I have been trying to come up with my own formula (that differs from the various CAFE formulas to have a weighting that suits me better). I found another interesting comparative formula, the AEI.
AEI = (W0*U0)/hp
where W0 is the gross weight in lbs
where U0 is the free stream velocity in ft/s
where the hp is the horse power required
In other words for Diamond DA40 this is:
W0=1200 kg = 2640 lbs
U0= 214 ft/s (127 kts cruise at 10000 ft)
hp_cruise=90
=>
AEI(Diamond DA40) = (2640*214)/90
AEI(Diamond DA40) = 6277
Unlike the CAFE formula this has no weighting for these parameters.
If I would like to make my formula based on this, I might want to weight the hp a bit more.
That is because the lower the hp figure gets, the better is the fuel economy if everything else remains constant.
Why this index is good? Because it isolates aerodynamics from structural engineering and does not care how much useful load the craft has. It only considers the aerodynamically important point, how much power is required to move the mass forwards and at the same time keep it on air.
12 comments:
The power required to overcome aerodynamic drag is
P ~ k*v^3
The formula shows that the aerodynamic drag increases very rapidly at some speed and eats all the power (what ever one might have). So I think you should either use the same speed to compare different planes or somehow compensate for the difference of the speed.
You cannot just use linear formula since the meaning of the speed is so important and would give false results if not using exactly the same speed for all planes when comparing.
The speed is also important feature of the plane in addition to economy. But it is secondary to economy, but still important, if you have very good economy but you are not getting anywhere, the plane is not very useful.
It is impossible to compare planes at the same speed without test flying them all at the same speed. Laws of physics don't make the comparison any more fair.
Every plane has different optimal economy speed because of wing loading, drag polar etc., so comparing them at the same speed does not tell about the efficiency of the plane. Going faster or slower makes the economy worse.
I don't really know two identical planes that would have exactly same parameters for being able to compare at the same speed, hence comparing at the same speed does not pay off.
However, planes shall be compared at the same design point: e.g. economy cruise power. The speed is different, but comparing planes from the point they are maximally efficient is a better way to do a comparison.
Actually I am already using that sort of - the DA40 specs in the sheet are the economy specs. At 75% power it cruises faster, but consumes 2 gallons per hour more gasoline. To get the optimal range, we fly it a bit slower and with lower power setting.
One alternative value for comparative performance is lambda. It was used for comparing the efficiency of different planes in some paper I recently looked at.
Ok...
Basicly you could convert all the speeds to some unit speed and recalculate that "AEI". Here is how:
Since your
AEI = Wo*Uo/hp
and on the other hand each plane will typically have
hp = k*Uo^3
First solve k for any plane:
k = hp/(Uo^3)
Set Uo to 1 and get
hp = k (since Uo = 1)
Now solve AEI
AEI = Wo*1/k = Wo/k
That will now give you a number that puts all the planes to the same speed 1.0 .. or use anys other same speed for all planes using that conversion method.
Remembering that if you know the speed and required hp at some point all other points can be calculated using:
Power = k*speed^3
Your formula ended up with
AEI = Wo*1/k = Wo/k
That can be also written:
W/P = weight to power ratio = power loading
What do you think this can indicate?
In other words, are you proposing solving weight to power ratio for each plane to be compared and then making some observations from that?
Or do you rather mean (where the engine size is not in consideration in relation to the weight, but the amount of power required):
gross weight / power required
Can you please elaborate how to interpret the results. Because I think that with your formula, the best score comes on a plane that has enormously large wings, very low wing loading and very low stall speed, which is optimal for flying at very low speed at high angle of attack at high cl (=sailplane). How do you extrapolate that to the requirement for going fast as efficiently as possible (low angle of attack, low cl), kilometers cruised per burnt fuel liter in a reasonable spent time?
That is not the same as W/P. What I have shown is Wo/P(U=1).. or Wo/P1.
But if the unit speed bothers you change that to anything else (for example 100 mph or 100 kts, which most planes can do) if that makes more sence. But use that formula to but all the planes to the same working point.
The point is that you can not make any reasonable linear comparsion when there is speed in the formula since the speed affects the performance nonlinear. The output of such comparsion would be totally random if the speed is random.
An example of such a matter is the sport car Bugatti Veyron. It needs 1000 hp, why? That was the only way to reach 250 mph after what ever they did to get the best aerodynamics in it. And why is that? Since the air resistance crows very rapidly when the speed increases. Why? Since the speed affcts as
P ~ k*v^3
and not
P ~ k*v
to the required power.
You have to give aerodynamic credit to those planes which fly at higher speed since they have to win MUCH LARGER resistance. Speed is not just a linear number.
But the point is not giving credit to the speed^3 but for giving the credit for producing optimum in relation to the use case.
I am looking at this because people come to say that "This plane X is much more efficient than that plane Y". I need some way to put them on line so I can compare. The issue is that speed is important, but also same thing goes for the fuel consumption. It would be great then to be able to find a formula that compares different aircraft so that the value of dozen variables becomes one number that is sortable in OpenOffice.org sheet. I have made my opinions which aircraft I prefer, but to explain that to others, it would be helpful if I can present that "here is the formula, do the math and see, that particular plane would not be good for me (i.e. not the compromise that I am looking for)".
I get a lot more comments than you can find from the blog, and to explain all of them what I am looking for is usually pretty tedious. I would like to find a faster way to tell what I want and what I do not want, and I believe that it can be represented with one single number, similar to CAFE formula, but changed so that it gives highest score to a compromise that is most beneficial for *my* use case.
Additional note about Bugatti Veyron:
- Airplanes have a magnitude lower drag always than even the most aerodynamic cars, because the cars have always turbulence underside and lots of friction from the tires.
Airplanes have higher drag on the ground too. That is why a lot more power is needed than is necessary to keep the plane level flight or climb, to take off. And also the fastest small aircraft that can fly faster than Bugatti Veyron can drive, will lose to Bugatti Veyron on the ground big time. Actually they will propably lose to even to Toyota Prius on the ground.
Example: Our Diamond DA40 accelerates long time on the ground at gross weight. The speed slowly rises to rotation speed. I have never tested the top speed of the aircraft (because that would be dangerous) on the ground, but I would estimate it would not be much more than 80 kts IAS. The plane on the other hand can achieve 140 kts IAS in flight at low altitude (1000 ft) but away from the ground. The difference of drag on the ground and in the air is significant, and thus the car comparison is not really the best one for aircraft.
To elaborate more: Raymer's takeoff parameter equation could indicate that aircraft always takes off, no matter how heavy it is. However, in reality the situation is worse. Where the takeoff parameter gets only high on paper, it can get infinite in reality. It is possible to design an aircraft that takes off on paper, but never takes off in reality because it can not achieve the rotation speed on the ground.
Karoliina, could you please (I'm assuming you have the time and energy) compare a couple of, say 5 different general aviation aeroplanes using your formula and post the spreadsheet with some conclusions here. Now, I'm not suspecting the usefulness of your formula (yet), but I have to say I think at the moment Exo has the upper hand in your debate...
Actually I believe that what Karoliina is trying to point out is that her efficiency index compares aircraft at their designed optimum flight speed.
What she's trying to do is not to compare power to weight or consumption to weight ratios, but compare the different aircraft as a complete entity (airframe, engine, propeller, retractable landing gear or not) to each other. As she pointed out, if you follow only Exo's formula you favor sailplanes for efficiency.
Which, in pure aerodynamics terms, is true. However she is trying to make the assertion that each aircraft has a fixed efficiency ratio based, not on speed, but on its airframe. The aerodynamics & power mount determine that optimum cruising speed.
Unless I'm somehow mistaken?
I mean, there are many different ways to compare efficiency: power or consumption to load weight (flying weight - weight of fueled aircraft), etc. There are very many ways in which efficiency can be measured. Pure aerodynamic comparisons are not a good measure of a practical powered aircraft.
Also, referring to the slow speed increase when on the ground - I believe that what you need to account for there is acceleration power, which varies for propeller and turbine powered vehicles, depending on the speed the vehicle is currently at. For instance a turbine's power output increases when the airspeed of the aircraft increases.
OK, so now I have two planes. They have AEI's of 4768 and 5268 (presuming my mathematical incompetence has not effected the calculations). Based on AEI, which one is better? Or rather, which one is better for what?
This comparison might prove whether AEI is usefull or not...
The planes can both be found on december issue of Flying magazine ; )
Juri,
If I'm reading this right, and I'd like Karoliina to confirm my understanding of this, the higher the AEI, the more efficient of an aircraft you have. The reason for that is that you transport a higher weight at cruise speed with less horsepower.
Thus the horsepower requirement per pound at cruise speed of said aircraft gives you a rough idea of how aerodynamically efficient it actually is compared to other aircraft.
The key here is how do you choose the U0 and hp. If you wanted to make a benchmark for a set speed, you'd be making an uneven comparison, again because each aircraft has a different cruise speed by design.
Goeland, your last paragraph points out the same problem that my example comparison was ment to display. Both the aircraft in the comparison are Diamond DA42, one of them with Austro diesels, the other with Lycoming gasoline engines.
Just by swicthing engines, you get different performance and therefore a different AEI. What is interesting is that in terms of performance, DA42 with Lycomings and lower AEI is better in everything else than cruise performance at higher altitudes...
And what actually makes the difference in this case is that the Austros installed are turbocharged, while Lycomings are not.
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