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How to calculate minimum takeoff speed due to distributed propulsion?

Is there a quick way or rule of thumb way to calculate effects of distributed propulsion?

I understand NASA is doing research in this area and is determined to prove an increase in efficiency of 500% in real world tests, although this may include the benefits of electric distributed propulsion.

I understand they swapped the wings out on their Dornier, from 17psf to a 45psf wing loading. That's an increase of almost 300% in just the aerodynamics!!

So....

I understand the bigger the prop disc, the more efficient the prop.

if I have a small ultralight with 25 hp, and now use two 12.5 hp motors instead with newly optimized props....

What is the approximate % increase in efficiency/thrust as the prop disc is now twice as big? ( assume only a small weight increase)

Now let's say I have a 10' span, with 2 motors with 6' props. The entire wing and tail is in the prop wash.

What is the formula for propeller exit speed(v2)?

Is this my new indicated airspeed(v+v2)?

In other words, if my stall speed is 40mph and my prop exit speed is 20 mph at zero ground speed distributed over the entire wing, can I take off at a 20mph ground speed? i.e IAS= 20mph ground speed + 20 mph prop exit speed= 40mph need for takeoff.

Is this correct, at least conceptually, after all, it's just like a headwind, no?

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The better way to look at predicting the performance of a distributed propulsion wing (specifically the NASA X-57 wing concept) is to think of the propellers along the span as a high-lift device that augments the lift of the wing through interaction with the trailing-edge flaps, rather than as headwind-generators. In other words, the propellers increase wing C_L, allowing you to have a smaller wing for the same landing and takeoff distances.

With many small propellers, you get performance that starts to resemble a jet flap or externally blown flap, but without the complexity of internal ducting or non-uniformity due to a small number of engines.

Optimistic rule of thumb from thin airfoil theory is that, for a 2D section $c_\ell = \frac{\partial c_\ell}{\partial\alpha}\alpha + \frac{\partial c_\ell}{\partial\delta_F}\delta_F$

$\frac{\partial c_\ell}{\partial\alpha} = 2\pi(1+0.151\sqrt{c_J}+0.219c_J)$

$\frac{\partial c_\ell}{\partial\delta_F} = 2\sqrt{\pi c_J}\left(1+0.151\sqrt{c_J}+0.139c_J\right)^{\frac{1}{2}}$

where $c_J = 2\frac{h_J}{c}\frac{V_J}{V_\infty}$

$V_J$ can be calculated from propeller momentum theory based on disk area, input power, and assumed efficiency.

Bear in mind this is for section $c_L$, and will need 3D corrections for the whole vehicle based on percentage of span with propellers, induced flow, etc. It is also reasonable only in the limit of small flap deflections and angles of attack. Wind tunnel data or flight test reports might be a better place to get a first-pass estimate.

https://www.researchgate.net/publication/299641308_Comparison_of_CFD_and_Experimental_Results_of_the_LEAPTech_Distributed_Electric_Propulsion_Blown_Wing

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740003719.pdf

https://arc.aiaa.org/doi/abs/10.2514/6.2019-3170

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    $\begingroup$ You have great content and great links. A helpful way to combine to two is by using the hyperlinks as references in line with the content. $\endgroup$ – Mark Jones Jr. Aug 7 at 16:09
  • $\begingroup$ If NASA swapped out the wings of their test plane, from a wing loading of 17psf to 45 psf, that's a factor of 2.6. I understand their test plane does not leading edge slats, but does have slotted flaps, so max cL is about 50% x 1.5 plus 50% 2.8 = max cL of about 2. What that tells me is that the new wings generate 2.6 x 2 = 5. So... distributed propulsion can raise the cL from 2 to 5. Is it really that much?!?!?!? $\endgroup$ – Fred Aug 10 at 13:48
  • $\begingroup$ Yes, maximum predicted CL of the X-57 is in the 4-5 range. But in principle with sufficient power it's possible to generate much higher values, up to 10 or more. $\endgroup$ – Chris Aug 10 at 23:34

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