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I am trying to run numbers on a concept VTOL fixed-wing aircraft, and I am looking at having fans or props positioned inside the wings, (which would therefore make the wings relatively thick for a fixed-wing aircraft). Naturally this would see the fans/props enjoying the benefits of a duct, but with minimal downsides such as the need to add weight/material as the wing is already going to be there.

I am using Blade Element Theory to trial propeller dimensions/numbers to see what kind of sizes I would need to achieve the VTOL (if I can minimize the radius of the VTOL fans then I can the wings thinner or with a smaller chord), but I don't know of any way to estimate the influence that the duct would have on the final thrust/power values.

If there is a rule of thumb for this, or a database, or some equation or anything, that would be a great help!

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There is info on power computations for ducted propellers in J. Gordon Leishman, Principles of Helicopter Aerodynamics. The beneficial contribution of the duct can be found in section 6.10.1 Fan-in-fin Design, and in section 3.3.10 Prandtl's Tip-Loss Function. In short:

enter image description here

  1. If the duct is designed to prevent wake contraction, extra thrust is generated and less power is required to drive the propeller. The effects are computed with momentum theory, which is a simplification. For thrust T and power P, with $T_D$ and $P_D$ = ducted, $T_U$ and $P_U$ = unducted, $a_w$ = ratio of outlet/propeller area.:

$$\frac{T_D}{T_U} = \frac{1}{2a_w} \tag{1}$$ $$\frac{P_D}{P_U} = \frac{1}{\sqrt{2a_w}} \tag{2}$$

  1. Tip loss. The duct prevents circulation loss at the tip if the clearance between tip and duct is small enough. Prandtl has been working on this in 1919 using momentum theory, an example for an untwisted blade is depicted below.

enter image description here

There is also this masters thesis, with measurements of small ducted rotors with low tip speeds.

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Hindawi Publishing Corporation International Journal of Rotating Machinery Volume 2012, Article ID 474785, 11 pages doi:10.1155/2012/474785 Research Article Open-Water Thrust and Torque Predictions of a Ducted Propeller System with a Panel Method J. Baltazar, J. A. C. Falc ao de Campos, and J. Bosschers

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