How to choose an attack angle?

Question

I'm writing a little piece of code for preliminary design that can calculate the ideal wing area for a given load, cruise speed and aspect ratio or span. It seemingly works well on the A380 data:

# A380
CONSTRAINS
Load: 475000
Cruise Speed: 907.48
Span: 80
Aspect Ratio: 7.5

Output:

GEOMETRY
Airfoil Lift Coef: 0.20400000000006546
Total Mass: 613525.1510617425
Lift Area: 851.9381984117002 m²               # <--- perfect 
Average Chord: 10.666666666666666 m

However, I have a little issue relating to the attack angle with small airplanes:

# Cessna 
CONSTRAINS
Load: 1157
Cruise Speed: 226
Span: 16
Aspect Ratio: 7.3

ERROR: Lift Coefficient was re-adjusted: 0.8950000000000661

GEOMETRY
Airfoil Lift Coef: 0.8950000000000661
Total Mass: 6796.908900200735
Lift Area: 34.68578659410046 m²                # <-- wrong supposed to be 16
Average Chord: 2.191780821917808 m             # <-- wrong

Now, we know the Cessna has a large attack angle, which increases the Lift coefficient to 1 and enables short distance takeoff capabilities at the cost of thrust. This incidentally reduces the Lift Area to 16m².

However, the same happens with the Piaggio and there is no explanation there.

My question is: How is the lift_coef and attack_angle chosen? There seems to be loads of variables at play.

Is it because the average chord is higher than what the wing can handle structurally?

Answer 1

This is a rather simple question if you look at it from a slightly different angle, mind the pun. If we assume cruise condition, that equates to straight and level flight. Here you know that:

$$ L = W = mg $$

By knowing the mass of the aircraft (m) you have the lift force (L), given g is a constant (9.81m/s2). You need to know your cruise altitude, which then gives you your density (p = rho). You can always look this up in a table for ISA (the international standard atmosphere). Hopefully where you got your other aircraft parameters, you also got a cruise speed. Looking at the equation for lift,

$$ L = \frac{1}{2} \times p \times v^2 \times C_l \times S $$

you have everything except Cl, which you can now calculate.

You can then use thin aerofoil theory to give you the corresponding angle of attack needed for this. If using radians, thin aerofoil theory gives the lift slope for a Cl v AoA plot as 2pi. In degrees this is approximately 0.11 "Cls per degree". If you don't want to assume a symmetric aerofoil, you can assume between 0.2 to 0.4 as the Cl at AoA = 0. Alternatively, if you have knowledge of what the aerofoil is, you can consult an online database of aerofoils. This should give you the lift slope and the coefficient of lift when AoA = 0.

If you want, you can also correct for the AR. That is, the lift for a wing is less than the lift that should be generated by the constituent aerofoil. That is, the angle of attack in 3D flow needs to be greater than for 2D flow. For this you need the correction equation, and you need to know the AoA for Cl = 0. Happy to provide a link to the equation if you feel you need to go that far.

Hope that helps.