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so I know that AR=b^2/S.

But let’s say that I don’t know b or S, how can I estimate the optimal AR if I know only this:

How many passengers are going to be onboard, cruise velocity (subsonic), altitude, range and a very rough configuration.

Endurance: around 500km worth Cruise: 180knots Altitude: 2km Payload: 220lb

Is there something that is used just to help estimate please, I read a book and it said ‘initial estimate made from historical data’?

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    $\begingroup$ Do you mean how would you estimate the optimal aspect ratio in such a case? If so, please clarify. The question as posted is not answerable. $\endgroup$ Jan 29, 2022 at 19:56
  • $\begingroup$ PS even just to suggest the optimal aspect ratio, it appears we might also need to know things like intended cruise altitude, intended range, etc- basically it appears you are asking us to design the whole airplane for you based only on the number of passengers, which appears to be way too broad a question. $\endgroup$ Jan 29, 2022 at 20:01
  • $\begingroup$ Hi, sorry I haven’t included numbers on purpose. I just wanted an indication of how to do it please so I could try myself first. Then come back if I need help $\endgroup$ Jan 29, 2022 at 20:18
  • $\begingroup$ Ok, I just suspect we may need more info. Perhaps not, perhaps someone will post an answer showing that what you have provided is all we need to know. But again that would be for some sort of optimal aspect ratio. (See first comment.). Well, we'll see how this goes... $\endgroup$ Jan 29, 2022 at 20:22
  • $\begingroup$ Hi, about the optimal AR, how is that obtained please? $\endgroup$ Jan 29, 2022 at 20:38

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What you do in this case is what every conceptual designer does: Use parametric data. This is a compilation of data from similar designs which yields numerical value equations (= the units on both sides don't match). Typical collections are the series of books published by Jan Roskam. But you could as well compile your own, using data from Jane's or Wikipedia.

As an example, here are the equations for wing mass $\mathsf{m_W}$, valid for small, propeller-driven airplanes, from Roskam's book:

$$\mathsf{Strutted:}\;\mathsf{m_W}\mathsf{=0,015\cdot\left(1,2-\frac{m_{Fuel}}{m}\right)\cdot\left(1,5\cdot n_{z_{max}}\right)^{0,611}\cdot S^{1,018}\cdot AR^{2,473}}$$

$$\mathsf{Cantilever:\;m_W\:}\mathsf{\mathsf{=0,068\cdot m^{0,397}\cdot\left(1,2-\frac{m_{Fuel}}{m}\right)\cdot\left(1,5\cdot n_{z_{max}}\right)^{0,397}\cdot S^{0,36}\cdot AR^{1,712}}}$$

Interestingly, the equation for the strutted wing only depends on fuel fraction $\mathsf{\frac{m_{Fuel}}{m}}$, load factor $\mathsf{n_{z_{max}}}$, wing area $\mathsf{S}$ and aspect ratio $\mathsf{AR}$, not the speed or range of the airplane. This shows that all data used for this equation came from designs with similar range and speed.

The equation for the cantilever wing already includes the take-off mass $\mathsf{m}$ by itself but is similarly from a small range of designs.

Since the units of both sides don't match, you need to be double sure to use the correct dimensions in this type of equations. Here, they are valid vor proper SI units: Kilogramm and Meter.

Starting with the number of passengers and a rough idea of the range, you can calculate the payload (100 kg/pax for short range or 161 kg per m³ of cargo volume) and from the payload fraction of similar airplanes you get a first estimate for take-off mass (MTOW).

Since you seem to have a fast, unpressurized single-seater in mind, we could as well calculate the mass of all parts. Fuselage mass $\mathsf{m_F}$ for small airplanes, according to Roskam, is

$$\mathsf{m_F=0,08378\cdot\left(n_{z_{max}}\cdot m\right)^{0,315}\cdot l_F^{0,943}\cdot\left(h_F+w_F\right)^{1,1}\cdot v_{D}^{0,372}}$$

Here we find length, height and width ($\mathsf{l_F, h_F, w_F}$) of the fuselage and the dive speed $\mathsf{v_D}$, so another unit to clarify is Seconds. $\mathsf{v_D}$ has to be in $\mathsf{\frac{m}{s}}$.

To complete the mass calculation you need to add empennage mass, systems (glass cockpit or only a few steam gauges? Retractable or fixed gear?) and engine. The power needed can be derived from the L/D of the plane at this speed, propeller efficiency and airplane mass. Let's assume an L/D of 8 and a propeller efficiency of 0.7 (follow the links to see how optimistic those estimates are!) and you have the installed power $\mathsf{P}$ needed. From that it is only a small step to engine mass $\mathsf{m_E}$:

$$\mathsf{m_E=0,68\cdot P\cdot\frac{2400}{n_{max}}}$$

with $\mathsf{n_{max}}$ the maximum RPM of the chosen engine. The range and fuel consumption of the engine tell you how much fuel to add. Once you know this, your parametric tank mass is

$$\mathsf{m_{Tank}=0,614\cdot m_{Fuel}^{0,667}}$$

Once you have an idea of the take-off mass, you look up typical span loadings (kg of MTOW per m of wingspan) and arrive at a first estimate for your wingspan. Next, you make some heroic assumptions about the wing's sweep (should be zero), airfoil and flap system to be used and, together with the maximum lift coefficient those provide, you calculate a wing area that both allows for a desired minimum speed and cruising altitude. With that information, you can go back to mass estimates and get a better idea of your wing weight and the thrust needed, allowing you to narrow down the estimate for engine weight. Given your relatively high cruising speed of 92.6 m/s, you should aim for a small wing with powerful flaps.

That helps you to update the previous calculations. Hopefully, the numbers converge in the successive steps and your idea of how the airplane and its components look like get more and more concrete.

Note that aspect ratio is not a design parameter at this stage, but merely the result of the chosen wingspan and area.

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  • $\begingroup$ Has the $\left( 1.2 - \frac{m_{\textrm{Fuel}}}{m} \right)$-term been added lately? In my edition of Part V - Component Weight Estimation that term is not there there $\endgroup$
    – Robe
    Feb 1, 2022 at 9:32
  • $\begingroup$ @Robe Hey, cool, someone who checks my equations! I converted the numbers to SI units and added the fuel part from the German LTH. For a 20% fuel fraction the results are the same. Note that I also replaced n$_{ult}$ with 1.5·n$_z$. $\endgroup$ Feb 1, 2022 at 14:25
  • $\begingroup$ Ah, good to know! Last year I did some mass estimation of FRP wings with a high aspect ratio. I ended up calculating my own fudge factors, but of the three formulas for GA planes provided in Roskam's book, this one had the largest error. Reading your answer, I though that it had perhaps been updated. $\endgroup$
    – Robe
    Feb 1, 2022 at 15:17
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AR isn’t really something that falls out. It’s something that’s set/chosen by the designer. Once you know how much lift you need to generate, and given the design lift coefficient you can compute the wing area required. At that point you get to choose whether you want long skinny wings or short stubby wings. Other design considerations may come into play here. Is wingspan limited by hangar constraints? Or airport terminal constraints. What kind of roll rates do I want to achieve? How much fuel will I need to store in the wings? For a given length how much will the wing weigh? You either decide what AR you want and go with span and chord that gives you or you decide what span you want and go with the chord and AR that gives you.
What your book probably meant was that you could look at existing planes with similar specs and choose something similar.

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