# How can I find the optimum chord wing length for this wing?

This Aviation SE forum looks the perfect place to ask this question: people here usually make maths the root of most solutions and I think this may solve my aerodynamic optimization problem!

I am having a very hard time optimizing the wing size of my rc plane. I already went to the excelent RC GROUPS community but, despite the generous help of many people, this subject got a little more confusing and personal opinions (practical experience) are conflicting.

My question is this: I already have an rc plane (with tail, fuselage...) and I want to optimize the wing size in order to get efficiency and the longer flight time. I already built dozens of rc planes but this time I want to make this one the most efficient as possible.

IMHO to get longer flight time I need to consume the least amount of energy from my lipo battery as possible - so I need to keep my brushless motor running in the slowest RPM possible, consuming less Amps. Having said that, my total plane weight (not considering the wing, of course) is around 300g (summed weight of ESC, servos, 1000kv motor, 11 inch prop, fuselage, tail, receiver and battery). The weight is fixed, 300g. My wingspan is also fixed, 90cm. I cant have a wing longer than that. The wing shape is rectangular (I know elipses would be the better shape but let's fix the wing shape to simplify this discussion).

If everthing is "fixed" so what is the variable? Well, I want to find the optimum chord wing length for this plane which is gonna bring me the highest flight duration.

I am using the CLARK Y airfoil but you can assume any other one you want. I wouldnt like to discuss airfoil, I would like to discuss the size of the chord and how to optimize it.

My experience and lots of study into this subject says that high aspect ratio wing are usually more efficient than low AR. My wingspan is fixed and all other variables are fixed (to make this discussion more specific) except chord.

So my dear friends, I am struggling a lot to find the optimium chord to get the best flight duration. It looks easy but I am almost 2 months trying to get this answer from many different sources and I still cant find a good solution.

PS: I know I could optimize many of my other fixed parameters, but to avoid having this question marked as too broad/global/generic, I really would like to only optimize the wing chord.

PS (2): the plane that I want to optimize the chord is in the image below. The wing that I show on the picture already provides good flight time (around 22 minutes) but I would like to optimize it in order to get better flight time.

• What if you left it as is, but did things to reduce drag? Like add fairing at right angle intersections, rounding the nose, getting rid of the square edges on wing/tail leading edges ... – CrossRoads Apr 18 '18 at 18:24
• @CrossRoads thanks for helping, but as I said this plane flies well. Despite that, I am looking to optimize every single part of this plane. But for now I would like to keep things fixed and only optimize chord. After that I will certainly optimize the other things. My plane is far from perfect, so assume my question is related to an already perfect/ideal plane, what would be the optimum chord for this ideal plane? – Samul Apr 18 '18 at 18:29
• Try the airfoil tool(s) here, airfoiltools.com/airfoil/index and m-selig.ae.illinois.edu/ads.html Might have to do some reading to find the best shape for gliding overall, and then experimenting with your specific wing length and depth. – CrossRoads Apr 18 '18 at 19:07
• @CrossRoads the airfoil database I used a lot, it's a great resource! But in this case I am researching about chord size specifically, not related to any specific airfoil. Assume no airfoil in my question if it would help to provide an answer! :) – Samul Apr 18 '18 at 19:15
• I don't know how you seperate airfoil from chord en.wikipedia.org/wiki/Chord_(aeronautics) Are you just looking for the best distance front to back with a perfectly flat wing? – CrossRoads Apr 18 '18 at 19:17

You want to fly at the polar point of the lowest energy consumption. If you describe the drag with the usual parabolic equation, you can use the optimum point derived in this answer. It is $$3\cdot c_{D0} = \frac{c^2_L}{\pi\cdot AR\cdot\epsilon}$$

Induced drag is three times the parasitic or zero-lift drag when you fly at the polar point for minimum energy loss. But the minimum induced drag at a certain speed is also fixed once wingspan and mass are given - you can only mess up by creating a non-elliptic circulation over span, but never get better than this limit.

Now get the zero-lift drag down as much as you can (pick a thin, highly cambered airfoil which produces a high lift coefficient at the given Reynolds number, definitely no Clark Y) and size chord such that this airfoil will work at the optimum lift coefficient.

Note that this will be an iterative process. Picking a chord will influence your aspect ratio and zero-lift drag, so the optimum lift coefficient will be depend on the chosen chord.

• I promise I read your answer almost 10 times and still didnt get it. One thing that I really cant agree is that you told me to get a high cambered profile BUT it generates lots of drag and I think this is not true cause most of the most efficient planes (gliders) are not cambered at all. – Samul Apr 19 '18 at 19:39
• @Samul: Those gliders are optimized for best $L/D$; you want minimum sink speed (best $L^3/D^2$). The drag coefficient is higher there, but the actual drag is smaller because of the lower flight speed. And gliders do have camber, some even flaps to increase camber for slow flight. Also, an airfoil with a high lift coefficient allows to use less chord, so there is less surface for the drag coefficient, and it is the product of drag coefficient and surface (times dynamic pressure) that makes the drag force. – Peter Kämpf Apr 19 '18 at 20:23
• thanks for your help. Now I understood better what you said but I still cant get the iterative process. If you could tell more the steps in the iterative process I would more than glad to accept your answer since it looks exactly what I am looking for! – Samul Apr 20 '18 at 19:17
• @Samul: It's not that hard. Pick a chord - any chord. Then compute AR and the optimum polar point. Now see where you are: I would expect a $c_L$ somewhere between 1.2 and 1.5 (depending on Reynolds number). If not, repeat. When you get close, select an airfoil that has a good figure of merit ($\frac{c_L^3}{c_D^2}$) and refine your work for that particular airfoil. Generally, a higher $c_L$ without separation will produce the best results. – Peter Kämpf Jul 18 '18 at 21:25

My final year design project for my undergrad (mechanical) looked at designing a UAV to get the best lift to drag ratio under several constraints (wingspan, max weight, etc.) set by our advisor. One of the tools we used to find the optimum reference area (S) for our preliminary design is a matching chart. An example of one is shown below:

The matching chart plots Thrust to weight ratio (T/W) against your Wing Loading (W/S). The lines you see are obtained by plotting your performance constraints. Any point on the hatched region does not meet your requirements.

The match point shown in the picture gives you your optimum T/W and W/S according to your inputs. This is the point where your T/W and W/S are at minimum, while still meeting your requirements.

From your match point you can get your reference area if you know the total weight of your plane.

Snorri Gudmunsson's textbook, General Aviation Aircraft Design - chapter 3, has a section on how you can make a matching chart.

• thanks for helping. After you teaching me about this "matching chart". I searched a lot about it on google but I still couldnt find a way to mathematicaly optimize the chord (or Area as you stated). I am amazed by how incredible important is this question that I am asking and the total lack of content online to help with that. I think this shouldnt be that hard, I just want to optimize one single parameter: chord. Do you know why is it so hard to find a bibliography about this optimization? – Samul Apr 18 '18 at 21:07
• Another option is to model your plane on XFLR5 and play around with the different chords until you obtain your desired performance. There's plenty of tutorials on youtube on how to use XFLR5. I guess there's not much bibliography because the optimization really depends on what type of flying you want to do (i.e. acrobatics or light flying) – LaVolpe Apr 18 '18 at 21:20
• XFLR5 is great but I dont know how to translate the analysis data into "efficiency". The lift/drag diagram is great but if I compare two exact planes with different chords, how will I know which one is better? – Samul Apr 18 '18 at 21:21

Since you are looking for maximum endurance, the general configuration you want would resemble a sailplane as far as possible. Wingspan is most important. Aspect ratio really only matters as to how it affects span.

Since you have already constrained weight and span, the next most significant factor is wing loading, directly related to your objective to select chord. Some RC modelers use Wing Cube Loading to determine the appropriate wing area since it works like wing loading but is less affected by scale. Here is a description:

http://www.theampeer.org/M1-outrunners/M1-outrunners.htm#CWL

and here is a calculator

http://www.ef-uk.net/data/wcl.htm

WCL for gliders is less than 4 and up to 8 for sailplanes. Using your 300g weight constraint translates to 12-18dm2 of wing area. Dividing by your 9dm span constraint means chord should be in the range 1.3-2dm.

• thanks, nice links you provided (even the calculator is great). However I would like to know the ideal/optimal chord (actually I would like to know the theory/formulas used) not a range. Imagine this: if I plot a graph of EFFICIENCY x CHORD then I am pretty sure that the graph will be someting like f(x) = -x^2 where there will be an optimum point where the chord gets the most efficiency posible. – Samul Apr 19 '18 at 0:10
• I doubt there is any such graph that does what you ask because aircraft are not designed a single parameter at a time. The interaction of lift, drag and other parameters have much more effect on the result than a few percent difference in chord. Your current construction methods cause considerable drag, so I would guess that the glider sized wing would be preferable. If you cleaned up your aero (at least the square leading edges) and carried an airfoil to the tip the sailplane size would likely be significantly better. – Pilothead Apr 19 '18 at 2:59
• I intend to do the optimizations you rightly said here, but first I want to get the chord right. When you reply me in the other thread I think I will get what you mean and I am pretty sure that's in the right direction! – Samul Apr 19 '18 at 19:30

Maybe you can get something from the Daedalus:

The MIT Aeronautics and Astronautics Department's Daedalus was a human-powered aircraft2 that, on 23 April 1988, flew a distance of 72.4 mi (115.11 km) in 3 hours, 54 minutes, from Iraklion on the island of Crete to the island of Santorini. The flight holds official FAI world records for total distance, straight-line distance, and duration for human-powered aircraft.... Airfoils Mark Drela had written the program XFOIL which enables the design of aerofoils and which can accurately predict the performance at any Reynolds number which is actually superior to a wind-tunnel test of a proposed section. The wing sections never existed on paper, either as lists of numbers or as drawings. The information came out of the design program onto disc, and this guided the cutter. The spar of the MLE (Michelob Light Eagle) was three tubes aligned vertically.

This scheme was abandoned for the final Daedalus design where the more usual single tube was used. Construction of Light Eagle took 15,000 hours work by 18 members of MIT. The airfoils which were used are: DAE11, DAE21, DAE31, and DAE51.

• Wish I knew what tags or whatever are being used to create the quotes and show the images like that, I'd try and start using them more. – CrossRoads Apr 18 '18 at 19:33
• You can click "edit" and see the markdown, or use the buttons at the top of the edit box. – fooot Apr 18 '18 at 20:12
• @CrossRoads thanks for researching and bringing this plane in but actualy it does not help me to discover the best chord possible. It is indeed a great performance plane but I have fixed contrainsts that I would like to keep fixed. BTW it was not me who downvoted. – Samul Apr 18 '18 at 21:03
• Here's a link to some of the basic markdowns. I have it bookmarked and refer to it all the time. – TomMcW Apr 18 '18 at 21:36
• Thanks TomMcW - I have bookmarked that now also. – CrossRoads Jul 19 '18 at 14:16