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After reading this article I would like to know if the lift produced by a wing with a 10° flap is equal to the lift created by a wing with 10° AOA but with no flap?

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No, there is a big difference between an entire wing being at an angle of attack and a small part of the wing being at an angle of attack.

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    $\begingroup$ On top of that, a good insight can be gained by referring to thin airfoil theory $\endgroup$ – ABCD Jun 14 '18 at 12:19
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    $\begingroup$ ... and what is that "big difference"? $\endgroup$ – BDLPPL Jun 15 '18 at 9:51
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No. While the article correctly mentions that extending flaps kind of has the same affect as increasing your AoA, the lift would not be the same.

First of all the flaps are only installed toward the root of the wing and not along the entire surface. So flaps only increase lift on a certain portion of the wing. Increasing the AoA increases lift on the entire wing so increasing your AoA would increase your lift more.

The main advantage of flaps is that it reduces your stall speed. An aerodynamic stall happens when there is a flow separation along the upper surface of the wing. This flow separation happens at the critical angle of attack so increasing your AoA too much is dangerous. Employing the flaps allows you to increase lift without increasing your AoA thus preventing the temptation of a pilot to stall his airplane during landing.

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