AE1110-I Introduction to Aerospace Engineering-I
Introduction to Aerospace
Engineering-I
AE1110-I
Delft University of Technology
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, AE1110-I Introduction to Aerospace Engineering-I
Table of Contents
Equations of State and Ballooning………………………………………………………………………………………………………3
International Standard Atmosphere (ISA) and Altitude……………………………………………………………………….5
Forces on an Aircraft…………………………………………………………………………………………………………………………..8
Stability and Control…………………………………………………………………………………………………………..............…11
Structural Concepts………………………………………………………………………………………………………………………….16
Materials and Exploring the Limits……………………………………………………………………………………………………17
Propulsion………………………………………………………………………………………………………………………………………..20
Cockpit instrumentation, Altitude, Speed and Control……………………………………………………………………..25
Special Vehicles and Future………………………………………………………………………………………………………………28
Aerodynamics…………………………………………………………………………………………………………………………………..31
Laminar and Turbulent Flow…………………………………………………………………………………………………………….38
Turbulent and Laminar Boundary Layers………………………………………………..………………………………………..44
Airfoils, Lift from Pressure Distribution and Critical Mach Number………………………………………….……….46
Finite Wings………………………………………………………………………..……………………………………………………………55
Equations of Motion…………………………………………………………………………………………………………………………60
Horizontal Flight Performance………………………………………………………………………………………………………….66
Climbing and Descending Flight Performance…………………………………………………………………………………..70
Altitude Effects on Aircraft Performance………………………………………………………………………………………….73
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, AE1110-I Introduction to Aerospace Engineering-I
Equations of State and Ballooning
Economics
72
𝑇2 =
𝑟
T2 = doubling time (in years)
r = growth percentage per year
Three ways to counter gravity
- Floating by being lighter
- Push air downwards
- Push something else downwards
Balloons
Pros: - very efficient lift
- can go very high
Cons: - slow
- high drag
How does a balloon lift
𝐽
𝑝 ∙𝑉 =𝑛 ∙ ℛ ∙𝑇 ℛ = 8.314462175 𝑚𝑜𝑙 𝐾
𝑚 𝑚 𝑚 𝑚
𝑝 ∙ 𝜌
= 𝑀
∙ ℛ ∙𝑇 𝜌= 𝑉
→𝑉 = 𝜌
ℛ 𝑚
𝑝= 𝜌 ∙ ∙𝑇 𝑛=
𝑀 𝑀
𝑝 = 𝜌 ∙𝑅 ∙𝑇 R = specific gas constant for air
ℛ 8.314462
𝑅 = 𝑀 = 0.02897 = 287.00 𝐽/𝑘𝑔 𝐾
𝑔 = 9.80665 𝑚/𝑠 2
𝑚 =𝜌∙𝑉
𝐿 = 𝑚𝑎𝑖𝑟 𝑔 − 𝑚𝑔𝑎𝑠 𝑔 = 𝑔 (𝑚𝑎𝑖𝑟 − 𝑚𝑔𝑎𝑠 )
𝐿 = 𝜌𝑎𝑖𝑟 𝑉𝑔 − 𝜌𝑔𝑎𝑠 𝑉𝑔
𝜌𝑔𝑎𝑠
𝐿 = 𝜌𝑎𝑖𝑟 𝑉𝑔 (1 − )
𝜌𝑎𝑖𝑟
ℛ
𝑝= 𝜌 ∙ ∙𝑇
𝑀
𝑝 𝑚
𝜌= ∙
𝑇 ℛ
𝜌1 𝑝1 𝑇2 𝑀1
= ∙ ∙
𝜌2 𝑝2 𝑇1 𝑀2
Helium balloon
𝜌1 𝑝1 𝑇2 𝑀1
= ∙ ∙
𝜌2 𝑝2 𝑇1 𝑀2
𝜌 𝑀𝐻𝑒
Assumptions → 𝐻𝑒 =
𝜌𝑎𝑖𝑟 𝑀𝑎𝑖𝑟
𝑀
𝐿𝐻𝑒 = 𝜌𝑎𝑡𝑚 𝑉𝑔(1 − 𝑀 𝐻𝑒 )
𝑎𝑖𝑟
4.00
𝐿𝐻𝑒 = 𝜌𝑎𝑡𝑚 𝑉𝑔(1 − 28.97)
𝐿𝐻𝑒 = 0.861926𝜌𝑎𝑡𝑚 𝑉𝑔
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Hot air balloon
𝜌1 𝑝1 𝑇2 𝑀1
= ∙ ∙
𝜌2 𝑝2 𝑇1 𝑀2
𝜌 𝑇
Assumptions → 𝜌 ℎ𝑜𝑡 = 𝑇𝑐𝑜𝑙𝑑
𝑎𝑡𝑚 ℎ𝑜𝑡
𝑇
𝐿ℎ𝑜𝑡 = 𝜌𝑎𝑡𝑚 𝑉𝑔(1 − 𝑇+ ∆𝑇)
𝑇+ ∆𝑇 𝑇
𝐿ℎ𝑜𝑡 = 𝜌𝑎𝑡𝑚 𝑉𝑔( − )
𝑇+ ∆𝑇 𝑇+ ∆𝑇
∆𝑇
𝐿ℎ𝑜𝑡 = 𝜌𝑎𝑡𝑚 𝑉𝑔 𝑇+ ∆𝑇
Comparison
6
𝐿𝐻𝑒 ≈ 𝜌 𝑉𝑔
7 𝑎𝑡𝑚
∆𝑇 6
Equal to He for hot → 𝑇+ ∆𝑇 = 1+6 → ∆T = 6T
𝐿𝐻2 = 0.93𝜌𝑎𝑡𝑚 𝑉𝑔 (this shows that it is not true that if the weight of the gas halves, the lift doubles)
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International Standard Atmosphere (ISA) and Altitude
Why a standard atmosphere?
- Meaningful aircraft performance specification
- Pressure altitude definition
- Model atmosphere for simulation and analysis
Definition standard atmosphere
Define three variables at every altitude:
- Pressure
- Density
- Temperature
Physically correct, has to obey:
- Gas law (relation 𝑝 − 𝜌 − 𝑇)
- Gravity law → pressure caused by mass of air above
Static equilibrium → no vertical winds
Standard sea level values
𝑝 = 101325.0 𝑃𝑎
𝑇 = 15℃ = 288.15 𝐾
𝜌 = 1.225 𝑘𝑔/𝑚3
Area A 𝑝 + ∆𝑝
ISA
𝑇 = 𝑇0 + 𝑎(ℎ − ℎ0 ) ℎ
Here 𝑎 is the temperature change per unit altitude. ∆ℎ
𝑝 = 𝜌𝑅𝑇
𝑑𝑝 = −𝜌𝑔𝑑ℎ
𝐹𝑔 = 𝑚 ∙ 𝑔 𝐹𝑔
𝑝
𝐹𝑔 = 𝜌 ∙ 𝑉 ∙ 𝑔
𝐹𝑔 = 𝜌 ∙ 𝐴 ∙ ∆ℎ ∙ 𝑔
(𝑝 + ∆𝑝)𝐴 + 𝜌𝐴∆ℎ𝑔 = 𝑝𝐴
𝑝𝐴 + ∆𝑝𝐴 + 𝜌𝐴∆ℎ𝑔 = 𝑝
∆𝑝 + 𝜌∆ℎ𝑔 = 0
∆𝑝 = −𝜌∆ℎ𝑔 (works really well for fluids)
𝑝
𝑝 = 𝜌𝑅𝑇 → 𝜌 =
𝑅𝑇
𝑝
𝑑𝑝 = − 𝑔𝑑ℎ
𝑅𝑇
∆𝑇 𝑑𝑇 𝑑𝑇
𝑇 = 𝑇0 + 𝑎(ℎ − ℎ0 ) → 𝑎 = = → 𝑑ℎ =
∆ℎ 𝑑ℎ 𝑎
1 𝑔 1
𝑑𝑝 = − ∙ 𝑑ℎ
𝑝 𝑅 𝑇
𝑝1 𝑇1
1 𝑔 1
∫ 𝑑𝑝 = − ∫ 𝑑𝑇 (unfortunately this formula cannot be used in isothermic layers)
𝑝 𝑎𝑅 𝑇
𝑝0 𝑇0
𝑔
ln 𝑝1 − ln 𝑝0 = − (ln 𝑇1 − ln 𝑇0 )
𝑎𝑅
𝑔
𝑒 ln 𝑝1 −ln 𝑝0 = 𝑒 −𝑎𝑅(ln 𝑇1 −ln 𝑇0 )
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𝑔
𝑒 ln 𝑝1 𝑝1 𝑇1 −𝑎𝑅
= = ( )
𝑒 ln 𝑝0 𝑝0 𝑇0
𝑔
𝑇1 −𝑎𝑅
𝑝1 = 𝑝0 ( )
𝑇0
𝑝1
𝜌1 =
𝑅𝑇1
Isothermic layer
1 𝑔 1
𝑑𝑝 = − ∙ 𝑑ℎ
𝑝 𝑅 𝑇
𝑇1 = 𝑇0
𝑝1 ℎ1
1 𝑔
∫ 𝑑𝑝 = − ∫ 𝑑ℎ
𝑝 𝑅𝑇
𝑝0 ℎ0
𝑔
ln 𝑝1 − ln 𝑝0 = − (ℎ1 − ℎ0 )
𝑔
𝑅𝑇
(ℎ )
𝑒 ln 𝑝1 −ln 𝑝0 = 𝑒 −𝑅𝑇 1 −ℎ0
𝑝1 𝑔
(ℎ )
= 𝑒 −𝑅𝑇 1 −ℎ0
𝑝0
𝑔
(ℎ1 −ℎ0 )
𝜌1 = 𝜌0 ∙ 𝑒 −𝑅𝑇
Absolute altitude and geometric altitude
Geometric altitude: real altitude with sea level = 0.
Absolute altitude: distance to centre of earth.
ℎ𝑎 = ℎ𝐺 + 𝑅
Earth radius: R = 6357 km
In a formula we always want to fill in the geopotential altitude. To go from the geometric altitude
𝑅𝑒
(HG) to the geopotential altitude (H), we use the formula 𝐻 = 𝑅 +𝐻 𝐻𝐺
𝑒 𝐺
Power and work
The work done by an object can be calculated with the formula: 𝑊 = 𝐹𝑠. In this formula the
displacement 𝑠 of course has to be done in the direction of the force.
∆𝑊 𝐹∙∆𝑥
Power can be calculated with the formula: 𝑃 = = = 𝐹𝑣.
∆𝑡 ∆𝑡
1
2𝐹∙ 𝑑∙∆𝜑
2
The shaft power of an engine can be calculated with the formula: 𝑃 = ∆𝑡
= 𝑀𝜔
Pitot tube and static pressure measurement
1 1
To calculate the total pressure use the formula: 𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑠𝑡𝑎𝑡𝑖𝑐 + 2 𝜌𝑉 2. We call 2 𝜌𝑉 2 also 𝑞.
Therefore the formula can also be 𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑠𝑡𝑎𝑡𝑖𝑐 + 𝑞.
2(𝑝𝑡𝑜𝑡𝑎𝑙 −𝑝𝑠𝑡𝑎𝑡𝑖𝑐 )
To calculate the velocity then, we can use the formula: 𝑉 = √ 𝜌
Propulsion
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To calculate the brake power of an engine use the formula: 𝑃𝑏𝑟 = 𝑇ℎ𝑟𝑜𝑡𝑡𝑙𝑒 ∙ 𝑃𝑏𝑟𝑚𝑎𝑥 . If we then
include the efficiency of the engine the formula for the power becomes: 𝑃𝑎 = 𝜂 ∙ 𝑃𝑏𝑟 = 𝑇 ∙ 𝑉.
Equivalent air speed (EAS) and true air speed (TAS)
Equivalent air speed is often the indicated air speed measured by the pitot tube. There is a difference
since the calculations so far dit not take into account the difference in density of the air. So to go
1 2 1 2
from equivalent air speed to indicated airspeed, use this formula: 2 𝜌𝑉𝑇𝐴𝑆 = 2 𝜌0 𝑉𝐸𝐴𝑆 . The true air
𝜌0
speed can therefore also be expressed as: 𝑉𝑇𝐴𝑆 = √ ∙ 𝑉𝐸𝐴𝑆 .
𝜌
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Forces on an Aircraft
Major forces on an aircraft
There are four major forces on an aircraft:
- Weight (W): is composed of three main components: aircraft empty weight, fuel and
payload.
- Lift (L): mainly generated by the wing (small contribution of the tail surfaces).
- Drag (D): is caused by the fuselage, wing, tail surfaces etc.
- Thrust (T): provided by the engines.
For a constant velocity and a constant altitude: W = L and D = T.
Lift
1
The formula for lift is: 𝐿 = 𝐶𝐿 𝜌 𝑉 2 𝑆
2
- 𝜌 = density of the air (kg/m3)
- 𝑉 = air speed (m/s)
- 𝑆 = wing area (m2)
- 𝐶𝐿 = lift coefficient
This is an aerodynamic coefficient, it depends on:
- shape of aircraft and wing
- angle between air speed vector and wing
The lift coefficient also depends on the angle of attack (𝛼). The air density (𝜌) depends on the
altitude and the temperature. The air speed (V) and the wing area (S) are design parameters.
Airfoils
There are different types of airfoils, and some have been given a code. For example NACA 2412
means:
- 2% camber (of
chord length) at
0.4 of the cord
(from the
leading edge)
- 12%
thickness/cord
ratio
If the two first numbers
of this four digit code
are both zero, this
means that the airfoil is symmetrical.
It can also happen that there is a five-digit code. This is for more complex airfoils. For example NACA
23014:
- 2: a design coefficient of lift, 𝐶𝐿 = 0.3 (2 × 0.15)
30
- 30: maximum camber located at ( 2 ) 15% chord
- 14: maximum thickness of 14% chord
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Viscosity/scale effect
𝜌𝑉𝐿 𝑉𝐿
The formula for the Reynolds number is: 𝑅𝑒 = =
𝜇 𝑣
- Re: Reynolds number
- 𝜌: density of the air (kg/m3)
- 𝑉: airspeed (m/s)
- 𝐿: typical length (chord c for airfoils)
- 𝜇: dynamic viscosity (Ns/m2)
𝜇
- 𝑣: kinematic viscosity = 𝜌
Bernoulli’s principle
There are several formulas for Bernoulli’s principle:
1
𝑝 + 𝜌 𝑉 2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 𝑠𝑡𝑟𝑒𝑎𝑚𝑙𝑖𝑛𝑒
2
𝑝𝑠𝑡𝑎𝑡𝑖𝑐 + 𝑝𝑑𝑦𝑛𝑎𝑚𝑖𝑐 = 𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 𝑠𝑡𝑟𝑒𝑎𝑚𝑙𝑖𝑛𝑒
𝑝𝑠𝑡𝑎𝑡𝑖𝑐 + 𝑞 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 𝑠𝑡𝑟𝑒𝑎𝑚𝑙𝑖𝑛𝑒
These are all only true for incompressible flow, this means that the 𝜌 does not change, for example
during low speeds.
Generation of lift
Lift is generated by a difference in pressure, this difference in pressure causes a force upward for
airfoils. By changing the angle of attack the lift is improved since it then generates more downward
flow.
Lift coefficient
There is a curve where the angle of attack is on the horizontal
axes and the coefficient of lift on the vertical axes. On this axes
we can thus find for which angle of attack the coefficient of lift
reaches its maximum value.
Airflow
We distinguish between two different types of boundary layers:
- Laminar boundary layer: thin, low friction.
- Turbulent boundary layer: thick, high friction.
If we have to fly slower, there are several things we can do in order to make sure the aircraft can
keep flying. We can increase the surface area of the wing, we can increase the coefficient of lift in
order to increase the lift.
Drag
1
The formula for drag is: 𝐷 = 𝐶𝐷 2 𝜌 𝑉 2 𝑆.
The drag coefficient consists of:
- Parasitic drag:
- profile drag of wing
- pressure and friction of tail, nacelle, fuselage etc
- Induced drag:
- due to lift being generated
Drag coefficient: 𝐶𝐷 = 𝐶𝐷0 + 𝐶𝐷𝑖
- 𝐶𝐷0 : zero-lift drag
- 𝐶𝐷𝑖 : induced drag
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𝐶2
𝐿
Drag polar: 𝐶𝐷 = 𝐶𝐷0 + 𝜋 𝐴𝑒
- A: aspect ratio
𝑠𝑝𝑎𝑛 𝑏 𝑠
A = 𝑐ℎ𝑜𝑟𝑑 = 𝑐 = 𝑐 2
- c: average chord
- E: Oswald efficiency factor
It is very important for an aircraft to have the
lowest possible 𝐶𝐷 for the highest 𝐶𝐿 (lower fuel
consumption), we can find this using a lift-drag
polar. It is very important in these kind of graphs
to keep in mind the scales.
2D and 3D differences
There are very big differences in results if we talk
about 3D situations compared to 2D situations.
This has to do with the fact that vortices are
created on the side of the wings due to the
pressure differences. This is the reason that
winglets lower the fuel consumption of an
airplane.
We also distinguish between infinite wings and finite wings. Infinite wings make use of a span of 1 m
(per meter span). A finite wing experiences the effect of an increasing 𝛼 since the lift increases too.
Thrust
To maintain a constant speed, the thrust has to be of the same magnitude as the drag (T = D).
The reason we fly at high altitude is since the air density is lower at higher altitudes. This means that
one of the parameters has to increase to maintain the lift force. We want to go faster since this
results in the fact that we arrive at our destination quicker. Therefore the reason we fly at higher
altitudes is since we fly faster there for lower drag compared to lower altitudes.
Weight
The weight of an aircraft consists of 3 components:
- Aircraft empty weight
- Payload
- Fuel
If you minimize the aircraft empty weight, the aircraft is capable to take more payload. If we take
away some weight of the structure of the aircraft, we need less thrust, therefore the engines are
smaller and weigh less, therefore the weight is lower. A saving of weight will result in a snowball-
effect meaning that a saving of weight will result in additional savings of weight after the original
saving.
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