan, B. M., \Universal Parametric Geometry Representation Method," Journal of Aircraft, Vol. 45, No. 1, 2008, DOI: 10.2514/1.29958. [10] Anderson, J. D., Fundamental of Aerodynamics, McGraw-Hill, 2007. [11] Venkataraman, P., \A New Procedure for Airfoil De nition," AIAA Paper 95-1875-CP, 13th Applied Aerodynamics Conference, 1995. [12] Rogalsky, T., Kocabiyik, S., and Derksen, R. W., \Di erential Evolution in Aerody- namic Optimization," Industrial Engineering, Vol. 46, No. 4, 2000, pp. 183{190. [13] Kulfan, B. M. and Bussoletti, J. E., \Fundamental Parametric Geometry Representa- tions for Aircraft Component Shapes," AIAA paper 2006-6948, 2006. [14] XFOIL User Guide 6.96 , MIT Aero & Astro, 2001. [15] Drela, M., \XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils," Low Reynolds Number Aerodynamics, Vol. 54, Springer-Verlag, 1989. 92 [16] Nelson, W. C., Airplane Propeller Principles, John Wiley and Sons, 1944. [17] Weick, F. E., Aircraft Propeller Design, McGraw-Hill, 1930. [18] Viterna, A. and Janetzke, D., \Theoretical and Experimental Power from Large Horizontal-Axis Wind Turbines," Proceedings from the Large Horizontal-Axis Wind Tur- bine Conference, July 1981, DOE/NASA-LeRC. [19] Kenjo, T. and Nagamori, S., Permanent-Magnet and Brushless DC Motors, Oxford University Press, 1985. [20] Sokira, T. J. and Ja e, W., Brushless DC Motors Electronic Commutation and Controls, TAB BOOKS, Inc., 1990. [21] Pittman Servo Motor Application Notes, Pittman. [22] Zhu, J. G. and Watterson, P., \Electromechanical Systems - Chapter 12. Brushless DC Motors," Lecture Notes. [23] Hepperle, M., 2008, http://www.mh-aerotools.de/airfoils/cox performance.htm. [24] Eberhart, R. and Kennedy, J., \A New Optimizer using Particle Swarm Theory," Pro- ceedings Sixth Symposium on Micro Machine and Human Science, IEEE Service Center, 1995. [25] Hooke, R. and Jeeves, T., Direct Search Solution of Numerical and Statistical Problems, 1961. [26] Jenkins, R. and Hart eld, R., \Hybrid Particle Swarm-Pattern Search Optimizer for Aerospace Propulsion Applications," AIAA Paper 2010-7078, 2010. [27] MATLAB User Guide R2010a, The MathWorks Inc., 2010. [28] J. C. Lagarias, J. A. Reeds, M. H. W. and Wright, P. E., \Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM Journal of Optimization, Vol. 9, No. 1, 1998, pp. 112{147. [29] William H. Press, Saul A. Teukolsky, W. T. V. and Flannery, B. P., Numerical Recipes in Fortarn 77 , Cambridge University Press, 1992. [30] Tangler, J. and Kocureki, J. D., \Wind Turbine Post-Stall Airfoil Performance Char- acteristics Guidelines for Blade-Element Momentum Methods," National Renewable Energy Laboratory, Report NREL/CP-500-36900, October 2004. [31] E ite Park 450 Brushless Outrunner Instructions, Horizon Hobby, Inc., 2009. [32] APCProp@AOL.com, 2012, Personal Correspondance with APC Propellers. [33] Ananda, G. et al., \UIUC Propeller Database," 2008, http://www.ae.illinois.edu/m- selig/props/propDB.html. 93 Appendices 94 Appendix A Xfoil Inputs and Outputs A.1 Example XFOIL Command Inputs The following is a list of commands that would be entered for a standard XFOIL session. More inputs can be found in the users manual [14]. plop (plot menu) G (disable plotting) (blank return) load (load menu) Point File.dat ( le containing x-y points) Airfoil Name (name of airfoil) ppar (paneling parameters menu) N (change number of panels) 140 (increase number of panels to 140) (blank return) (blank return) oper (direct operating points menu) visc (toggle to viscous mode) 150000 (enter a Reynolds number) iter (change the maximum iteration limit) 500 (new iteration limit) pacc (toggle to auto point accumulation to active polar menu) Airfoil data.dat ( le for polar to be saved to) Airfoil data dump.dat ( le for information to be dumped to) aseq -5 20 1 (run an angle of attack sweep in 1 degree increments from 5 to 20 (blank return) quit (exit XFOIL) 95 A.2 Example XFOIL Point File The following is a example input for an airfoil which would be placed in a .dat or .txt le. The rst column is the x-coordinates and the second column is the y-coordinates. The coordinates are sepearated by a space not a tab. 1.0000000 0.0015000 0.8998800 0.0331400 0.7998101 0.0537800 0.6997400 0.0714200 0.5997000 0.0840600 0.4996700 0.0917000 0.3996600 0.0953400 0.2996600 0.0957800 0.1996700 0.0906200 0.0997300 0.0742600 0.0498100 0.0540800 0.0248700 0.0359900 0.0000000 0.0000000 0.0250200 -.0050100 0.0500200 -.0049200 0.1000200 -.0047400 0.2000200 -.0043800 0.3000100 -.0040200 0.4000100 -.0036600 0.5000100 -.0033000 0.6000100 -.0029400 0.7000100 -.0025800 0.8000100 -.0022200 0.9000100 -.0018600 1.0000000 -.0015000 96 A.3 Example XFOIL Output File The following is an example output le from XFOIL. XFOIL Version 6.96 Calculated polar f o r : Airfoil name 1 1 Reynolds number f i x e d Mach number f i x e d x t r f = 1.000 ( top ) 1.000 ( bottom ) Mach = 0.000 Re = 0.150 e 6 Ncrit = 9.000 alpha CL CD CDp CM Top Xtr Bot Xtr 5.000 0.2971 0.06867 0.06517 0.0435 0.9774 0.0423 4.000 0.1719 0.05244 0.04819 0.0617 0.9602 0.0537 3.000 0.0243 0.04023 0.03482 0.0746 0.9400 0.0797 1.000 0.3144 0.02246 0.01434 0.0905 0.8947 0.0633 0.000 0.4805 0.01534 0.00969 0.0980 0.8748 1.0000 1.000 0.6333 0.01355 0.00722 0.1036 0.8420 1.0000 2.000 0.7748 0.01237 0.00566 0.1071 0.7816 1.0000 3.000 0.8683 0.01245 0.00503 0.1012 0.6140 1.0000 4.000 0.9332 0.01526 0.00624 0.0917 0.4074 1.0000 5.000 1.0193 0.01757 0.00800 0.0876 0.3374 1.0000 6.000 1.1143 0.01973 0.01000 0.0854 0.2994 1.0000 7.000 1.2096 0.02195 0.01239 0.0834 0.2708 1.0000 8.000 1.3077 0.02469 0.01528 0.0823 0.2469 1.0000 9.000 1.3956 0.02764 0.01878 0.0794 0.2243 1.0000 10.000 1.4389 0.02869 0.02017 0.0687 0.1916 1.0000 11.000 1.4503 0.02956 0.02164 0.0538 0.1456 1.0000 12.000 1.4317 0.03907 0.03050 0.0408 0.0300 1.0000 13.000 1.4107 0.05045 0.04293 0.0330 0.0256 1.0000 14.000 1.3559 0.06895 0.06228 0.0327 0.0235 1.0000 15.000 1.2904 0.09461 0.08865 0.0415 0.0229 1.0000 16.000 1.2540 0.11515 0.10958 0.0479 0.0219 1.0000 17.000 1.2702 0.12359 0.11825 0.0443 0.0204 1.0000 18.000 1.2434 0.14474 0.14006 0.0535 0.0200 1.0000 19.000 1.2073 0.17196 0.16791 0.0713 0.0204 1.0000 20.000 0.7811 0.17806 0.17484 0.0597 0.0285 1.0000 97 Appendix B Derivation of Adkins-Liebeck Di erential Coe cients This appendix is for the derivation of Equations 3.47 and 3.48 which comes from Adkins and Liebeck in Reference [3]. This is to clear any questions of their derivation process and to correct an error in dCtd equation in Reference [3] even though the error is not found in the original paper [6]. First the momentum propeller theory and Figure 3.8 is used to nd the thrust per unit radius. dFt dr = (massperunitnondimensionalradius) (velocityincreaseaxialdirection) = (2 r V0 (1 +a)) (2V0aF) = 4 r V20 (1 +a)aF (B.1) The torque per unit radius is found using the same process. dQ rdr = (massperunitnondimensionalradius) (velocityincreaserotationdirection) = (2 r V0 (1 +a)) (4 nra0F) = 8 2r2 V0n(1 +a)a0F (B.2) If Cy, Cx, Cl, and Cd are substituted for dFt, dF, dL, and dD respectively in Figure 3.8. Cy and Cx and then de ned by the following equations. Cy = Clcos( ) Cdsin( ) = Cl (cos( ) + sin( )) (B.3) Cx = Clsin( ) +Cdcos( ) = Cl (sin( ) + cos( )) (B.4) The blade element theory can then be used to de ne the thrust and torque per unit radius where b is the chord of the element. dFt dr = 1 2 V 2 relBbCy (B.5) dQ rdr = 1 2 V 2 relBbCx (B.6) 98 With Equations B.1 and B.5 set equal to each other the axial interference factor, a, can be found. Using Vrel = V0(1+a)sin( ) from Figure 3.8. dFt dr = 1 2 V 2 relBbCy = 4 r V 2 0 (1 +a)aF 1 2 V 0 (1 +a) sin( ) 2 BbCy = 4 r V20 (1 +a)aF 1 2 V20 (1 +a)2 sin2 ( ) BbCy = 4 rV 2 0 (1 +a)aF a = 1F Bb2 r Cy4sin2 ( ) (1 +a) a = 1 F Bb 2 r Cy 4sin2 ( ) + 1 F Bb 2 r Cy 4sin2 ( ) a a 1 1F Bb2 r Cy4sin2 ( ) = 1 F Bb 2 r Cy 4sin2 ( ) a = h 1 F Bb 2 r Cy 4sin2( ) i h 1 1F Bb2 r Cy4sin2( ) i where the solidity is = Bb2 r and a constant K = Cy4sin2( ) a nal simpli ed equation can be found. a = K(F K) (B.7) The same process is repeated with Equations B.2 and B.6 to nd the rotational interference factor, a0, but before this is found a geometric relationship from Figure 3.8 for needs to be expressed. tan( ) = V0(1 +a)2 nr(1 a0) (B.8) a? is then found as follows. dQ rdr = 1 2 V 2 relBbCx = 8 2r2 V0n(1 +a)a0F 1 2 V 0(1 +a) sin( ) 2 BbCx = 8 2r2 V0n(1 +a)a0F 1 2 V20 (1 +a)2 sin2( ) BbCx = 8 2r2V0n(1 +a)a0F Bb 2 r V0(1 +a) sin2( ) Cx = 8 rna 0F Bb 2 r 2 nr(1 a0)tan( ) sin2( ) Cx = 8 rna 0F 99 1 F Bb 2 r Cx 4cos( )sin( )(1 a 0) = a0 a0 = 1 F Bb 2 r Cx 4cos( )sin( ) 1 F Bb 2 r Cx 4cos( )sin( ) a0 a0 + 1 F Bb 2 r Cx 4cos( )sin( ) a0 = 1 F Bb 2 r Cx 4cos( )sin( ) a0 = h 1 F Bb 2 r Cx 4cos( )sin( ) i 1 + h 1 F Bb 2 r Cx 4cos( )sin( ) i where K0 = Cx4cos( )sin( ), a0 = K 0 F + K0 (B.9) The di erential forms of thrust coe cient with respect to can now be found using Equation B.10, the thrust coe cient equation and Equation B.1, the thrust equation. Ct = Ft n2D4 (B.10) dCt d = dFt d n2D4 dCt d = 4 r V20 (1 +a)aFR n2D4 dCt d = 4 rV20 1 + K(F K) K (F K)FR n2D4 dCt d = 4 rRV20 F 16n2R4 K (F K) + 2K2 (F K)2 dCt d = V20 F 4n2R2 ( K) (F K) + 2K2 (F K)2 dCt d = V20 F 4n2R2 KF (F K)2 dCt d = V20 F2 4n2R2 K (F K)2 dCt d = V20 F2 4n2R2 (F K)2 C y 4sin2( ) 100 Equation B.8 must be modi ed now before the derivation can continue. tan( ) = V0(1 +a)2 nr(1 a0) tan( ) = V02 nr 1 + K(F K) 1 K0 F+ K0 tan( ) = V02 nr F K+ K F K F+ K0 K0 F+ K0 tan( ) = V02 nrF (F + K 0) (F K)F sin( ) cos( ) = V0 2 nr (F + K0) (F K) (F K)sin( ) = V02 nr (F + K0)cos( ) (B.11) Equation B.11 can now be substituted into the derivation. dCt d = 16 CyV20 F3 n2R2 1 (F K)2sin2 ( ) dCt d = 16 CyV20 F3 n2R2 1 V20 4 2n2r2 (F + K 0)2cos2 ( ) dCt d = 3 4 3F2 Cy [(F + K0)cos( )]2 (B.12) The di erential form of torque with respect to follows the same procedure as the previous with using B.13 and B.2. Cp = P n3D5 (B.13) dCp d = dP d n3D5 = dQ d 2 n n3D5 = (2 n) (8 2r3 V0n(1 +a)a0F)R n3D5 dCp d = 16n2 3r3V0 (1 +a)a0FR n3D5 dCp d = 16 3r3V0 (1 +a)a0FR 32nR5 dCp d = 3 3V0Fa0(1 +a) 2nR dCp d = 3 3V0F 2nR K0 F + K0 1 + K(F K) 101 dCp d = 3 3V0F 2nR K0 F + K0 + K0 (F + K0) K (F K) dCp d = 3 3V0F 2nR ( K0) (F K) + 2K0K (F + K0) (F K) dCp d = 3 3V0F 2nR FK0 (F + K0) (F K) dCp d = 3 3V0F 2nR F (F + K0) (F K) Cx 4cos( )sin( ) Using the relationship found in Equation B.11, dCp d = 3 3V0F 2nR F 4cos( ) (F + K0) Cx V0 2 nr (F + K 0)cos( ) dCp d = 3 4 3F2 Cy [(F + K0)cos( )]2 Cx Cy dCp d = dCt d Cx Cy (B.14) 102 Appendix C Airfoil Optimization Codes C.1 AirfoilMaker() function [ c l cd angles c l s cds ] = AirfoilMaker ( alpha , Atop , Abottom ) Order = 6 ; T r a i l z = 0 . 0 0 1 ; numPoints = 50; N1 = 0 . 5 ; N2 = 1 . 0 ; %Figure Out A i r f o i l f i l e name ( A i r f o i l I D ) temp = ? ?; f o r n=1: length ( Atop ) i f round ( Atop (n) 100) < 10 && round ( Atop (n) 100) >= 0.0 temp = s t r c a t (temp , ? 0 ? , num2str ( round ( Atop (n ) 1 0 0 ) ) ) ; e l s e temp = s t r c a t (temp , num2str ( round ( Atop (n ) 1 0 0 ) ) ) ; end end f o r n=1: length ( Abottom ) i f round ( Abottom (n) 100) < 10 && round ( Abottom (n) 100) >= 0.0 temp = s t r c a t (temp , ? 0 ? , num2str ( round ( Abottom (n ) 1 0 0 ) ) ) ; e l s e i f round ( Abottom (n) 100) < 0 . 0 . . . && round ( Abottom (n) 100) > 10.0 temp = s t r c a t (temp , ? 0 ? , num2str( 1 round ( Abottom (n ) 1 0 0 ) ) ) ; e l s e i f round ( Abottom (n) 100) <= 10 temp = s t r c a t (temp , ? ? , num2str( 1 round ( Abottom (n ) 1 0 0 ) ) ) ; e l s e temp = s t r c a t (temp , num2str ( round ( Abottom (n ) 1 0 0 ) ) ) ; end end A i r f o i l I D = temp ; %Get the points f o r the A i r f o i l [ x , y , isGood ] = . . . P a r a m e t r i c A i r f o i l ( Atop , Abottom , Order , Trailz , numPoints , N1, N2 ) ; 103 %Get the l i f t and drag c o e f f i f isGood == 1 [ cl , cd , angles , cls , cds ] = ClCdFinder (x , y , alpha , A i r f o i l I D ) ; e l s e c l = 10; cd = 100; angles = l i n s p a c e ( 0 , 1 5 , 1 5 ) ; c l s = 1 l i n s p a c e ( 1 5 , 2 0 , 1 5 ) ; cds = 100 l i n s p a c e ( 1 5 , 2 0 , 1 5 ) ; end C.2 ParametricAirfoil() function [ x , y , isGood ] = . . . P a r a m e t r i c A i r f o i l ( Atop , Abottom , Order , Trailz , numPoints , N1, N2) PolyOrder = 6 ; xoc = l i n s p a c e (0 ,1 , numPoints ) ; %f i n d the K i values f o r n=0:Order Ki (n+1) = f a c t o r i a l ( Order )/( f a c t o r i a l (n) f a c t o r i a l ( Order n ) ) ; end %f i n d c l a s s and shape function values . Finished with the z/c values f o r n=1: length ( xoc ) C(n) = xoc (n)^N1 (1 xoc (n ))^N2 ; i f xoc (n) == 0.0 Stop (n) = 0 . 0 ; Sbottom (n) = 0 . 0 ; e l s e i f xoc (n) == 1.0 Stop (n) = 0 . 0 ; Sbottom (n) = 0 . 0 ; e l s e Stop (n) = 0 . 0 ; Sbottom (n) = 0 . 0 ; f o r m=0:Order Stop (n) = Stop (n) + . . . Atop (m+1) Ki (m+1) xoc (n)^m (1 xoc (n ) ) ^ ( Order m) ; Sbottom (n) = Sbottom (n) + . . . Abottom (m+1) Ki (m+1) xoc (n)^m (1 xoc (n ) ) ^ ( Order m) ; end 104 end zocTop (n)=C(n) Stop (n)+xoc (n) T r a i l z ; zocBottom (n)=C(n) Sbottom (n) xoc (n) T r a i l z ; end %Check to see i f the bottom curve %i n t e r s e c t s or passes the upper curve check = 0 ; f o r n=1: length ( zocTop ) i f zocTop (n) zocBottom (n) < 0.0 check = check + 1 ; end end i f check == 0 isGood = 1 ; e l s e isGood = 0 ; end %End of i n t e r s e c t i o n check %combine upper and lower s u r f a c e %Atop == Abottom f o r n=1: length ( xoc ) x (n) = xoc ( length ( xoc) n+1); y (n) = zocTop ( length ( xoc) n+1); i f n>1 x (n+length ( xoc) 1) = xoc (n ) ; y (n+length ( xoc) 1) = zocTop (n ) ; end end end C.3 ClCdFinder() function [ cl , cd , angles , cls , cds ] = ClCdFinder (x , y , alpha , A i r f o i l I D ) %This function takes a s e t of points f o r an a i r f o i l and %runs x f o i l . exe with the points and a s e t alpha . I t then %returns the l i f t and drag c o e f f i c i e n t s warning o f f a l l %User Defined S e t t i n g s 105 DirectoryName = ? OptimumAirfoils ? ; N = 120; %number of segments f o r x f o i l AoAstart = 0 ; AoAend = 15; AoAstep = 0 . 5 ; isGood = 1 ; %Check f o r Main Directory CheckMDir = i s d i r ( DirectoryName ) ; i f CheckMDir == 0 mkdir ( DirectoryName ) ; DoesFileExist = 0 ; e l s e %Check f o r A i r f o i l fidTemp = fopen ( s t r c a t ( DirectoryName , ?n? , AirfoilID , ? . dat ? ) , ? r ? ) ; i f fidTemp ~= 1 f c l o s e ( fidTemp ) ; DoesFileExist = 1 ; e l s e DoesFileExist = 0 ; end end % %i f the A i r f o i l I D f i l e does not e x i s t i t w i l l be created i f DoesFileExist == 0 stop = 1 ; while stop == 1 d e l e t e ( ? points . dat ? ) d e l e t e ( ?dump . dat ? ) d e l e t e ( ? BatchInstr . inp ? ) %Create Points F i l e f i d = fopen ( ? points . dat ? , ?w? ) ; f o r m=1: length ( x ) f p r i n t f ( fid , ?%8.4 f %8.4 fnrnn ? , x (m) , y (m) ) ; end f c l o s e ( f i d ) ; %Create I n s t r u c t i o n F i l e f o r Batch F i l e f i d = fopen ( ? BatchInstr . inp ? , ?w? ) ; f p r i n t f ( fid , ? plopnrnnG nrnn nrnn ? ) ; f p r i n t f ( fid , ? loadnrnn ? ) ; f p r i n t f ( fid , ? points . datnrnn%snrnn ? , A i r f o i l I D ) ; f p r i n t f ( fid , ? pparnrnnNnrnn%inrnn nrnn nrnn ? ,N) ; f p r i n t f ( fid , ? opernrnn ? ) ; f p r i n t f ( fid , ? v i s cnrnn150000nrnn ? ) ; f p r i n t f ( fid , ? i t e rnrnn400nrnn ? ) ; f p r i n t f ( fid , ? paccnrnn%snn%s . datnrnndump . datnrnn ? , . . . 106 DirectoryName , A i r f o i l I D ) ; f p r i n t f ( fid , ? aseq %4.2 f %4.2 f %4.2 fnrnn ? , . . . AoAstart , AoAend , AoAstep ) ; f p r i n t f ( fid , ?nrnnquitnrnn ? ) ; f c l o s e ( f i d ) ; %Run Batch F i l e and d e l e t e some unwanted f o l d e r s [ sh ,wow]=dos ( ? x f o i l t a b l e f i l e c r e a t e . bat ? ) ; c l e a r sh c l e a r wow %Check f o r NaN filenameNaN = s t r c a t ( DirectoryName , ?n? , AirfoilID , ? . dat ? ) ; fidNaN = fopen ( filenameNaN , ? r ? ) ; i f fidNaN == 1 Nplus = 1 ; e l s e currentLine = 1 ; c=1; Nplus = 0 ; while ~ f e o f ( fidNaN ) temp = f g e t l ( fidNaN ) ; Angles = 0 . 0 ; i f currentLine >= 13 a i r f o i l D a t a = str2num ( temp ) ; Angles = a i r f o i l D a t a ( 1 ) ; Cl = a i r f o i l D a t a ( 2 ) ; Cd = a i r f o i l D a t a ( 3 ) ; i f Nplus == 0 i f isnan ( Cl ) jj isnan (Cd) N=N+10; Nplus = 1 ; end end end currentLine = currentLine +1; end i f Nplus == 0 i f Angles <= 0.6667 AoAend N=N+10; Nplus = 1 ; end end f c l o s e ( fidNaN ) ; i f Nplus ~= 1 jj N==170; i f N>165 && Nplus==1 isGood = 0 ; 107 end stop = 0 ; e l s e d e l e t e ( filenameNaN ) end end end d e l e t e ( ? points . dat ? ) d e l e t e ( ?dump . dat ? ) d e l e t e ( ? BatchInstr . inp ? ) end % %Open f i l e and get the l i f t and drag curves i f isGood == 1 f i d = fopen ( s t r c a t ( DirectoryName , ?n? , AirfoilID , ? . dat ? ) , ? r ? ) ; currentLine = 1 ; n=1; while ~ f e o f ( f i d ) temp = f g e t l ( f i d ) ; i f currentLine >= 13 a i r f o i l D a t a = str2num ( temp ) ; angles (n) = a i r f o i l D a t a ( 1 ) ; c l s (n) = a i r f o i l D a t a ( 2 ) ; cds (n) = a i r f o i l D a t a ( 3 ) ; n=n+1; end currentLine=currentLine +1; end e l s e f i d = fopen ( s t r c a t ( DirectoryName , ?n? , AirfoilID , ? . dat ? ) , ?w? ) ; f o r n=1:40 f p r i n t f ( fid , ?% i 10000 10000nrnn ? , n ) ; end f c l o s e ( f i d ) ; c l = 10; cd = 100; angles = l i n s p a c e ( 0 , 1 5 , 1 5 ) ; c l s = 1 l i n s p a c e ( 1 5 , 2 0 , 1 5 ) ; cds = 100 l i n s p a c e ( 1 5 , 2 0 , 1 5 ) ; end %Find c l and cd at d e s i r e d alpha using i n t e r p o l a t i o n i f isGood == 1 AoA = alpha ; %Begin Table Lookup 108 stopLookup = 0 ; nn=1; while stopLookup == 0 i f AoA < 0 stopZero = 0 ; m=1; while stopZero == 0 i f angles (m) >= 0 ZeroLoc = m; stopZero = 1 ; end m=m+1; end s h i f t = 10; c l = c l s ( ZeroLoc+s h i f t ) . . . ( angles ( ZeroLoc+s h i f t ) AoA) ( c l s ( ZeroLoc+s h i f t ) . . . c l s ( ZeroLoc ) ) / ( angles ( ZeroLoc+s h i f t ) . . . angles ( ZeroLoc ) ) ; cd = cds ( ZeroLoc+s h i f t ) ... ( angles ( ZeroLoc+s h i f t ) AoA) ( cds ( ZeroLoc+s h i f t ) . . . cds ( ZeroLoc ) ) / ( angles ( ZeroLoc+s h i f t ) . . . angles ( ZeroLoc ) ) ; stopLookup = 1 ; e l s e i f AoA == angles (nn) c l = c l s (nn ) ; cd = cds (nn ) ; stopLookup = 1 ; e l s e i f AoA < angles (nn+1) && AoA > angles (nn) c l = c l s (nn+1) ( angles (nn+1) AoA ) . . . ( c l s (nn+1) c l s (nn ) ) / ( angles (nn+1) angles (nn ) ) ; cd = cds (nn+1) ( angles (nn+1) AoA ) . . . ( cds (nn+1) cds (nn ) ) / ( angles (nn+1) angles (nn ) ) ; stopLookup = 1 ; e l s e i f AoA > angles ( length ( angles ) ) c l = c l s ( length ( angles ) ) . . . ( angles ( length ( angles )) AoA ) . . . ( c l s ( length ( angles )) c l s ( length ( angles ) 1 ) ) . . . /( angles ( length ( angles ) ) . . . angles ( length ( angles ) 1)); cd = cds ( length ( angles ) ) . . . ( angles ( length ( angles )) AoA ) . . . ( cds ( length ( angles )) cds ( length ( angles ) 1 ) ) / . . . ( angles ( length ( angles ) ) . . . angles ( length ( angles ) 1)); stopLookup = 1 ; 109 e l s e i f AoA < angles (1) c l = c l s (2) ( angles (2) AoA ) . . . ( c l s (2) c l s ( 1 ) ) / ( angles (2) angles ( 1 ) ) ; cd = cds (2) ( angles (2) AoA ) . . . ( cds (2) cds ( 1 ) ) / ( angles (2) angles ( 1 ) ) ; stopLookup = 1 ; end nn=nn+1; end %End Table Lookup end f c l o s e ( ? a l l ? ) ; 110 Appendix D Propeller Optimization Codes D.1 BrushlessMotor() function [ Output ] = . . . BrushlessMotorV2 ( Input ,Kv, I n t e r n a l R e s i s t a n c e , . . . numPoles , numPhases , IdleCurrent , WhichCase ) %Cases %WhichCase = 1 : INPUTS : [ Voltage , Current ] OUTPUTS: [ Torque , RPM, Eta ] %WhichCase = 2 : INPUTS : [RPM, Torque ] OUTPUTS: [ Voltage , Current , Eta ] %Constants Kt = 1000/Kv 1 . 3 4 5 ; VoltageStep = 0 . 0 1 ; RPMtol = 0 . 1 ; %+, rpm t o l e r a n c e range %Solve f o r Unknowns i f WhichCase == 1 %Voltage and Current are inputs Voltage = Input ( 1 ) ; Current = Input ( 2 ) ; %f i n d RPM Max RPM = Voltage Kv; lambda = 2/( numPoles Max RPM ) . . . ( Voltage I n t e r n a l R e s i s t a n c e IdleCurrent ) ; Tem = ( numPhases numPoles )/2 lambda Current ; RPM = ( Voltage /( numPoles lambda /2) ... I n t e r n a l R e s i s t a n c e /( numPhases ( numPoles lambda /2)^2) Tem) ; %f i n d Torque Torque = Kt Current ; %Find E f f i c i e n c y Power In = Voltage Current ; Power Out =(Voltage Current . . . I n t e r n a l R e s i s t a n c e ) ( Current IdleCurrent ) ; Eta = Power Out/Power In ; 111 %Set Outputs Output (1) = Torque ; Output (2) = RPM; Output (3) = Eta ; e l s e i f WhichCase == 2 %RPM and Torque are inputs RPM = Input ( 1 ) ; Torque = Input ( 2 ) ; %Find the Current f o r the Motor given the Torque Current = Torque/Kt ; %Find the Voltage required f o r the given Torque and RPM Voltage = 1 . 0 ; stop = 0 ; while stop == 0 Max RPM = Voltage Kv; lambda = 2/( numPoles Max RPM ) . . . ( Voltage I n t e r n a l R e s i s t a n c e IdleCurrent ) ; Tem = ( numPhases numPoles )/2 lambda Current ; RPMVolt = ( Voltage /( numPoles lambda /2) ... I n t e r n a l R e s i s t a n c e /( numPhases . . . ( numPoles lambda /2)^2) Tem) ; i f abs (RPM RPMVolt) < RPMtol jj RPMVolt > RPM stop = 1 ; e l s e Voltage = Voltage+VoltageStep ; end end %Solve f o r E f f i c i e n c y Power In = Voltage Current ; Power Out =(Voltage Current I n t e r n a l R e s i s t a n c e ) . . . ( Current IdleCurrent ) ; Eta = Power Out/Power In ; %Set Outputs Output (1) = Voltage ; Output (2) = Current ; Output (3) = Eta ; end D.2 PropellerPerformance() 112 function [ Ct , Cp, Eta , J ] = PropellerPerformance ( Beta , . . . Diameter , Chord , Position , Props ) %This i s ?The ? function f o r c a l c u a t i n g performance %parameters f o r p r o p e l l e r s . I t uses the Adkins/ Glauert %method to f i n d thrust /power/ speed curves load A i r f o i l I D l i s t . mat RPM = Props ( 1 ) ; FreeStreamMPH = Props ( 2 ) ; numBlades = Props ( 3 ) ; Position = Position Diameter /2; MaxIter = 150; % Units % Beta = Radians % RPM = Revolutions per Minute % Diameter = inches % Chord = inches % Position = inches % FreeStreampMPH = Miles per Hour % numBlades = non dim % rho = Slugs per cubic f o o t % OutputData = non dim %constants damp = 0 . 5 ; %t h i s i s the damping c o e f f i c i e n t f o r convergence J = (FreeStreamMPH 5280/3600)/((RPM/60) ( Diameter / 1 2 ) ) ; numSections = length ( Position ) ; % % MAIN % % I n i t i a l i z e Arrays alpha = zeros ( numSections ) ; c l = zeros ( numSections ) ; cd = zeros ( numSections ) ; a = zeros ( numSections ) ; a prime = zeros ( numSections ) ; phi new = zeros ( numSections , 1 ) ; cdcl = zeros ( numSections ) ; Cy = zeros ( numSections ) ; 113 K = zeros ( numSections ) ; Cx = zeros ( numSections ) ; Kprime = zeros ( numSections ) ; sigma = zeros ( numSections ) ; xi = zeros ( numSections ) ; p h i t = zeros ( numSections ) ; f = zeros ( numSections ) ; F = zeros ( numSections ) ; Ct prime = zeros ( numSections ) ; Cp prime = zeros ( numSections ) ; W = zeros ( numSections ) ; % Solve f o r i n i t i a l inflow angles f o r n=1: numSections i n i t i a l p h i (n , 1 ) = atan ( ( FreeStreamMPH 5 2 8 0 / 3 6 0 0 ) / . . . (2 pi Position (n)/12 RPM/ 6 0 ) ) ; sigma (n) = ( numBlades Chord (n )/12)/(2 pi Position (n ) / 1 2 ) ; xi (n) = Position (n )/( Diameter / 2 ) ; end % Run the I n t e r a t i o n Part of the Code %Adkins/ Glauert Method phi = i n i t i a l p h i ; stop = 0 ; numIt = 1 ; angles = l i n s p a c e ( 5 ,15 ,21); % angles = l i n s p a c e ( 0 , 1 5 , 1 6 ) ; while stop == 0 ; %f i n d alpha /cd/ c l /Cy/Cx/ everything f o r n=1: numSections alpha (n) = r e a l ( Beta (n) phi (n ) ) ; i f n == numSections alpha (n) = 0 . 0 ; c l (n) = c l (n 1); cd (n) = cd (n 1); e l s e i f alpha (n) >= 19 pi /180 %Flat plate Theory c l (n) = 2 s i n ( alpha (n )) cos ( alpha (n ) ) ; cd (n) = 2 ( s i n ( alpha (n ) ) ) ^ 2 ; e l s e AoA = round ( alpha (n) 180/ pi ) ; i f AoA <= 5 idNum = 1 ; e l s e i f AoA >= 15 idNum = 21; 114 e l s e stopAoA = 0 ; c=1; while stopAoA == 0 i f AoA<= angles ( c+1) && AoA>angles ( c ) idNum = c ; stopAoA=1; end c=c+1; end end [ c l (n) cd (n ) ] = ClCdFinderAirfoilID ( . . . num2str ( A i r f o i l I D l i s t (idNum , : ) ) , alpha (n ) . . . 180/ pi ) ; end end cdcl (n) = cd (n)/ c l (n ) ; Cy(n) = c l (n ) ( cos ( phi (n)) cdcl (n) s i n ( phi (n ) ) ) ; K(n) = Cy(n )/(4 s i n ( phi (n )) s i n ( phi (n ) ) ) ; Cx(n) = c l (n ) ( s i n ( phi (n))+ cdcl (n) cos ( phi (n ) ) ) ; Kprime (n) = Cx(n )/(4 cos ( phi (n )) s i n ( phi (n ) ) ) ; p h i t (n) = atan ( xi (n) tan ( phi (n ) ) ) ; f (n) = ( numBlades/2) (1 xi (n ))/ s i n ( p h i t (n ) ) ; F(n) = (2/ pi ) acos ( exp( 1 f (n ) ) ) ; a (n) = ( sigma (n) K(n ) ) / (F(n) sigma (n) K(n ) ) ; i f isnan ( a (n ) ) a (n) = 0 . 0 ; e l s e i f a (n) > 0.7 a (n) = 0 . 7 ; e l s e i f a (n) < 1.0 a (n) = 0 . 7 ; end a prime (n) = ( sigma (n) Kprime (n ) ) / . . . (F(n)+sigma (n) Kprime (n ) ) ; i f isnan ( a prime (n ) ) a prime (n) = 0 . 0 ; e l s e i f a prime (n)>0.7 a prime (n) = 0 . 7 ; end phi new (n , 1 ) = atan2 ( ( ( FreeStreamMPH 5 2 8 0 / 3 6 0 0 ) . . . (1+a (n ) ) ) / (RPM/60 2 pi Position (n ) / . . . 12 (1 a prime (n ) ) ) , 1 ) ; i f isnan ( phi new (n ) ) phi new (n , 1 ) = 0 . 0 ; e l s e i f n == numSections 115 phi new (n , 1 ) = phi new (n 1 ,1); end end i f ( abs ( phi new phi ) <= 10^ 4) stop = 1 ; e l s e phi = phi damp +(1 damp) phi new ; end i f ( numIt >= MaxIter ) stop = 1 ; end numIt = numIt+1; end % Finished with Interating %Find Ct , Cp, Eta f o r n=1: numSections W(n) = (FreeStreamMPH 5280/3600) (1+a (n ))/ s i n ( phi (n ) ) ; i f isnan (W(n ) ) W(n) = W(n 1); end Ct prime (n) = ( pi ^3/4) sigma (n) Cy(n) xi (n ) ^ 3 . . . F(n )^2/((F(n)+sigma (n) Kprime (n )) cos ( phi (n ) ) ) ^ 2 ; i f isnan ( Ct prime (n ) ) Ct prime (n) = 0 . 0 ; end Cp prime (n) = Ct prime (n) pi xi (n) Cx(n)/Cy(n ) ; i f isnan ( Cp prime (n ) ) Cp prime (n) = 0 . 0 ; end end Ct = 0 ; Cp = 0 ; %sum up Thrust and Torque f o r n=1: numSections i f n==1 Ct=Ct+Ct prime (n)/2 xi (n ) ; Cp=Cp+Cp prime (n)/2 xi (n ) ; e l s e Ct = Ct + ( Ct prime (n)+Ct prime (n 1))/2 ( xi (n) xi (n 1)); Cp = Cp + ( Cp prime (n)+Cp prime (n 1))/2 ( xi (n) xi (n 1)); end end 116 Eta = Ct J/Cp; D.3 ClCdFinderAirfoilID() function [ cl , cd ] = ClCdFinderAirfoilID ( AirfoilID , alpha ) DirectoryName = ? OptimumAirfoils ? ; f i d = fopen ( s t r c a t ( DirectoryName , ?n? , AirfoilID , ? . dat ? ) , ? r ? ) ; currentLine = 1 ; n=1; while ~ f e o f ( f i d ) temp = f g e t l ( f i d ) ; i f currentLine >= 13 a i r f o i l D a t a = str2num ( temp ) ; angles (n) = a i r f o i l D a t a ( 1 ) ; c l s (n) = a i r f o i l D a t a ( 2 ) ; cds (n) = a i r f o i l D a t a ( 3 ) ; n=n+1; end currentLine=currentLine +1; end f c l o s e ( f i d ) ; AoA = alpha ; %Begin Table Lookup stopLookup = 0 ; nn=1; while stopLookup == 0 i f AoA < 0 stopZero = 0 ; m=1; while stopZero == 0 i f angles (m) >= 0 ZeroLoc = m; stopZero = 1 ; end m=m+1; end s h i f t = 10; c l = c l s ( ZeroLoc+s h i f t ) ( angles ( ZeroLoc+s h i f t ) AoA ) . . . ( c l s ( ZeroLoc+s h i f t ) c l s ( ZeroLoc ) ) / . . . 117 ( angles ( ZeroLoc+s h i f t ) angles ( ZeroLoc ) ) ; cd = cds ( ZeroLoc+s h i f t ) ( angles ( ZeroLoc+s h i f t ) AoA ) . . . ( cds ( ZeroLoc+s h i f t ) cds ( ZeroLoc ) ) / . . . ( angles ( ZeroLoc+s h i f t ) angles ( ZeroLoc ) ) ; stopLookup = 1 ; e l s e i f AoA == angles (nn) c l = c l s (nn ) ; cd = cds (nn ) ; stopLookup = 1 ; e l s e i f AoA < angles (nn+1) && AoA > angles (nn) c l = c l s (nn+1) ( angles (nn+1) AoA ) . . . ( c l s (nn+1) c l s (nn ) ) / ( angles (nn+1) angles (nn ) ) ; cd = cds (nn+1) ( angles (nn+1) AoA ) . . . ( cds (nn+1) cds (nn ) ) / ( angles (nn+1) angles (nn ) ) ; stopLookup = 1 ; e l s e i f AoA > angles ( length ( angles ) ) c l = c l s ( length ( angles ) ) ( angles ( length ( angles ) ) . . . AoA) ( c l s ( length ( angles )) c l s ( length ( angles ) 1 ) ) / . . . ( angles ( length ( angles )) angles ( length ( angles ) 1)); cd = cds ( length ( angles ) ) ( angles ( length ( angles ) ) . . . AoA) ( cds ( length ( angles )) cds ( length ( angles ) 1 ) ) / . . . ( angles ( length ( angles )) angles ( length ( angles ) 1)); stopLookup = 1 ; e l s e i f AoA < angles (1) c l = c l s (2) ( angles (2) AoA) ( c l s (2) c l s ( 1 ) ) / . . . ( angles (2) angles ( 1 ) ) ; cd = cds (2) ( angles (2) AoA) ( cds (2) cds ( 1 ) ) / . . . ( angles (2) angles ( 1 ) ) ; stopLookup = 1 ; end nn=nn+1; end %End Table Lookup 118 Appendix E Optimized Airfoil Data 119 Table E.1: Coe cients for Bernstein Polynomial for of Upper Surface of Optimized Airfoils Angle ( ) Aupper0 Aupper1 Aupper2 Aupper3 Aupper4 Aupper5 Aupper6 -5 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 -4 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 -3 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 -2 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 -1 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 0 0.1507 0.3571 0.2964 0.1991 0.3068 0.3724 0.1995 1 0.1484 0.3410 0.3030 0.2072 0.3134 0.3827 0.2060 2 0.1532 0.3209 0.2909 0.2036 0.3152 0.3787 0.1995 3 0.1505 0.3043 0.2892 0.2008 0.3207 0.3847 0.1994 4 0.1517 0.2783 0.2942 0.2085 0.3084 0.3876 0.1992 5 0.1579 0.2671 0.2583 0.1967 0.3297 0.3845 0.2032 6 0.1558 0.2443 0.2630 0.2046 0.3419 0.3804 0.2000 7 0.1790 0.2279 0.2994 0.2307 0.3354 0.2938 0.1671 8 0.1999 0.2495 0.3054 0.2365 0.3235 0.2938 0.1550 9 0.2234 0.2509 0.3607 0.2605 0.2886 0.2383 0.1525 10 0.2373 0.2707 0.3762 0.2573 0.2763 0.2354 0.1551 11 0.2564 0.2855 0.3963 0.2463 0.2657 0.2461 0.1572 12 0.2780 0.3070 0.4063 0.2324 0.2557 0.2499 0.1613 13 0.3016 0.3325 0.3953 0.2201 0.2497 0.2645 0.1683 14 0.3181 0.3450 0.3522 0.2162 0.2496 0.2739 0.1719 15 0.3181 0.3450 0.3522 0.2162 0.2496 0.2739 0.1719 120 Table E.2: Coe cients for Bernstein Polynomial for of Lower Surface of Optimized Airfoils Angle ( ) Alower0 Alower1 Alower2 Alower3 Alower4 Alower5 Alower6 -5 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2100 0.1865 -4 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2100 0.1865 -3 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2100 0.1865 -2 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2100 0.1865 -1 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2000 0.1865 0 -0.0524 0.0904 0.0602 0.2000 0.1988 0.2000 0.1776 1 -0.0509 0.0907 0.0599 0.1918 0.1945 0.2038 0.1829 2 -0.0515 0.0910 0.0599 0.2032 0.2011 0.2040 0.1852 3 -0.0531 0.0954 0.0606 0.2006 0.2030 0.2016 0.1781 4 -0.0518 0.0915 0.0598 0.2049 0.2019 0.2066 0.1927 5 -0.0520 0.0948 0.0609 0.1953 0.2099 0.2209 0.1873 6 -0.0522 0.0960 0.0607 0.1983 0.2078 0.2275 0.1911 7 -0.0487 0.1015 0.0650 0.2017 0.2073 0.2285 0.1777 8 -0.0479 0.1022 0.0671 0.1943 0.2065 0.2177 0.1715 9 -0.0485 0.1033 0.0656 0.1978 0.2010 0.2085 0.1744 10 -0.0481 0.1042 0.0655 0.1966 0.1999 0.2046 0.1655 11 -0.0482 0.0921 0.0655 0.2058 0.1961 0.1996 0.1624 12 -0.0490 0.0916 0.0647 0.1889 0.1943 0.2012 0.1578 13 -0.0501 0.0876 0.0611 0.1796 0.1943 0.1935 0.1579 14 -0.0505 0.0887 0.0615 0.1782 0.1994 0.1836 0.1644 15 -0.0505 0.0887 0.0615 0.1782 0.2069 0.1859 0.1644 121 Table E.3: Lift Coe cients, and Drag to Lift Ratios for Optimized Airfoils Angles ( ) Cl ClC d-5 0.3255 54.2500 -4 0.4299 58.0946 -3 0.5343 60.7159 -2 0.6386 62.6078 -1 0.7430 63.2821 0 0.8404 64.3985 1 0.9425 77.1324 2 1.0599 85.3908 3 1.1620 88.6870 4 1.2625 91.0250 5 1.3636 94.1172 6 1.4540 93.2293 7 1.4929 88.6897 8 1.4575 81.8492 9 1.6282 79.0274 10 1.6905 73.6789 11 1.7652 67.2100 12 1.8412 59.9403 13 1.9087 50.8843 14 1.9264 41.1357 15 1.9274 31.0871 122 Appendix F Optimized Cruise Propeller Table F.1: Propeller Properties for Cruise Case 1 (Objective Function = Cp) Element: Position (in) Chord (in) ( ) Pitch 1 0.720 0.522 40.946 3.93 2 1.307 0.565 40.737 7.07 3 1.893 0.505 28.958 6.58 4 2.480 0.589 27.518 8.12 5 3.067 0.544 26.356 9.55 6 3.653 0.474 22.756 9.63 7 4.240 0.353 20.156 9.78 8 4.827 0.268 18.071 9.90 9 5.413 0.152 13.832 8.37 10 6.000 0.000 8.704 5.77 Pitch at 3/4 radius 9.83 J 0.8000 Diameter (in) 12.0 Ct 0.0244 RPM 5500 Cp 0.0218 Motor Power (Watts) 66.20 prop 89.54% Torque (oz-in) 12.87 Thrust (oz) 7.55 Motor Voltage (Volts) 7.78 Motor Current (Amps) 8.51 motor 71.70% system 62.20% 123 Table F.2: Propeller Properties for Cruise Case 2a (Objective Function = PropellerPower) Element: Position (in) Chord (in) ( ) Pitch 1 0.720 0.577 57.746 7.17 2 1.307 0.593 43.346 7.75 3 1.893 0.597 39.937 9.96 4 2.480 0.553 34.422 10.68 5 3.067 0.525 24.889 8.94 6 3.653 0.441 22.895 9.69 7 4.240 0.341 18.898 9.12 8 4.827 0.262 14.973 8.11 9 5.413 0.143 11.845 7.13 10 6.000 0.000 1.526 1.00 Pitch at 3/4 radius 8.67 J 0.8085 Diameter (in) 12.0 Ct 0.0248 RPM 5442 Cp 0.0220 Motor Power (Watts) 64.62 prop 91.14% Torque (oz-in) 12.72 Thrust (oz) 7.51 Motor Voltage (Volts) 7.69 Motor Current (Amps) 8.4036 motor 71.63% system 65.29% 124 Table F.3: Propeller Properties for Cruise Case 2a (Objective Function = PropellerPower) Element: Position (in) Chord (in) ( ) Pitch 1 0.698 0.527 54.353 6.11 2 1.266 0.550 43.204 7.47 3 1.834 0.612 35.056 8.09 4 2.971 0.551 33.987 10.18 5 3.539 0.530 25.063 8.73 6 4.108 0.451 24.838 10.29 7 4.108 0.353 21.876 10.36 8 4.676 0.280 18.775 9.99 9 5.245 0.128 12.768 7.47 10 5.813 0.000 4.803 3.07 Pitch at 3/4 radius 10.20 J 0.8359 Diameter (in) 11.6 Ct 0.0283 RPM 5433 Cp 0.0261 Motor Power (Watts) 65.35 prop 90.64% Torque (oz-in) 12.84 Thrust (oz) 7.52 Motor Voltage (Volts) 7.70 Motor Current (Amps) 8.49 motor 71.53% system 64.83% 125 Appendix G Optimized Climb Propeller Table G.1: Propeller Properties for Climb Case 3 (Objective Function = 1Ct) Element: Position (in) Chord (in) ( ) Pitch 1 0.487 1.409 58.049 4.908 2 0.884 1.463 45.207 5.595 3 1.281 1.578 31.079 4.852 4 1.678 1.517 28.190 5.651 5 2.075 1.534 20.501 4.875 6 2.472 1.308 20.296 5.744 7 2.869 1.056 15.991 5.166 8 3.266 0.805 14.601 5.345 9 3.663 0.474 12.378 5.051 10 4.060 0.000 4.885 2.180 3=4 ( ) 15.38 Pitch at 3/4 radius 5.25 J 0.2452 Diameter (in) 8.12 Ct 0.2072 RPM 7956 Cp 0.0982 Motor Power (Watts) 190.71 prop 51.74% Torque (oz-in) 17.21 Thrust (oz) 28.10 Motor Voltage (Volts) 11.1 Motor Current (Amps) 17.18 motor 74.62% system 38.61% 126 Table G.2: Propeller Properties for Climb Case 4 (Objective Function = 1Ft) Element: Position (in) Chord (in) ( ) Pitch 1 0.584 1.420 42.671 3.381 2 1.059 1.376 33.071 4.334 3 1.535 1.374 23.345 4.162 4 2.010 1.395 18.384 4.198 5 2.486 1.337 12.619 3.497 6 2.962 1.270 10.490 3.446 7 3.437 1.139 9.783 3.724 8 3.913 0.853 6.348 2.735 9 4.388 0.420 3.700 1.783 10 4.864 0.000 1.465 0.781 3=4 ( ) 8.26 Pitch at 3/4 radius 3.29 J 0.2067 Diameter (in) 9.73 Ct 0.1054 RPM 7876 Cp 0.0.0415 Motor Power (Watts) 131.10 prop 52.51% Torque (oz-in) 17.60 Thrust (oz) 28.87 Motor Voltage (Volts) 11.1 Motor Current (Amps) 11.81 motor 74.05% system 38.88% 127 Appendix H Optimized Climb-Cruise Propeller 128 Table H.1: Propeller Properties for Climb Cruise Case 5 (Objective Function = 1 prop motor) Element: Position (in) Chord (in) ( ) Pitch 1 0.639 0.994 53.738 5.476 2 1.160 1.023 38.633 5.827 3 1.681 1.084 36.985 7.956 4 2.202 1.103 30.661 8.203 5 2.723 1.096 26.319 8.464 6 3.244 0.994 21.798 8.152 7 3.765 0.908 20.314 8.758 8 4.286 0.694 18.726 9.129 9 4.807 0.357 17.575 9.567 10 5.328 0.000 16.416 9.863 Cruise Case 5 3=4 ( ) 19.61 Pitch at 3/4 radius 8.92 J 0.9009 Diameter (in) 10.6559 Ct 0.0457 RPM 5500 Cp 0.0470 Motor Power (Watts) 82.20 prop 87.60% Torque (oz-in) 15.33 Thrust (oz) 8.79 Motor Voltage (Volts) 8.11 Motor Current (Amps) 10.14 motor 69.82% system 61.16% Climb Case 5 J 0.2684 Ct 0.1413 RPM 5538 Cp 0.0739 Motor Power (Watts) 151.45 prop 51.32% Torque (oz-in) 24.43 Thrust (oz) 27.54 Motor Voltage (Volts) 9.37 Motor Current (Amps) 16.16 motor 62.66% system 32.16% 129 Table H.2: Propeller Properties for Climb Cruise Case 6 (Objective Function = PowerCruiseThrust Climb ) Element: Position (in) Chord (in) ( ) Pitch 1 0.627 0.850 46.855 4.206 2 1.139 0.896 45.400 7.255 3 1.650 0.976 36.412 7.646 4 2.161 1.049 33.671 9.046 5 2.672 1.027 26.021 8.197 6 3.183 0.906 26.021 9.765 7 3.695 0.794 22.309 9.525 8 4.206 0.645 16.611 7.884 9 4.717 0.415 15.770 8.370 10 5.228 0.000 13.792 8.064 Cruise Case 6 3=4 ( ) 19.78 Pitch at 3/4 radius 8.80 J 0.9345 Diameter (in) 10.4566 Ct 0.0441 RPM 5404 Cp 0.0469 Motor Power (Watts) 68.78 prop 87.87% Torque (oz-in) 13.43 Thrust (oz) 7.59 Motor Voltage (Volts) 7.74 Motor Current (Amps) 8.89 motor 70.94% system 62.34% Climb Case 6 J 0.2230 Ct 0.1244 RPM 6792 Cp 0.0578 Motor Power (Watts) 190.41 prop 48.00% Torque (oz-in) 26.15 Thrust (oz) 33.82 Motor Voltage (Volts) 11.00 Motor Current (Amps) 17.31 motor 65.74% system 31.55% 130