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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
- Veröffentlichungsjahr: 2016
The pilot's guide to aeronautics and the complex forces of flight Flight Theory and Aerodynamics is the essential pilot's guide to the physics of flight, designed specifically for those with limited engineering experience. From the basics of forces and vectors to craft-specific applications, this book explains the mechanics behind the pilot's everyday operational tasks. The discussion focuses on the concepts themselves, using only enough algebra and trigonometry to illustrate key concepts without getting bogged down in complex calculations, and then delves into the specific applications for jets, propeller crafts, and helicopters. This updated third edition includes new chapters on Flight Environment, Aircraft Structures, and UAS-UAV Flight Theory, with updated craft examples, component photos, and diagrams throughout. FAA-aligned questions and regulatory references help reinforce important concepts, and additional worked problems provide clarification on complex topics. Modern flight control systems are becoming more complex and more varied between aircrafts, making it essential for pilots to understand the aerodynamics of flight before they ever step into a cockpit. This book provides clear explanations and flight-specific examples of the physics every pilot must know. * Review the basic physics of flight * Understand the applications to specific types of aircraft * Learn why takeoff and landing entail special considerations * Examine the force concepts behind stability and control As a pilot, your job is to balance the effects of design, weight, load factors, and gravity during flight maneuvers, stalls, high- or low-speed flight, takeoff and landing, and more. As aircraft grow more complex and the controls become more involved, an intuitive grasp of the physics of flight is your most valuable tool for operational safety. Flight Theory and Aerodynamics is the essential resource every pilot needs for a clear understanding of the forces they control.
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Table of Contents
Cover
Title Page
Copyright
Preface
About the Authors
Chapter 1: Introduction
The Flight Environment
Basic Quantities
Forces
Mass
Scalar and Vector Quantities
Moments
Equilibrium Conditions
Newton's Laws of Motion
Linear Motion
Rotational Motion
Work
Energy
Power
Friction
Symbols
Equations
Problems
Chapter 2: Atmosphere, Altitude, and Airspeed Measurement
Properties of the Atmosphere
ICAO Standard Atmosphere
Altitude Measurement
Continuity Equation
Bernoulli's Equation
Airspeed Measurement
Symbols/Abbreviations
Equations
Problems
Chapter 3: Structures, Airfoils, and Aerodynamic Forces
Aircraft Structures
Airfoils
Development of Forces on Airfoils
Aerodynamic Force
Aerodynamic Pitching Moments
Aerodynamic Center
Symbols
Problems
Chapter 4: Lift
Introduction to Lift
Angle of Attack Indicator
Boundary Layer Theory
Reynolds Number
Adverse Pressure Gradient
Airflow Separation
Stall
Aerodynamic Force Equations
Lift Equation
Airfoil Lift Characteristics
High Coefficient of Lift Devices
Lift During Flight Manuevers
Symbols
Equations
Problems
Chapter 5: Drag
Drag Equation
Induced Drag
Ground Effect
Laminar Flow Airfoils
Parasite Drag
Total Drag
Lift to Drag Ratio
Drag Reduction
Symbols
Equations
Problems
Chapter 6: Jet Aircraft Basic Performance
Thrust-Producing Aircraft
Principles of Propulsion
Thrust-Available Turbojet Aircraft
Specific Fuel Consumption
Fuel Flow
Thrust-Available-Thrust-Required Curves
Items of Aircraft Performance
Symbols
Equations
Problems
Chapter 7: Jet Aircraft Applied Performance
Variations in the Thrust-Required Curve
Variations of Aircraft Performance
Equations
Problems
Chapter 8: Propeller Aircraft: Basic Performance
Power Available
Principles of Propulsion
Power-Required Curves
Items of Aircraft Performance
Symbols
Equations
Problems
Chapter 9: Propeller Aircraft: Applied Performance
Variations in the Power-Required Curve
Variations in Aircraft Performance
Equations
Problems
Chapter 10: Takeoff Performance
Definitions Important to Takeoff Planning
Aborted Takeoffs
Linear Motion
Factors Affecting Takeoff Performance
Improper Liftoff
Symbols
Equations
Problems
Chapter 11: Landing Performance
Prelanding Performance
Improper Landing Performance
Landing Deceleration, Velocity, and Distance
Landing Equations
Hazards of Hydroplaning
Symbols
Equations
Problems
Chapter 12: Slow-Speed Flight
Stalls
Region of Reversed Command
Spins
Low-Level Wind Shear
Aircraft Performance in Low-Level Wind Shear
Effect of Ice and Frost
Wake Turbulence
Problems
Chapter 13: Maneuvering Performance
General Turning Performance
Equations
Problems
Chapter 14: Longitudinal Stability and Control
Definitions
Oscillatory Motion
Airplane Reference Axes
Static Longitudinal Stability
Dynamic Longitudinal Stability
Pitching Tendencies in a Stall
Longitudinal Control
Symbols
Equations
Problems
Chapter 15: Directional and Lateral Stability and Control
Directional Stability and Control
Static Directional Stability
Directional Control
Multi-Engine Flight Principles
Lateral Stability and Control
Static Lateral Stability
Lateral Control
Dynamic Directional and Lateral Coupled Effects
Symbols
Equations
Problems
Chapter 16: High-Speed Flight
The Speed of Sound
High-Subsonic Flight
Design Features for High-Subsonic Flight
Transonic Flight
Supersonic Flight
Symbols
Equations
Problems
Chapter 17: Rotary-Wing Flight Theory
Momentum Theory of Lift
Airfoil Selection
Forces on Rotor System
Thrust Development
Hovering Flight
Ground Effect
Rotor Systems
Dissymmetry of Lift in Forward Flight
High Forward Speed Problems
Helicopter Control
Helicopter Power-Required Curves
Power Settling, Settling with Power, and Vortex Ring State
Autorotation
Dynamic Rollover
Problems
Answers to Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
References
Government Publications
Periodicals
Personal Interview
Index
Wiley End User License Agreement
List of Illustrations
Chapter 1: Introduction
Figure 1.1: Forces on an airplane in steady flight.
Figure 1.2: Resolved forces on an airplane in steady flight.
Figure 1.3: Vector of an eastbound aircraft.
Figure 1.4: Vector of a north wind.
Figure 1.5: Vector addition.
Figure 1.6: Vector of an aircraft in a climb.
Figure 1.7: Vectors of groundspeed and rate of climb.
Figure 1.8: Seesaw in equilibrium.
Figure 1.9: Coefficients of friction for airplane tires on a runway.
Chapter 2: Atmosphere, Altitude, and Airspeed Measurement
Figure 2.1: Standard pressure.
Figure 2.2: Field elevation versus pressure altitude.
Figure 2.3: Density altitude chart.
Figure 2.4: Flow of air through a pipe.
Figure 2.5: Pressure change in a Venturi tube.
Figure 2.6: Velocities and pressures on an airfoil superimposed on a Venturi tube.
Figure 2.7: Flow around a symmetrical object.
Figure 2.8: Schematic of a pitot–static airspeed indicator.
Figure 2.9: Air data computer and pitot‐static sensing.
Figure 2.10: Compressibility correction chart.
Figure 2.11: Altitude and EAS to TAS correction chart.
Chapter 3: Structures, Airfoils, and Aerodynamic Forces
Figure 3.1: Modern transport category control surfaces.
Figure 3.2: Differential ailerons.
Figure 3.3: Frise‐type ailerons.
Figure 3.4: Elevator movement.
Figure 3.5: Adjustable horizontal stabilizer.
Figure 3.6: Rudder movement.
Figure 3.7: Common flap designs.
Figure 3.8: Trim tabs.
Figure 3.9: Antiservo tab.
Figure 3.10: Airfoil section.
Figure 3.11: Airfoil terminology.
Figure 3.12: Examples of airfoil design.
Figure 3.13: NACA airfoils (NACA data).
Figure 3.14: Effect of pressure disturbances on airflow around an airfoil.
Figure 3.15: Velocity changes around an airfoil.
Figure 3.16: Static pressure on an airfoil (a) at zero AOA, and (b) at a positive AOA.
Figure 3.17: Components of aerodynamic force.
Figure 3.18: Pressure forces on (a) nonrotating cylinder and (b) rotating cylinder.
Figure 3.19: Pitching moments on a symmetrical airfoil (a) at zero AOA and (b) at positive AOA.
Figure 3.20: Pitching moments on a cambered airfoil: (a) zero lift, (b) developing lift.
Chapter 4: Lift
Figure 4.1: Critical angle of attack, stall, and angle of attack indications.
Figure 4.2: Boundary layer composition.
Figure 4.3: Smoke pattern.
Figure 4.4: Boundary layer velocity profile.
Figure 4.5: Laminar and turbulent velocity profiles.
Figure 4.6: Reynolds number effect on airflow on a smooth flat plate.
Figure 4.7: Adverse pressure gradient.
Figure 4.8: Airflow separation velocity profiles.
Figure 4.9: Sphere wake drag: (a) smooth sphere, (b) rough sphere.
Figure 4.10:
C
L
vs. AOA for a symmetrical airfoil.
Figure 4.11: C
L
vs. AOA for a cambered airfoil.
Figure 4.12: Thickness effect.
Figure 4.13: Camber effect.
Figure 4.14: High‐C
L
devices.
Figure 4.15: Common leading edge high‐C
L
devices.
Figure 4.16: Effect of a camber changer on the
curve.
Figure 4.17: Vortex generators.
Figure 4.18: Fixed slot at (a) low AOA and (b) high AOA.
Figure 4.19: Effect of an energy adder on the
curve.
Figure 4.20: Forces in a banked turn.
Figure 4.21: Coordinated versus uncoordinated level turns.
Figure 4.22: Lift during climb entry.
Chapter 5: Drag
Figure 5.1:
C
L
vs. AOA and
C
D
vs. AOA.
Figure 5.2: Wing planform terminology.
Figure 5.3: Wingtip vortices.
Figure 5.4: Airflow about an infinite wing.
Figure 5.5: Vertical velocity vectors of an infinite wing.
Figure 5.6: Vertical velocity vectors of a finite wing.
Figure 5.7: Airflow about a finite wing.
Figure 5.8: Relative wind and force vectors on a finite wing.
Figure 5.9: Induced drag versus velocity.
Figure 5.10: Airflow in ground effect.
Figure 5.11: T
r
and C
L
curves in ground effect.
Figure 5.12: Ground effect.
Figure 5.13: Comparison of drag characteristics of conventional and laminar flow airfoils.
Figure 5.14: Form drag.
Figure 5.15: Interference drag at the wing root.
Figure 5.16: Example total drag curve for a helicopter.
Figure 5.17: Parasite drag–airspeed curve.
Figure 5.18: Drag vector diagram.
Figure 5.19: Total drag curve.
Figure 5.20: Typical lift‐to‐drag ratios.
Figure 5.21: Wingtip vortex reduction methods.
Figure 5.22: Winglets.
Chapter 6: Jet Aircraft Basic Performance
Figure 6.1: Aircraft in equilibrium flight.
Figure 6.2: Turbojet engine.
Figure 6.3: Turbofan engine.
Figure 6.4: T‐38 drag curve.
Figure 6.5: T‐38 thrust required.
Figure 6.6: Engine thrust schematic.
Figure 6.7: Propulsion efficiency.
Figure 6.8: T‐38 installed thrust.
Figure 6.9: T‐38 thrust variation with altitude.
Figure 6.10: T‐38
c
t
–rpm.
Figure 6.11: T‐38
c
t
–altitude.
Figure 6.12: T‐38 fuel flow–altitude.
Figure 6.13: T‐38 thrust available–thrust required.
Figure 6.14: Forces acting on a climbing aircraft.
Figure 6.15: Velocity for maximum climb angle.
Figure 6.16: Wind effect on climb angle to the ground.
Figure 6.17: Obstacle clearance for jet takeoff.
Figure 6.18: Climb angle and rate of climb.
Figure 6.19: Rate of climb velocity vector.
Figure 6.20: Velocity for maximum rate of climb.
Figure 6.21: Finding maximum endurance velocity.
Figure 6.22: Finding the maximum specific range velocity.
Figure 6.23: Wind effect on specific range.
Figure 6.24: Total range calculation.
Chapter 7: Jet Aircraft Applied Performance
Figure 7.1: Effect of weight change on induced drag.
Figure 7.2: Effect of weight change on the
curve.
Figure 7.3: Effect of configuration on parasite drag.
Figure 7.4: Effect of configuration on the
curve.
Figure 7.5: Effect of altitude on
and
, curves.
Figure 7.6: Effect of weight change on specific range.
Figure 7.7: T‐38 effect of altitude on specific range.
Figure 7.8: Range improvement using cruise–climb.
Chapter 8: Propeller Aircraft: Basic Performance
Figure 8.1: Airfoil sections of a propeller blade.
Figure 8.2: Propeller blade angle.
Figure 8.3: Blade angle in flight.
Figure 8.4: Various blade angle ranges.
Figure 8.5: Thrust from a propeller.
Figure 8.6: Propeller pitch angle configurations.
Figure 8.7: (a) Thrust‐required and (b) power‐required curves.
Figure 8.8: Power required.
Figure 8.9: Power required and power available.
Figure 8.10: Fixed shaft turboprop engine.
Figure 8.11: Turbocharged engine.
Figure 8.12: Forces on a climbing aircraft.
Figure 8.13: Thrust versus climb angle.
Figure 8.14: Climb angle versus velocity.
Figure 8.15: Rate of climb velocity vector.
Figure 8.16: Power versus climb rate.
Figure 8.17: Finding the maximum rate of climb.
Figure 8.18: Finding the maximum endurance and range.
Figure 8.19: Effect of wind on range.
Chapter 9: Propeller Aircraft: Applied Performance
Figure 9.1: Effect of weight change on a
curve.
Figure 9.2: Effect of configuration on the
curve.
Figure 9.3: Effect of altitude on a
curve.
Figure 9.4:
versus
with altitude.
Figure 9.5: Effect of weight change on specific range.
Figure 9.6: Effect of altitude on specific range.
Chapter 10: Takeoff Performance
Figure 10.1: Takeoff distance graph.
Figure 10.2: Accelerate‐stop distance, accelerate‐go distance, and climb gradient.
Figure 10.3: Balanced field length.
Figure 10.4: Segmented one‐engine climb graph.
Figure 10.5: Single‐engine velocity–distance profiles.
Figure 10.6: Multi‐engine velocity–distance profiles.
Figure 10.7: Forces on an airplane during takeoff.
Figure 10.8: Effect of wind on takeoff.
Figure 10.9: U.S. Chart Supplement information.
Figure 10.10: Crosswind takeoff.
Figure 10.11: Premature takeoff.
Chapter 11: Landing Performance
Figure 11.1: Landing distance graph.
Figure 11.2: Forces in equilibrium.
Figure 11.3: Forces acting in a power‐off glide.
Figure 11.4: Glide ratio vector diagram.
Figure 11.5: FAR landing field length required.
Figure 11.6: Approach glide paths.
Figure 11.7: Lift from (a) propellers and (b) turbojets.
Figure 11.8: Effect of flaps on final approach.
Figure 11.9: High roundout.
Figure 11.10: Porpoising during roundout.
Figure 11.11: Improper drift correction.
Figure 11.12: Sideslip during crosswind.
Figure 11.13: Forces acting on an airplane during landing.
Figure 11.14: Aerodynamic braking and wheel braking.
Figure 11.15: Normal and friction forces.
Figure 11.16: Coefficient of friction versus wheel slippage.
Figure 11.17: Effect of runway condition on coefficient of friction.
Figure 11.18: Effect of wind on landing.
Figure 11.19: Effect of headwind during landing approach.
Figure 11.20: Forces on tire: (a) static condition, (b) rolling tire.
Figure 11.21: Hydroplaning forces on tire: (a) low speed, (b) medium speed, (c) high speed.
Chapter 12: Slow‐Speed Flight
Figure 12.1: Effect of sweepback on
curves.
Figure 12.2: Spanwise lift distribution.
Figure 12.3: Wing spanwise lift distribution.
Figure 12.4: Stall patterns.
Figure 12.5: Power‐off stall and recovery.
Figure 12.6: Secondary stall due to improper stall recovery.
Figure 12.7: Regions of normal and reversed command.
Figure 12.8: Constant airspeed climb. Stick or throttle?
Figure 12.9: Spin entry and recovery.
Figure 12.10: Stall hitting the horizontal tail.
Figure 12.11: Swept wings stall at tips first.
Figure 12.12: Aerodynamics of spin for straight‐wing aircraft.
Figure 12.13: Aerodynamics of spin for swept‐wing aircraft.
Figure 12.14: Wind shear caused by a downdraft.
Figure 12.15: “Bursts” caused by a thunderstorm.
Figure 12.16: Thunderstorm gust front.
Figure 12.17: Temperature inversion LLWS.
Figure 12.18: Tailwind wind shear encountered on takeoff: (a) flight path, (b) forces and moments.
Figure 12.19: Tailwind shear encountered in landing approach: (a) flight path, (b) forces and moments.
Figure 12.20: Effect of ice and frost on wings.
Figure 12.21: Wingtip vortices behind an aircraft.
Figure 12.22: Helicopter vortices.
Figure 12.23: Wake turbulence avoidance.
Figure 12.24: Lateral movement of tip vortices with (a) no wind and (b) crosswind.
Chapter 13: Maneuvering Performance
Figure 13.1: Increasing bank angle and related forces.
Figure 13.2: Forces on an aircraft in a coordinated level turn.
Figure 13.3: Vector diagram of forces on an aircraft in a turn.
Figure 13.4: Load factors at various bank angles.
Figure 13.5: Forces on an aircraft during a 90° roll.
Figure 13.6: Load factor and stall speed.
Figure 13.7: First‐stage construction of a
V–G
diagram.
Figure 13.8: Second‐stage construction of
V–G
diagram.
Figure 13.9: Antisymmetrical loading.
Figure 13.10: Maneuver (corner) speed.
Figure 13.11: Ultimate load factors.
Figure 13.12: Stall speed and turn radius with varying angle of bank.
Figure 13.13: Rate and radius of a turn.
Figure 13.14: Constant altitude turn performance.
Figure 13.15: Forces on the complete aircraft.
Figure 13.16: Thrust‐limited turn radius.
Figure 13.17: Perfect and normal loop.
Figure 13.18: Centrifugal forces in a vertical loop.
Figure 13.19: Loading on an example aircraft.
Chapter 14: Longitudinal Stability and Control
Figure 14.1: Types of static stability.
Figure 14.2: Dynamic stability.
Figure 14.3: Positive static and negative dynamic stability.
Figure 14.4: Positive static and neutral dynamic stability.
Figure 14.5: Positive static and positive dynamic stability.
Figure 14.6: Airplane reference axes.
Figure 14.7: Establishing positive moment direction.
Figure 14.8: Airplane axes and moment directions.
Figure 14.9: Movement of the longitudinal axis in pitch.
Figure 14.10: Positive static longitudinal stability.
Figure 14.11: Types of static longitudinal stability.
Figure 14.12: Degrees of positive static stability.
Figure 14.13: Aircraft static longitudinal stability.
Figure 14.14: Effect of CG and AC location on static longitudinal stability.
Figure 14.15: Static longitudinally stable flying wing in equilibrium.
Figure 14.16: Airplane with static longitudinal stability.
Figure 14.17: Pressure distribution about a body of revolution.
Figure 14.18: Thrust line and longitudinal stability.
Figure 14.19: Power changes and longitudinal stability.
Figure 14.20: Engine nacelle location contribution to pitch stability.
Figure 14.21: Lift of horizontal stabilizer produces a stabilizing moment.
Figure 14.22: Effect of speed on tail‐down force.
Figure 14.23: Typical buildup of aircraft components.
Figure 14.24: Effect of CG location on static longitudinal stability.
Figure 14.25: Stick‐free–stick‐fixed stability.
Figure 14.26: Phugoid longitudinal dynamic mode.
Figure 14.27: Short‐period dynamic mode.
Figure 14.28: Forces on a pitching plane.
Figure 14.29: Wing wake influences on a low‐tail aircraft.
Figure 14.30: Wing wake influences on a swept‐wing T‐tail aircraft.
Figure 14.31: Change in pressure distribution at stall.
Figure 14.32: Swept‐wing stall characteristics.
Figure 14.33: Forces producing moments during takeoff.
Chapter 15: Directional and Lateral Stability and Control
Figure 15.1: (a) Negative yawing moment, (b) positive yawing moment.
Figure 15.2: (a) Unstable, (b) stable in yaw.
Figure 15.3: Static directional stability.
Figure 15.4: Static directional stability at high sideslip angles.
Figure 15.5: Effect of wing sweepback on directional stability.
Figure 15.6: Directional instability of fuselage.
Figure 15.7: Vertical tail is stabilizing in yaw.
Figure 15.8: Dorsal fin decreases drag and increases stability.
Figure 15.9: Typical buildup of component effects on static directional stability.
Figure 15.10: Rudder‐fixed rudder‐free yaw stability.
Figure 15.11: Loss of directional stability at high AOA.
Figure 15.12: Asymmetrical loading.
Figure 15.13: Slipstream rotation causes yaw.
Figure 15.14: Yawing moment due to asymmetrical thrust.
Figure 15.15: Yawing moment due to critical engine.
Figure 15.16: Forward slip.
Figure 15.17: Propeller drag contribution.
Figure 15.18: Effect of rearward CG on yaw.
Figure 15.19: Relationship of V
MC
to V
S
.
Figure 15.20: Rolling moment caused by sideslip.
Figure 15.21: (a) Stable, (b) neutral, and (c) unstable static lateral stability.
Figure 15.22: Static lateral stability.
Figure 15.23: Dihedral angle.
Figure 15.24: Dihedral producing static lateral stability.
Figure 15.25: Dihedral effect of sweepback.
Figure 15.26: Vertical tail effect on lateral stability.
Figure 15.27: Adverse yaw.
Figure 15.28: High AOA: (a) upgoing wing, (b) downgoing wing.
Figure 15.29: Coupled ailerons and rudder.
Figure 15.30: Flight paths due to coupled dynamic effects: (a) spiral divergence, (b) directional divergence, (c) Dutch roll.
Chapter 16: High‐Speed Flight
Figure 16.1: (a) Subsonic flow, (b) supersonic flow.
Figure 16.2: Airflow over a wing section.
Figure 16.3: Comparison of supercritical and laminar flow airfoils at Mach 0.75.
Figure 16.4: Effect of wing sweep on a
curve.
Figure 16.5: Vortex generators.
Figure 16.6: Force divergence effect on
C
D
.
Figure 16.7: Force divergence effect on
C
L
.
Figure 16.8: Normal shock wave on bottom of wing.
Figure 16.9: Aerodynamic center location shift.
Figure 16.10: Stick forces versus Mach number.
Figure 16.11: Normal shock waves move to trailing edge.
Figure 16.12: Unattached bow wave at transonic speed.
Figure 16.13: Formation of an oblique shock wave.
Figure 16.14: Formation of an expansion wave.
Figure 16.15: Summary of supersonic wave characteristics.
Figure 16.16: Double‐wedge airfoil in supersonic airflow: (a) wave pattern, (b) pressure distribution.
Figure 16.17: Double‐wedge airfoil developing lift: (a) wave pattern, (b) pressure distribution.
Figure 16.18: Circular arc airfoil in supersonic flow: (a) wave pattern, (b) pressure distribution.
Figure 16.19: Effect of wing sweep on
C
D
.
Figure 16.20: Mach cone.
Figure 16.21: Swept wing in supersonic flight.
Figure 16.22: Subsonic control surface.
Figure 16.23: Supersonic control surface.
Figure 16.24: Normal shock engine inlet.
Figure 16.25: “Spike” oblique shock engine inlets.
Figure 16.26: Stagnation temperatures.
Figure 16.27: Effect of temperature on tensile strength of metals after half‐hour exposure.
Chapter 17: Rotary‐Wing Flight Theory
Figure 17.1: Momentum theory airflow: (a) schematic, (b) pressure and velocity distribution.
Figure 17.2: NACA 0012 airfoil.
Figure 17.3: Location of critical forces on an airfoil.
Figure 17.4: Centrifugal force straightens rotor blade.
Figure 17.5: Lift force and centrifugal force.
Figure 17.6: Resultant of lift and centrifugal forces.
Figure 17.7: Forces acting on a lifting blade.
Figure 17.8: Entire lifting rotor system.
Figure 17.9: Hovering helicopter at light weight.
Figure 17.10: Hovering helicopter at heavy weight.
Figure 17.11: Forward flight forces.
Figure 17.12: Lift component of 10,000‐lb total thrust at 15°.
Figure 17.13: Rotor velocity distribution in hover.
Figure 17.14: Lift distribution on a twisted/untwisted blade.
Figure 17.15: Hovering out of ground effect.
Figure 17.16: Hovering in ground effect.
Figure 17.17: Anti‐torque rotor.
Figure 17.18: Correction for anti‐torque rotor drift.
Figure 17.19: Rigid rotor system.
Figure 17.20: Semi‐rigid rotor system.
Figure 17.21: Articulated rotor system.
Figure 17.22: Rotor tip velocities in a hover.
Figure 17.23: Blade‐tip velocity in forward flight.
Figure 17.24: Rigid rotor rolling moment in forward flight.
Figure 17.25: Angle of attack and flight path changes: (a) advancing blade, (b) retreating blade.
Figure 17.26: CG radius change with flapping motion.
Figure 17.27: Hunting motion of a fully articulated blade.
Figure 17.28: AOA distribution during a retreating blade stall.
Figure 17.29: Gyroscopic precession.
Figure 17.30: Swash plate schematic.
Figure 17.31: Rotor flapping caused by cyclic stick movement.
Figure 17.32: Tail rotor dissymmetry of lift.
Figure 17.33: Helicopter power available and power required.
Figure 17.34: Running/rolling takeoff.
Figure 17.35: Vortex ring state.
Figure 17.36: Height/velocity diagram.
Figure 17.37: Induced flow velocity in a hover.
Figure 17.38: Airflow and force vectors in forward flight.
Figure 17.39: Airflow and forces in steady‐state descent.
Figure 17.40: Airflow and forces during autorotative flare.
List of Tables
Chapter 2: Atmosphere, Altitude, and Airspeed Measurement
Table 2.1: Standard Atmosphere Table
Chapter 13: Maneuvering Performance
Table 13.1: Load Factors at Various Bank Angles
Guide
Cover
Table of Contents
Begin Reading
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Advanced Engineering Thermodynamics
Fourth Edition
Adrian BejanJ. A. Jones Distinguished ProfessorDuke UniversityDurham, North Carolina
This book is printed on acid-free paper.
Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Dole, Charles E. (Charles Edward), 1916— author. | Lewis, James E., 1946— author. | Badick, Joseph R. (Joseph Robert), 1952— author. | Johnson, Brian A. (Brian Andrew), 1975— author.
Title: Flight theory and aerodynamics : a practical guide for operational safety / Charles E. Dole, James E. Lewis, Joseph R. Badick, Brian A. Johnson.
Description: Third edition. | Hoboken, New Jersey : John Wiley & Sons Inc., [2017] | Includes bibliographical references and index.
Identifiers: LCCN 2016025499 | ISBN 9781119233404 (hardback) | ISBN 9781119233411 (epub) | ISBN 9781119233428 (epdf)
Subjects: LCSH: Aerodynamics—Handbooks, manuals, etc. | Airplanes—Piloting—Handbooks, manuals, etc. | Aeronautics—Safety measures—Handbooks, manuals, etc. | BISAC: TECHNOLOGY & ENGINEERING / Mechanical.
Classification: LCC TL570 .D56 2017 | DDC 629.132—dc23 LC record available at https://lccn.loc.gov/2016025499
Cover Design: Wiley
Cover Images: Blue sky © Fabian Rothe/Getty Images, Inc.; Airplane frontview © Laksone/iStockphoto;
Isometric flying plane © aurin/iStockphoto
Preface
The third edition of Flight Theory and Aerodynamics was revised to further enhance the book's use as an introductory text for colleges and universities offering an aeronautical program. The publisher conducted a survey with aviation schools to determine what was needed in an updated text. The result is this third edition that meets not only classroom requirements but also practical application.
All seventeen chapters have some level of updating and additional content. The revision retains mathematical proofs, but also seeks to provide a non-mathematical discussion of aerodynamics geared toward a more practical application of flight theory. As such, it is a how to handbook as well as one about the theory of flying. It was written for all participants in the aviation industry: Pilots, aviation maintenance technicians, aircraft dispatchers, air traffic controllers, loadmasters, flight engineers, flight attendants, meteorologists, avionics technicians, aviation managers, as all have a vested interest in both safety and operational efficiency.
Updates in the third edition:
New sequence of chapters for better flow of topics
Extensive upgrade to the helicopter chapter, including discussion of other types of rotorcraft
Added modern graphics, including correlation with current FAA publications
Added detail in subject matter emphasizing practical application
Additional terms and abbreviations
The authors would like to thank our contacts at Wiley for their support throughout this revision as well as the support of our colleagues and families. In particular the authors would like to thank Steven. A. Saunders for his technical contribution to this revision, employing over 50 years of military, airline, and general aviation experience in the process. Finally, the authors would like to gratefully acknowledge the previous work of Charles E. Dole and James E. Lewis for their contribution to improving aviation safety throughout the aviation industry.
Joseph R. BadickBrian A. Johnson
About the Authors
A former marine, the late CHARLES E. DOLE taught flight safety for twenty-eight years to officers of the U.S. Air Force, Army, and Navy, as well as at the University of Southern California.
The late JAMES E. LEWIS was an associate professor of Aeronautical Science at Embry Riddle Aeronautical University in Florida, former aeronautical engineer for the Columbus Aircraft Division of Rockwell International, and retired Ohio National Guard military pilot.
JOSEPH R. BADICK has over forty years of flight experience in single, multi-engine, land/seaplane aircraft. Rated in commercial rotor-craft and gliders, with the highest rating of (A.T.P.) Airline Transport Pilot. A licensed airframe and powerplant mechanic, with inspection authorization (I.A.), he has installed numerous aircraft aerodynamic performance (S.T.C's) Supplemental Type Certificates, with test flight checks. He holds a Ph.D. (ABD) in Business from Northcentral University of Arizona and a Master's degree in Aeronautical Science. He was a Naval Officer for 30 years as an Aeronautical Engineer Duty Officer (AEDO), involved in all aspects of aircraft maintenance, logistics, acquisition, and test/evaluation. Currently he is a professor of aviation at a community college in the Career Pilot/Aviation Management degree programs.
BRIAN A. JOHNSON is a former airline and corporate pilot who holds a multi-engine Airline Transport Pilot certificate, in addition to Commercial pilot single-engine land/sea privileges. He is an active instrument and multi-engine Gold Seal flight instructor with an advanced ground instructor rating. He holds a Master's degree in Aeronautical Science from Embry-Riddle Aeronautical University and currently serves in a faculty position for a two-year Career Pilot/Aviation Management degree program, in addition to serving as an adjunct faculty member in the Aeronautical Science department of a major aeronautical university.
Chapter 1Introduction
A basic understanding of the physical laws of nature that affect aircraft in flight and on the ground is a prerequisite for the study of aerodynamics. Modern aircraft have become more sophisticated, and more automated, using advanced materials in their construction, requiring pilots to renew their understanding of the natural forces encountered during flight. Understanding how pilots control and counteract these forces better prepares pilots and engineers for the art of flying, and for harnessing the fundamental physical laws that guide them.
Perhaps your goal is to be a pilot, who will “slip the surly bonds of earth,” as John Gillespie Magee wrote in his classic poem “High Flight.” Or maybe you aspire to build or maintain aircraft as a skilled technician. Or possibly you wish to serve in another vital role in the aviation industry, such as manager, dispatcher, meteorologist, engineer, teacher, or another capacity. Whichever area you might be considering, this textbook will attempt to build on previous material you have learned, and hopefully will prepare you for a successful aviation career.
THE FLIGHT ENVIRONMENT
This chapter begins with a review of the basic principles of physics and concludes with a summary of linear motion, mechanical energy, and power. A working knowledge of these areas, and how they relate to basic aerodynamics, is vital as we move past the rudimentary “four forces of flight” and introduce thrust and power‐producing aircraft, lift and drag curves, stability and control, maneuvering performance, slow‐speed flight, and other topics.
Up to this point you have seen that there are four basic forces acting on an aircraft in flight: lift, weight, thrust, and drag. Now we must understand how these forces change as an aircraft accelerates down the runway, or descends on final approach to a runway and gently touches down even when traveling twice the speed of a car on the highway. Once an aircraft has safely made it into the air, what effect does weight have on its ability to climb, and should the aircraft climb up to the flight levels or stay lower and take “advantage” of the denser air closer to the ground?
By developing an understanding of the aerodynamics of flight, how design, weight, load factors, and gravity affect an aircraft during flight maneuvers from stalls to high speed flight, the pilot learns how to control the balance between these forces. This textbook will help clarify these issues, among others, hopefully leaving you with a better understanding of the flight environment.
BASIC QUANTITIES
An introduction to aerodynamics must begin with a review of physics, and in particular, the branch of physics that will be presented here is called mechanics. We will examine the fundamental physical laws governing the forces acting on an aircraft in flight, and what effect these natural laws and forces have on the performance characteristics of aircraft. To control an aircraft, whether it is an airplane, helicopter, glider, or balloon, the pilot must understand the principles involved and learn to use or counteract these natural forces.
We will start with the concepts of work, energy, power, and friction, and then build upon them as we move forward in future chapters.
Because the metric system of measurement has not yet been widely accepted in the United States, the English system of measurement is used in this book. The fundamental units are
Force
pounds (lb)
Distance
feet (ft)
Time
seconds (sec)
From the fundamental units, other quantities can be derived:
Velocity (distance/time)
ft/sec (fps)
Area (distance squared)
square ft (ft
2
)
Pressure (force/unit area)
lb/ft
2
(psf)
Acceleration (change in velocity)
ft/sec/sec (fps
2
)
Aircraft measure airspeed in knots (nautical miles per hour) or in Mach number (the ratio of true airspeed to the speed of sound). Rates of climb and descent are measured in feet per minute, so quantities other than those above are used in some cases. Some useful conversion factors are listed below:
Multiply
by
to get
knots
1.69
feet per second (fps)
fps
0.5925
knots
miles per hour (mph)
1.47
fps
fps
0.6818
mph
mph
0.8690
knots
knots
1.15
mph
nautical miles (nm)
6076
feet (ft)
nm
1.15
statute miles (sm)
sm
0.869
nm
knots
101.3
feet per minute (fpm)
FORCES
A force is a push or a pull tending to change the state of motion of a body. Typical forces acting on an aircraft in steady flight are shown in Fig. 1.1. Figure 1.2 shows the resolution of the aerodynamic forces during straight‐and‐level, unaccelerated flight and is separated into four components. The component that is 90° to the flight path and acts toward the top of the airplane is called lift. The component that is parallel to the flight path and acts toward the rear of the airplane is called drag; while the opposing forward force is thrust and is usually created by the engine. Weight opposes lift and as we will see is a function of the mass of the aircraft and gravity.
Figure 1.1 Forces on an airplane in steady flight.
Figure 1.2 Resolved forces on an airplane in steady flight.
U.S. Department of Transportation Federal Aviation Administration, Pilot's Handbook of Aeronautical Knowledge, 2008
MASS
Mass is a measure of the amount of material contained in a body. Weight, on the other hand, is a force caused by the gravitational attraction of the earth (), moon, sun, or other heavenly bodies. Weight will vary, depending on where the body is located in space. Mass will not vary with position.
Rearranging gives
This mass unit is called the slug.
SCALAR AND VECTOR QUANTITIES
A quantity that has size or magnitude only is called a scalar quantity. The quantities of mass, time, and temperature are examples of scalar quantities. A quantity that has both magnitude and direction is called a vector quantity. Forces, accelerations, and velocities are examples of vector quantities. Speed is a scalar, but if we consider the direction of the speed, then it is a vector quantity called velocity. If we say an aircraft traveled 100 nm, the distance is a scalar, but if we say an aircraft traveled 100 nm on a heading of 360°, the distance is a vector quantity.
Scalar Addition
Scalar quantities can be added (or subtracted) by simple arithmetic. For example, if you have 5 gallons of gas in your car's tank and you stop at a gas station and top off your tank with 9 gallons more, your tank now holds 14 gallons.
Vector Addition
Vector addition is more complicated than scalar addition. Vector quantities are conveniently shown by arrows. The length of the arrow represents the magnitude of the quantity, and the orientation of the arrow represents the directional property of the quantity. For example, if we consider the top of this page as representing north and we want to show the velocity of an aircraft flying east at an airspeed of 300 knots, the velocity vector is as shown in Fig. 1.3. If there is a 30‐knot wind from the north, the wind vector is as shown in Fig. 1.4.
Figure 1.3 Vector of an eastbound aircraft.
Figure 1.4 Vector of a north wind.
To find the aircraft's flight path, groundspeed, and drift angle, we add these two vectors as follows. Place the tail of the wind vector at the arrow of the aircraft vector and draw a straight line from the tail of the aircraft vector to the arrow of the wind vector. This resultant vector represents the path of the aircraft over the ground. The length of the resultant vector represents the groundspeed, and the angle between the aircraft vector and the resultant vector is the drift angle (Fig. 1.5).
Figure 1.5 Vector addition.
The groundspeed is the hypotenuse of the right triangle and is found by use of the Pythagorean theorem :
The drift angle is the angle whose tangent is , which is 5.7° to the right (south) of the aircraft heading.
Vector Resolution
It is often desirable to replace a given vector by two or more other vectors. This is called vector resolution. The resulting vectors are called component vectors of the original vector and, if added vectorially, they will produce the original vector. For example, if an aircraft is in a steady climb, at an airspeed of 200 knots, and the flight path makes a 30° angle with the horizontal, the groundspeed and rate of climb can be found by vector resolution. The flight path and velocity are shown by vector in Fig. 1.6.
Figure 1.6 Vector of an aircraft in a climb.
In Fig. 1.7 to resolve the vector into a component parallel to the horizontal, which will represent the groundspeed, and a vertical component, , which will represent the rate of climb, we simply draw a straight line vertically upward from the horizontal to the tip of the arrow . This vertical line represents the rate of climb and the horizontal line represents the groundspeed of the aircraft. If the airspeed is 200 knots and the climb angle is 30°, mathematically the values are
Figure 1.7 Vectors of groundspeed and rate of climb.
MOMENTS
If a mechanic tightens a nut by applying a force to a wrench, a twisting action, called a moment, is created about the center of the bolt. This particular type of moment is called torque (pronounced “tork”). Moments, , are measured by multiplying the amount of the applied force, , by the moment arm,:
The moment arm is the perpendicular distance from the line of action of the applied force to the center of rotation. Moments are measured as foot‐pounds (ft‐lb) or as inch‐pounds (in.‐lb). If a mechanic uses a 10‐in.‐long wrench and applies 25 lb of force, the torque on the nut is 250 in.‐lb.
The aircraft moments that are of particular interest to pilots include pitching moments, yawing moments, and rolling moments. If you have ever completed a weight and balance computation for an aircraft you have calculated a moment, where weight was the force and the arm was the inches from datum. Pitching moments, for example, occur when an aircraft's elevator is moved. Air loads on the elevator, multiplied by the distance to the aircraft's center of gravity (CG), create pitching moments, which cause the nose to pitch up or down. As you can see from Eq. 1.2, if a force remains the same but the arm is increased, the greater the moment.
Several forces may act on an aircraft at the same time, and each will produce its own moment about the aircraft's CG. Some of these moments may oppose others in direction. It is therefore necessary to classify each moment, not only by its magnitude, but also by its direction of rotation. One such classification could be by clockwise or counterclockwise rotation. In the case of pitching moments, a nose‐up or nose‐down classification seems appropriate.
Mathematically, it is desirable that moments be classified as positive (+) or negative (−). For example, if a clockwise moment is considered to be a + moment, then a counterclockwise moment must be considered to be a − moment. By definition, aircraft nose‐up pitching moments are considered to be + moments.
EQUILIBRIUM CONDITIONS
Webster defines equilibrium as “a state of balance or equality between opposing forces.” A body must meet two requirements to be in a state of equilibrium:
There must be no unbalanced forces acting on the body. This is written as the mathematical formula
, where
(cap sigma) is the Greek symbol for “sum of.”
Figures 1.1
and
1.2
illustrate situations where this condition is satisfied (lift = weight, thrust = drag, etc.)
There must be no unbalanced moments acting on the body. Mathematically,
(
Fig. 1.8
).
Figure 1.8 Seesaw in equilibrium.
Moments at the fulcrum in Fig. 1.8 are 50 ft‐lb clockwise and 50 ft‐lb counterclockwise. So, . To satisfy the first condition of equilibrium, the fulcrum must press against the seesaw with a force of 15 lb. So, .
NEWTON'S LAWS OF MOTION
Sir Isaac Newton summarized three generalizations about force and motion. These are known as the laws of motion.
Newton's First Law
In simple language, the first law states that a body at rest will remain at rest and a body in motion will remain in motion, in a straight line, unless acted upon by an unbalanced force. The first law implies that bodies have a property called inertia. Inertia may be defined as the property of a body that results in its maintaining its velocity unchanged unless it interacts with an unbalanced force, as with an aircraft at rest on a ramp without unbalanced forces acting upon it. The measure of inertia is what is technically known as mass.
Newton's Second Law
The second law states that if a body is acted on by an unbalanced force, the body will accelerate in the direction of the force and the acceleration will be directly proportional to the force and inversely proportional to the mass of the body. Acceleration is the change in motion (speed) of a body in a unit of time, consider an aircraft accelerating down the runway, or decelerating after touchdown. The amount of the acceleration , is directly proportional to the unbalanced force, , and is inversely proportional to the mass, , of the body. These two effects can be expressed by the simple equation
or, more commonly,
Newton's Third Law
The third law states that for every action force there is an equal and opposite reaction force. Note that for this law to have any meaning, there must be an interaction between the force and a body. For example, the gases produced by burning fuel in a rocket engine are accelerated through the rocket nozzle. The equal and opposite force acts on the interior walls of the combustion chamber, and the rocket is accelerated in the opposite direction. As a propeller aircraft pushes air backwards from the propeller, the aircraft moves forward.
LINEAR MOTION
Newton's laws of motion express relationships among force, mass, and acceleration, but they stop short of discussing velocity, time, and distance. These are covered here. In the interest of simplicity, we assume here that acceleration is constant. Then,
where
(cap delta) means “change in”
=
velocity at time
t
=
velocity at time
t
0
If we start the time at and rearrange the above, then
If we start the time at and (brakes locked before takeoff roll) and rearrange the above where can be any velocity given (for example, liftoff velocity), then
The distance s traveled in a certain time is
The average velocity is
Therefore,
Solving Eqs. 1.4 and 1.5 simultaneously and eliminating , we can derive a third equation:
Equations 1.3, 1.4 and 1.5 are useful in calculating takeoff and landing factors. They are studied in some detail in Chapters 10 and 11.
ROTATIONAL MOTION
Without derivation, some of the relationships among tangential (tip) velocity, ; radius of rotation, ; revolutions per minute, rpm; centripetal forces, ; weight of rotating parts, ; and acceleration of gravity, , are shown below. The centripetal force is that force that causes an airplane to turn. The apparent force that is equal and opposite to this is called the centrifugal force.
WORK
In physics, work has a meaning different from the popular definition. You can push against a solid wall until you are exhausted but, unless the wall moves, you are not doing any work. Work requires that a force must move an object in the direction of the force. Another way of saying this is that only the component of the force in the direction of movement does any work:
Work is measured in ft‐lb.
ENERGY
Energy is the ability to do work. There are many kinds of energy: solar, chemical, heat, nuclear, and others. The type of energy that is of interest to us in aviation is mechanical energy.
There are two kinds of mechanical energy. The first is called potential energy of position, or more simply potential energy, . No movement is involved in calculating . A good example of this kind of energy is water stored behind a dam. If released, the water would be able to do work, such as running a generator. As a fighter aircraft zooms to a zenith point it builds ; once it starts to accelerate downward it converts PE to KE. PE equals the weight, , of an object multiplied by the height, , of the object above some base plane:
The second kind of mechanical energy is called kinetic energy, . As the name implies, kinetic energy requires movement of an object. It is a function of the mass, , of the object and its velocity, :
The total mechanical energy, , of an object is the sum of its and :
The law of conservation of energy states that the total energy remains constant. Both potential and kinetic energy can change in value, but the total energy must remain the same: Energy cannot be created or destroyed, but can change in form.
POWER
In our discussion of work and energy we have not mentioned time. Power is defined as “the rate of doing work” or work/time. We know:
and
James Watt defined the term horsepower (HP) as 550 ft‐lb/sec:
If the speed is measured in knots, , and the force is the thrust, , of a jet engine, then
Equation 1.13 is very useful in comparing thrust‐producing aircraft (turbojets) with power‐producing aircraft (propeller aircraft and helicopters).
FRICTION
If two surfaces are in contact with each other, then a force develops between them when an attempt is made to move them relative to each other. This force is called friction. Generally, we think of friction as something to be avoided because it wastes energy and causes parts to wear. In our discussion on drag, we will discuss the parasite drag on an airplane in flight and the thrust or power to overcome that force. Friction is not always our enemy, however, for without it there would be no traction between an aircraft's tires and the runway. Once an aircraft lands, lift is reduced and a portion of the weight is converted to frictional force. Depending on the aircraft type, aerodynamic braking, thrust reversers, and spoilers will be used to assist the brakes and shorten the landing, or rejected takeoff distance.
Several factors are involved in determining friction effects on aircraft during takeoff and landing operations. Among these are runway surfacing material, condition of the runway, tire material and tread, and the amount of brake slippage. All of these variables determine a coefficient of friction (mu). The actual braking force, , is the product of this coefficient (Greek symbol mu) and the normal force, , between the tires and the runway:
Figure 1.9 shows typical values of the coefficient of friction for various conditions.
Figure 1.9 Coefficients of friction for airplane tires on a runway.
SYMBOLS
Acceleration (ft/sec
2
)
Centrifugal force (lb)
Energy (ft‐lb)
Kinetic energy
Potential energy
Total energy
Force (lb)
Braking force
Acceleration of gravity (ft/sec
2
)
Height (ft)
Horsepower
Moment arm (ft or in.)
Mass (slugs, lb‐sec
2
/ft)
Moment (ft‐lb or in.‐lb)
Normal force (lb)
Radius (ft)
Revolutions per minute
Distance (ft)
Thrust (lb)
Time (sec)
Speed (ft/sec)
Speed (knots)
Initial speed
Tangential (tip) speed
Weight (lb)
Coefficient of friction (dimensionless)
EQUATIONS
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
Chapter 2Atmosphere, Altitude, and Airspeed Measurement
PROPERTIES OF THE ATMOSPHERE
The aerodynamic forces and moments acting on an aircraft in flight are due, in great part, to the properties of the air mass in which the aircraft is flying. By volume the atmosphere is composed of approximately 78% nitrogen, 21% oxygen, and 1% other gases. The most important properties of air that affect aerodynamic behavior are its static pressure, temperature, density, and viscosity.
Static Pressure
The static pressure of the air, , is simply the weight per unit area of the air above the level under consideration. For instance, the weight of a column of air with a cross‐sectional area of 1 ft2 and extending upward from sea level through the atmosphere is 2116 lb. The sea level static pressure is, therefore, 2116 psf (or 14.7 psi). Static pressure is reduced as altitude is increased because there is less air weight above. At 18,000 ft altitude the static pressure is about half that at sea level. Another commonly used measure of static pressure is inches of mercury. On a standard sea level day the air's static pressure will support a column of mercury (Hg) that is 29.92 in. high (Fig. 2.1). Weather reports use a third method of measuring static pressure called millibars, standard pressure here is 1,013.2 mb. In addition to these rather confusing systems, there are the metric measurements in use throughout most of the world. For the discussion of performance problems later in this textbook, we will primarily use the measurement of static pressure in inches of mercury.
Figure 2.1 Standard pressure.
U.S. Department of Transportation Federal Aviation Administration, Pilot's Handbook of Aeronautical Knowledge, 2008
In aerodynamics it is convenient to use pressure ratios, rather than actual pressures. Thus the units of measurement are canceled out:
where is the sea level standard static pressure (2116 psf or 29.92 in. Hg). Thus, a pressure ratio of 0.5 means that the ambient pressure is one‐half of the standard sea level value. At 18,000 ft, on a standard day, the pressure ratio is 0.4992.
Temperature
The commonly used measures of temperature are the Fahrenheit, , and Celsius, (formerly called centigrade) scales. Aviation weather reports for pilots, as well as performance calculation tables, will usually report the temperature in °C. Neither of these scales has absolute zero as a base, so neither can be used in calculations. Absolute temperature must be used instead. Absolute zero in the Fahrenheit system is −460° and in the Celsius system is −273°. To convert from the Fahrenheit system to the absolute system, called Rankine, , add 460 to the °F. To convert from the Celsius system to the absolute system, called Kelvin, , add 273 to the °C. The symbol for absolute temperature is and the symbol for sea level standard temperature is :
By using temperature ratios, instead of actual temperatures, the units cancel. The temperature ratio is the Greek letter theta, :
At sea level, on a standard day,
