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
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AoAuni0020
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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
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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
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[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