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I know on the 737, the leading edge slats deploy at the first flap setting, and the trailing edge flaps deploy after that at higher flaps settings. Why do the slats deploy before the trailing edge flaps?

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The slats help to protect the outer wing from stalling when the flaps deploy. The outer wing carries the ailerons, so there are no flaps, but the flaps inboard will induce more lift on the whole wing, raising the effective angle of attack even in the aileron section. Slats push the stall angle of attack up. Without the protection provided by the slats, the outer wing could stall, and if this happens on one wing first (very likely, there are many reasons for flow asymmetry), the aircraft will roll violently. To do so during an approach is not good at all.

Look at aircraft with powerful flaps: They all have slats at least on the outboard wing to avoid outer wing stall. One goes with the other.

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liftcoefficant

What you can see here is that as the slats deploy, the lift coefficient in theory for the same angle of attack remains relatively unaffected. Only the angle of attack at which stall is reached is increased.

This may be sensible because it increases the safety margin before deploying flaps particularly as these will increase drag which if left with the same power setting will increase the angle of attack to maintain the same vertical speed.

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While maximum coefficient of lift is higher with flaps (and no slats), as far as I remember the maximum angle of attack is less than clean configuration while your picture still shows slightly more. You are right that slats increase the stall angle of attack in either case. –  Jan Hudec May 2 at 20:51
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Been long time since I learned this, but if memory serves me the slats produce more lift for less drag (more "bang for your buck"), whereas flaps are the reverse. On typical airliner (I'm a 767 captain at AAL) slats reduce your stall speed an average of 30 knots.

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