### How much More Thrust Is Produced By A Shrouded Propeller Compared To An Unshrouded Propeller When the Power Remains Constant?

There may have been more than one discussion of shrouded props, but after reading the threads and a great deal of the reference material on prop design, I am more interested in a variation on previous discussions  I also have more to ask that I have never seen discussed about the actual prop design when it is used in a shroud.

If we hold the prop size (length) constant due to design constraints, and we hold the power input constant, how much will the static lifting force of this prop change?

We all know that a larger (longer) prop is more efficient than a shorter one. But this is not the question that interests me. For a given design of a multicopter, can it be made to produce more static lifting force if we add a shroud to the propellers? Once we know the improvement of static lifting force at the same power input, we can determine if this added force is great enough to lift the added mass of the shroud.

Let's say that the shroud is an integral part of the design for safety reasons.

Let's say that the flying speed of the muticopter is slow. This design is to be used for hovering and more specifically for inspections when large and fast movement is not the goal. Stable smooth hovering is the intended flight characteristic. other than for the purposes of maneuverability for the sake of safety and collision avoidance, the normal flight speed of about 1 MPH to 3 MPH will be adequate. This translates to a speed range from 1.5 fps to 4.5 fps. (0.46 m/s to 1.34 m/s). I add these details because I know that lift will be lost as the multicopter flies laterally in any direction.

If the weight of the shroud is greater than the added lift, that is the cost of the added safety in terms of flight duration due to a higher power consumption for a given flight.

And now, here is the last part of the question. How is the optimal design of a prop to be used in a shrouded propeller design different than a standard or typical multirotor prop design. Could I use a standard multirotor prop in a shroud and not suffer any unexpected changes to the performance of the prop? Would it be simple enough to take a standard 6 inch prop and simply use it in a shroud of the proper internal diameter?