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First a definition that you do not think of the wrong thing.
 
 
There are 3types of casings for props:
1) Prop Protection
Something that you do not hurt your fingers and the prop will not break on hit.

2) Ducted Fans  (Auf Deutsch: Impeller)
Very high rpm, very loud, very ineffective for hovering.


3) Shrouded props
They need to have a special shape.
And the distance between prop and shround needs to be as close as possible. About 1mm.


 
 
Shrouded props:

 
Papers:
Design and Testing of a Quad Shrouded Rotor Micro Air Vehicle in Hover
 
hover and wind-tunnel testing of shrouded rotors for improved micro air vehicle design
2008
Hover: Performance
At the same power consumption, increases in thrust
over the open rotor by up to 94% were observed, or, conversely, up to 49%
reductions in power at the same thrust
Within the range of values of the shrouded-rotor parameters tested, the optimal
configuration was that with tip = 0.1%Dt, rlip = 13%Dt, d = 10 and
Ld = 50% to 72%Dt.
Translational flight:
In axial flight ( = 0), the net thrust of the shrouded rotor was initially
greater than that of the open rotor, but rapidly decreased to being lower than
the open-rotor thrust as airspeed was increased, possibly because of the drag.
In edgewise-flow conditions ( = 90), the thrust of both the open and shrouded
rotors increased with increasing airspeed, with the thrust of the shrouded rotor
always remaining higher than that of the open rotor.
Hover Tests of Micro Aerial Vehicle-Scale Shrouded Rotors  
Experimental Investigation of a Shrouded Rotor Micro Air Vehicle in Hover and in Edgewise Gusts 1. Airfoil: In both the shrouded and unshrouded rotor congurations, the use of a sharpened leading edge (LE) with a circular camber airfoil yielded better hover performance over an unsharpened leading edge airfoil.
3. Shrouded rotor performance comparison: At an operating thrust of 300 grams, the power loading of the shrouded rotor was approximately 30% higher than the unshrouded rotor
 
 
 
 
 
 
 
 
 
 
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