I was trying to calculate the forces generated by an uncommanded flap deployment during cruising flight ($M_\infty=0.85, Alt.=36kft$) in comparison to the forces say, during landing. Assuming similar geometries, the forces will be proportional to the freestream dynamic pressure. If $q_\infty=\frac{\gamma}{2}p_\infty {M_\infty}^2$, the Mach number will be higher at cruise but the static pressure will be lower due to altitude. Thus, it is not immediately obvious to me which flight condition has the higher dynamic pressure. Does anyone have reasonable numbers for $q_\infty [lbf/ft^2]$ at landing vs cruise for a modern commercial airliner so that I can compare?
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2$\begingroup$ $lbf/ft^2$? xkcd.com/526 $\endgroup$– FedericoCommented Jun 6, 2014 at 19:25
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$\begingroup$ I don't understand the problem. I want pounds-force per foot squared. $\endgroup$– Bryson S.Commented Jun 6, 2014 at 19:30
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2$\begingroup$ The equivalent speed is about twice as high (the calculator linked by fooot gives 266 knots for your case while 130 is reasonable landing speed) and since equivalent speed is proportional to square root of dynamic pressure, it means about four times higher dynamic pressure at cruise. $\endgroup$– Jan HudecCommented Jun 6, 2014 at 21:59
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2 Answers
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Here is what I am getting:
Static pressure:
Sea level: 101300 Pa
30,000 feet: 30100 Pa
Mach:
Sea level1 (150 kt): 0.22
Cruise2: 0.85
Using the equation:
Landing: $3432.044 Pa = 73.68 lbf/ft^2$
Cruise: $15223.075 Pa = 317.94 lbf/ft^2$
So the high Mach in cruise more than compensates for the lower static pressure. This calculator agrees.
1Landing speeds for large jets are generally between 120-150 knots (Mach 0.18-0.22). Smaller planes around 60 knots (Mach 0.091). You can convert to Mach by hand or using a calculator like here.
2Cruise speeds are generally between 240-280 kias (knots indicated airspeed) for large jets, which translates to around 450-500 knots ground speed and Mach 0.75-0.85 (depending on altitude and conditions).
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2$\begingroup$ Well for planes that do Mach 0.85 at 30,000 feet it's generally around landing speed... $\endgroup$– foootCommented Jun 6, 2014 at 20:16
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$\begingroup$ This is why I was asking the question, I really don't know what a reasonable landing Mach is to two significant figures... $\endgroup$ Commented Jun 6, 2014 at 20:33
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$\begingroup$ @fooot: Cruise speed is not around landing speed, it's about twice as high. The calculator gives for the listed conditions equivalent airspeed (the one most closely related to dynamic pressure) 266 knots. Which is about exactly double (130 knots is reasonable landing speed) and that means four times as high dynamic pressure. $\endgroup$ Commented Jun 6, 2014 at 21:57
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The aircraft needs flaps in order to be slow enough for take-off and landing. This means the lift coefficient $c_l$ is larger in the landing configuration, and the dynamic pressure is proportionally lower, because lift is the product of $c_l$, the wing area and the dynamic pressure. Since the mass of the aircraft is roughly the same at the end of cruise and during landing, this product must be the same in both conditions.
Real-world examples: Double slotted flaps plus slats will give you a $c_{l\,max}$ of maybe 3.2, so your flight $c_l$ is around 2 to have enough safety margin. Cruise $c_l$ for the same airplane would be between 0.3 and 0.4.
Now with your more precise question I can give a more precise answer. Let's say wing loading is 500 $kg/m^2$, then your dynamic pressure in the landing phase will be 2600 $N/m^2$ versus 14000 $N/m^2$ in cruise. I guess this has not been obvious because you were stuck with imperial units. Do it with metric units, and things should be immediately clear. Formula: $$q = \frac{m \cdot g}{S \cdot c_l}$$
answered Jun 6, 2014 at 19:32
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$\begingroup$ I already know all of this. Essentially what I am asking is whether or not the flaps would be ripped off the airplane if they were deployed during cruise. It is not intuitively obvious for the reasons I outlined in the original post... $\endgroup$ Commented Jun 6, 2014 at 19:48
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$\begingroup$ @BrysonS. That's really more a structural question than an aerodynamic one (and the answer is different for each aircraft - one may be able to survive deploying flaps at cruise airspeeds, while another design may have the control surface torn from the aircraft). $\endgroup$– voretaq7Commented Jun 6, 2014 at 19:53
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1$\begingroup$ Normally, the maximum flap operation speed is 1.7 x stall speed. Speed here means calibrated speed, and your calibrated speed in cruise is higher by a factor of 5. So yes, the flaps will be torn off, no doubt, or they were designed much too heavy. $\endgroup$ Commented Jun 6, 2014 at 19:54
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$\begingroup$ @Peter Kampf. Yes I think this is generally what will happen, but I was trying to put a number to it. I will use the 1976 ISA to get a reasonable static pressure for both landing and cruise, but I still need a reasonable number for landing Mach (with full flaps deployed). Thoughts? $\endgroup$ Commented Jun 6, 2014 at 20:01
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$\begingroup$ @Bryson S. Don't make it more complicated than necessary. Mach in the landing configuration is irrelevant, because it is so small (warmer air plus slower speed). If you need a value, use 0.15. Static pressure is also not needed, only density, and that is part of the dynamic pressure already. $\endgroup$ Commented Jun 6, 2014 at 20:12