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I've heard many times that the smaller the gap between the fan blade and the duct, the better the efficiency/thrust is.

Is there a formula or graph available to see how much efficiency is lost as you increase the distance from the fan blade to the duct?

duct fan

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    $\begingroup$ The answers here might help. $\endgroup$
    – Wyatt
    Feb 6 at 4:22
  • $\begingroup$ You may look at figure 10 - variations in efficiency and leakage with increased clearance, in this document. $\endgroup$
    – mins
    Feb 16 at 14:14

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Even for just a simple propulsive fan it'll be hard to find one formula incorporating the many practical input variables: blade radius, blade count, blade twist, rpm, airspeed, air density, rotor position in duct, duct narrowing, etc. I find only measurements and simulations of particular families of these, e.g.

Performance Investigation of Ducted Aerodynamic Propulsors, Bi et al, SMP 2009,

whose conclusion, supported by measurements shown in Fig. 16 (for the same thrust, a gap of 1% to 3% wastes about 10% more power than zero gap), dares to claim nothing more rigorous than:

Propulsor efficiency is insensitive to blade tip clearance when the tip gap is between 1%R and 3%R. However, as the tip clearance approaches zero, propulsor efficiency can be quite sensitive to gap size.

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