What would be the best fan duct design to achieve maximum thrust?
It's not easy to find research on small ducted fans. I did come across this interesting masters thesis from 2009, which does show some relevant results - at low propeller speeds.

Duct design

The document describes three duct designs, one with a variable bypass inlet and two simpler ones. The simplest one is made from PVC tuning and a foam inlet lip:




Geometric information is given as:


Inner diameter = 14.5"
Duct Chord = 4"
Duct Length (C/D) = 0.28


Maximum Thrust

One of the results of the project is that a ducted propeller delivers more thrust than an un-ducted one. Here is one of the results, listing $C_T$ as a function of tip Mach numbers - which are really low and therefore hard to generalise into practical rotor and fan configurations.



Tip Clearance

Best results were obtained with a propeller with square tips with the smallest gap between propeller and duct that could be achieved. This was a 3-blade propeller, but the research project was severely limited by the small motor used, and therefore low maximum torque that could be exerted on the propellers.