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Lifting off

Issue 1, 18 July 2022

By: Anthony O. Ives

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Fixed wing aircraft are a good place to start when to learning how to size small remote controlled aircraft. Fixed wing design is also relevant in the design of winged helicopters and tilt rotor aircraft. Fixed wing aircraft are also simpler to build than rotary wing aircraft.

Fixed and rotary wing aircraft both generate a force called lift which is produced as an airfoil moves through the air. In fixed wing aircraft this airflow is produced by maintaining a forward speed without which the aircraft would stall and essentially fall out of sky. This critical speed is called stall speed. Rotary wing aircraft produce the airflow by rotating a wing (the rotor blades are the wings) through the air, this principle is also used in propellers which in this case the lift is used to produce thrust.

Aircraft Forces

The above picture illustrates the forces acting on a fixed wing aircraft in flight. Lift is generally equal to the aircraft's weight in steady level flight such as in the cruise phase. Thrust is generally equal to drag in steady level flight, thrust and drag are discussed in a later article.

In both cases of fixed and rotary wing aircraft the lift coefficient equation can be used to calculate the size of wings to produce the required lift. However in the case of rotary wings the equation takes a more complicated form which will also be discussed in a later article.

The lift coefficient equation in its primary form is:

\[L=q C_{L} S \]

Where q=(1/2) ρ (V²) and is also known as dynamic pressure, ρ is air density, V is the velocity of the airflow around the wing

C_L is lift coefficient, S is the area of the wing, L is the lift produced by the wing. The lift coefficient equation gives good summary of all the factors influencing an aircraft lifting performance or simply how much weight it can carry. Interesting points to note:

Air density is something we have no control over as it depends on weather conditions of particular day. A hot, humid day will give a low air density hence reducing the amount of weight the aircraft can carry as it will increase the stall speed if the aircraft's weight is not reduced.
Typically a good design would have low lift coefficient in cruise flight and in most phases of its flight as it is safer by reducing stall speed and is more efficient by reducing lift dependent drag. During approach to land is really the only time a aircraft should be flying with a high lift coefficient as it wants to be flying slow during this phase of it's flight.
From design point of view increasing wing area is the most effective way of increasing an aircraft's lifting performance, it can also increase it's turning performance among other things.

The table below gives typical lift coefficients for various types of aircraft in cruise or steady level flight:

Property General Aviation Aircraft Light Utilty Aircraft Miltary Combat Aircraft Miltary Transport Aircraft Commercial Airliner WW2 Miltary Combat Aircraft

Velocity, V/ms^-1 62.8 58.1 388.9 250 242 94.3

Wing Area, S/m² 16.2 16.58 78.8 300 112.3 31.9

Altitude, h/ft 5000 5000 35000 35000 35000 5000

Density, ρ/kgm^-3 1.056 1.056 0.379 0.379 0.379 1.056

Dynamic Pressure, q/Pa 2082 1782 28660 11844 11098 4695

Mass, m;/kg 1111 794 35000 195000 63100 4336

Lift Coefficient C_L 0.323 0.264 0.152 0.538 0.497 0.284

Property	General Aviation Aircraft	Light Utilty Aircraft	Miltary Combat Aircraft	Miltary Transport Aircraft	Commercial Airliner	WW2 Miltary Combat Aircraft
Velocity, V/ms^-1	62.8	58.1	388.9	250	242	94.3
Wing Area, S/m²	16.2	16.58	78.8	300	112.3	31.9
Altitude, h/ft	5000	5000	35000	35000	35000	5000
Density, ρ/kgm^-3	1.056	1.056	0.379	0.379	0.379	1.056
Dynamic Pressure, q/Pa	2082	1782	28660	11844	11098	4695
Mass, m;/kg	1111	794	35000	195000	63100	4336
Lift Coefficient C_L	0.323	0.264	0.152	0.538	0.497	0.284

Lift of the wings produced for steady level flight should be equal to the aircraft's weight, which is aircraft mass, m mutipled by gravity, g also known as the acceleration of freefall hence:

L=W=m g

Gravity is typically assumed to be 9.81ms^-1. The lift coefficient equation can then rearranged to calculate wing loading which is W/S:

\[\frac{W}{S}=q C_{L} \]

Wing loading is the weight a wing can carry for a unit area, it is very useful when designing a wing because it is a very simple way of working out different wing areas for different weights.

The following table shows how to calculate wing area using using some assumed values.

Symbol Property Example Value Units

ρ Air density 1.2256 kgm^-1

V Aircraft velocity 15 ms^-1

V² Velocity squared 15x15=225 m²s^-2

q Dynamic pressure (1/2)x1.2256x225=137.88 Pa

C_L Lift coefficent 0.2 None

W/S Wing loading 0.2x137.88=27.6 Pa

w₀ Total aircraft mass 2.07 kg

S Wing area (2.07x9.81)/27.6=0.736 m²

Symbol	Property	Example Value	Units
ρ	Air density	1.2256	kgm^-1
V	Aircraft velocity	15	ms^-1
V²	Velocity squared	15x15=225	m²s^-2
q	Dynamic pressure	(1/2)x1.2256x225=137.88	Pa
C_L	Lift coefficent	0.2	None
W/S	Wing loading	0.2x137.88=27.6	Pa
w₀	Total aircraft mass	2.07	kg
S	Wing area	(2.07x9.81)/27.6=0.736	m²

The lift coefficient can also be rearranged to calculate the aircraft stall speed:

\[V_{stall} = \left( \frac{2 m g}{\rho C_{L} S} \right) ^{\frac{1}{2}} \]

\[ = \sqrt \frac{2 m g} { \rho C_{L(Max)} S} \]

The aircraft stall speed is the speed the aircraft wings stops producing lift hence why its important pilots maintain their airspeed well above the stall speed. C_L(Max) is the maximum lift coefficient the aircraft is capable of, typically C_L(Max)=1.0 if the aircraft is not equipped with high lift devices. Based on a C_L(Max)=1.0 and using the same values used previously an example of a stall speed calculation is given below:

\[V_{stall}=\sqrt \frac {2 \times 2.07 \times 9.8} {1.2256 \times 1.0 \times 0.736} \]

=6.71ms^-1

You should now have good understanding of the main factors which influence an aircraft's lifting performance and how lift coefficient can be used to calculate wing area, stall speed, etc. A later article hopes to discuss the principle of lift generation and how lift coefficient can be determined from airfoil geometry and angle of attack. Further information on this topic can also be found in Ref [1].

Please leave a comment on my facebook page or via email and let me know if the lift coefficient equation has helped you understand the factors influencing aircraft flight performance or if there is anything I can do to make this article more helpful.

References:

[1] Fundamentals of Aerodynamics, John D. Anderson Jr., 3rd Edition, 2001, McGraw Hill

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