Flight physics problem

I have a question regarding flight physics, and I wonder if not someone here on these forums are qualified to help.

Take two airplanes A & B, where everything about them is identical except their mass, and B has twice the mass of A.

They fly at the same altitude under the same conditions and use the same engine thrust to achieve their respective maximum speeds.

Will the lighter airplane go faster and if so by how much?

According to my friend airplane A will always go much faster being half the mass, but according to me the difference will be negligible since air resistance is not dependent on mass at all, only the shapes which are identical.

Basically I am leaning on this equation: http://en.wikipedia.org/wiki/Drag_equation( where mass is not a factor ).

My friend is claiming though that mass is always multiplied into the final airspeed equation somewhere since the craft needs lift to maintain altitude.

I don't think that's right though since the force required to lift the airplane and maintain constant altitude should be constant regardless of speed, and the force required to overcome air resistance should quadruple if speed is doubled. So at maximum speeds where equilibrium is reached for most airplanes the force needed to maintain altitude should be very small in comparison to the force needed to overcome air resistance.

Note that this is not about takeoff or acceleration here, just maximum speed.

Thanks

Hell of a complex topic - to complex for a detailed answer.

But just this:

Lift induces drag; lift needs to be increased when mass increases. Lift can be increased (at given setup) by increasing airspeed or by increasing AoA. So here’s the latter. Ever seen (or heard) an aircraft’s engines when flying near max. AoA -> lots of drag. I think our F-4 needed 60% thrust just to maintain airspeed with the AoA alert being on.

So, simple answer:

Your friend is right.

Thanks for the answer, however I still struggle to understand why. Is really slow flight at max AoA alert relevant for this?

Isn't the angle of attack when flying at maximum speed generally very small?

According to this wikipedia graph it seems like (lift) induced drag at maximum airspeeds normally is negligible compared to form drag from air resistance. This also suggests that I am right.

Thanks for the answer, however I still struggle to understand why. Is really slow flight at max AoA alert relevant for this?

 

Isn’t the angle of attack when flying at maximum speed generally very small?

 

…

No, was just meant as an example to what extend AoA is responsible for induced drag.

The faster you fly, the less AoA you’ll have, sure. But it’s a tradeoff with airspeed when it comes to lift/drag.

And B naturally needs to have higher AoA when airspeed and setup remains const.

Generally your friend is as right as u are. Take cars as an example and you will find out, that u are right. The higher the weight of one car, the slower it accelerates to top speed. But top speed is the same in both cars, the heavier and the lighter one.

With planes it is almost the same, but for the mentioned AoA "problem". For the easyness of explanation in most learning books for ATPL/CPL training the drag part of the the AoA-factor of weight is neglegated in comparison to total drag. At least that's the case for the commercial part of pilot training. Obviously this is different in military training as AK seems to have learned a more precise definition than I did (or I ws asleep in PoF  :P ).

Generally your friend is as right as u are. Take cars as an example and you will find out, that u are right. The higher the weight of one car, the slower it accelerates to top speed. But top speed is the same in both cars, the heavier and the lighter one.

With planes it is almost the same, but for the mentioned AoA “problem”. For the easyness of explanation in most learning books for ATPL/CPL training the drag part of the the AoA-factor of weight is neglegated in comparison to total drag. At least that’s the case for the commercial part of pilot training. Obviously this is different in military training as AK seems to have learned a more precise definition than I did (or I ws asleep in PoF ).

I’d be interested in seeing an actual fuel calculation for a 747F for example. One for minimum weights, one for max., on a route A-B. The fuel burn for cruise overhead A - overhead B should be signifcantly different, not?

I wonder as you sound like there’s only neglectable difference.

I wasn't talking about fuel at all  ;)  ... of course, there will be a huge difference in fuel burn when simply calculating overhead to overhead. Not to mention the burn during climb and the lower max. initial cruise altitude. I bet a fully loaded 747F at MTOW won't get much higher than FL320 if at all, where an empty 747F will most probably reach FL390 with no prob.

I wasn't talking about fuel at all ;) ... of course, there will be a huge difference in fuel burn when simply calculating overhead to overhead. Not to mention the burn during climb and the lower max. initial cruise altitude. I bet a fully loaded 747F at MTOW won't get much higher than FL320 if at all, where an empty 747F will most probably reach FL390 with no prob.

When everything is constant, but only weight is double then we hardly have a case that’s realistic as your post shows.

The only thing we can say that will differ is needed lift for B. And increasing lift increases drag if we do the math.

That’s what you confirm with your post. A heavily loaded 747 can’t reach higher FL as it would need higher airspeed to generate enough lift. It would either overstress or stall (due to excessive AoA) - falling out of coffin corner so to say - granted its engines would deliver the required thrust at all.

In the end it all boils down to an increased lift-induced drag which can be seen in the higher fuel consumption. And I doubt the higher consumption is only the result of flying at lower altitude.

Back to the initial theoretical question.

If we only alter weight, then the only thing that generates the required lift is an increased AoA.

That's my main problem with the initial question.

When everything is constant, but only weight is double then we hardly have a case that’s realistic as your post shows.

 

Well in this argument we were only interested in theoretical maximum speeds possible to reach ( or if turned into a fuel consumption question how much extra is needed to maintain the same fast cruise speed with more weight ).

If using a 747F as example I think alot of the extra consumption would also come from the extra work needed to lift the thing up to cruising altitude, and to keep it up during takeoff, ascent, decent and landing when higher AoA are needed and lift induced drag will be a higher portion of the total drag.

So for that comparison to make sense in the context of our disagreement we only need to know the momentous consumption cruising at high speed at the same altitude and compare these.

So how much more would the fully loaded (say 2x mass) 747 consume once cruising? I doubt it would be 2x as much, but probably not the same either. If it's closer to 2x I'd concede that my friend is right, but below 50% extra consumption for twice weight and then drag from lift is the smaller portion of drag and I would say I was more right.

Yes but how much of the total drag is lift induced and how much is from the form drag? For a fairly realistic setting at either maximum level flight speeds or faster cruising speeds.

Heres a little insight from a sim junkie (who also has an ATP and CFI).

The only way someone is going to be answer this question specfically is if they possess both the mathmatical knowledge to accurately solve the equation AND they also have the specific numbers available for the particular example aircraft. These numbers are generally proprietary and are generally not found easily (coefficient of drag, coefficient of lift, etc). You could substitute these for some generic numbers if you could find them. I am not that well versed in mathematics to do so, even if I did have the numbers.

Generally speaking the answer is as indicated above: a heavier aircraft will fly slower due to the increased induced drag. In order to answer your specific question of “how much” slower or faster aircraft A or B will fly I conducted a little experiment using microsoft flight simulator. You can of course do the same if you have the software. This is a good way to bypass the need for the math as the simulator is solving the equations for you in real time.

Take any aircraft and load it to maximum weight and climb to cruise altitude. In order to keep this as realistic as possible I used a payware PMDG 737-900ER (with winglets) and climbed to FL280. I didnt climb higher as I wanted to stay away from the problems associated with the “coffin corner” and the backside of the power curve and I wanted to make sure I had a good buffer until hitting maximum mach speed (MMO) for the sake of the experiment,

Now make sure you have autothrottles disengaged if applicable and note the power setting and airspeed.

Next take your aircraft and bring it down to about 10% fuel this will bring the weight down considerably. In the case of my 737, about 50,000 lbs lighter, which is roughly 30% of the aircrafts MTOW.

What you will notice is that (as long as you kept the aircrafts throttle in the same position and autothrottles off) that the aircrafts speed will steadily increase. In the case of my 737 28 kts faster or about a 10% increase in airspeed (took about a minute).

This serves to illustrate that the induced drag created by a higher weight is definitely not neglible.

So back to your point of mass not being a factor in the drag equation. It acually is, it is just compensated for underneath drag, if you expanded the equation out you would see that mass is most certainly accounted for. Furthermore I see your point about induced drag becoming less of a factor as airspeed increases. This is much more difficult to answer, however I suspect it is because the “drag curves” chart you are looking at is a very large scope, and at the airspeeds on the extreme right of the X axis (airspeed) are so high that eventually induced drag becomes a negligible factor. Also keep in mind that Newtonian “lift” also increases with dynamic pressure (again the square of the airspeed). Another factor becomes the vertical component of thrust. At airspeeds in the extreme region, you would need so much thrust that even a 1 or 2 degree angle of attack could probably account of the entireity of the crafts weight simply from the vertical component of thrust from the engine.

Thats all I think I care to elaborate on at the moment but if you have any other questions I’ll go grab my aerodynamics books and try to explain in a different manner. I also know a few aerodynamics teachers I can contact if I cant answer better.

EDIT: The AOA during my experiment only changed about 1/2 degree or so. So I can only assume small changes in AOA result in significant changes in induced drag.

If you have the same shape and the same volume on the planes your friend is right, if you have different volume and shape you are right.

Thanks for the detailed answer OffDutyFO ( and everyone else that contributed earlier ). I think I understand the problem a bit more now.

There are still some things that are not clear for me though.

If lift induced drag was the only factor the same airplane that is 30% lighter should go sqrt(1/0.7) = 20% faster.

This tells us that your 737-900 example very roughly form drag should be about equally large as lift induced is since the observed speed increase was 10%.

But according to wikipedia we have this:

“The combined overall drag curve therefore shows a minimum at some airspeed - an aircraft flying at this speed will be at or close to its optimal efficiency. Pilots will use this speed to maximize endurance (minimum fuel consumption), or maximize gliding range in the event of an engine failure.”

At this minimum drag we can see in the graph that induced and form drag are equal.

Shouldn’t the airplane be going significantly slower then your example if it was attempting to “maximize gliding range in the event of an engine failure”?

The disagreement we originally had was also somewhat related to military combat airplanes. My gut feeling is that lift induced drag in speeds around or well above mach 1 should be if not negligible at least less then 10-20% of total drag. But I lack software like simulators ( and knowledge of how they work ) to be able to test this.

After the discussions here I think that what I am looking for actually is some real power curves ( like the wikipedia graph ) but with actual airspeed numbers on them for a few thypical airplane models.