I'm trying to design a miniature ducted fan engine for a small project, but I'm not too sure how much contraction should happen between the fan and the end of the engine, if at all.
Engines like this or this look like they have very little contraction at all, but why not use the pressure created by the fan and contract the space making the output velocity higher?
This is the same sort of principle behind the bypass air of a turbo fan engine and I don't see why not to use it.
but why not use the pressure created by the fan and contract the space making the output velocity higher?
That is exactly how the pressure energy added by the fan is converted into thrust. Thrust, after all, is the product of the speed increase between free stream and exit flow, mass flow and nozzle area.
That real jet engines don't show so much contraction is due to the expansion of the core gas flow from heating. If you don't add heat, your exit area should be quite a bit smaller than the entry area.
If by "normal ducted fan" you mean one where the duct is simply a cylinder of constant diameter, then yes, a properly designed nozzle will create more thrust.
Depending what you mean by a smaller unit. For some time now I have been studying and testing 'electric ducted fans' where gas velocities are less than 100m/s and RE less than 150k. I have manufactured my own and tested numerous units from various sources. I am a retired production engineer with the best part of 30 years dedicated to gas turbines including pure jet, low and high bypass, shaft engines both for flight and static. These little horrors have left me at times baffled. Nothing I have read pans out. First measuring gas velocities from the entry to the exit, not once have I found a significant change in gas velocity after the entry to the fan. Just decay. They do not operate in any way like a full size high bypass fan. It appears only Guy Lussacs gas laws can be applied. Any nozzle other than the vane section reduces thrust for a given power input. More power higher rpm, creating a greater reduction in pressure at the entry to the fan compensates. So far, I have found, only the shape of the entry to the duct can improve static thrust for a given power input. I realise now that I have years of work to provide a comprehensive guide to small low velocity fans. The next step is to build a small dynamometer to ensure I know the power input to the fan and then a small wind tunnel. There is much more I could say but at this stage all I will say is that they are pressure difference propulsive units with very little momentum change. I have thousands of measurements of thrust, gas velocities, power requirements, pressure measurements, etc. Finally good luck in designing a minature high bypass unit. I imagine you are trying to produce a minature gas turbine driving your fan. There is a good deal out there to help you but the bypass fan may not operate in quite the way you expect.
It all depends upon the configuration of the diffuser for the fan but the answer is not necessarily. For a typical high bypass turbofan on a transonic aircraft application, it will use a fixed geometry diffuser (nozzle) for both fan and gas core exhaust with the nacelle. P&W and GE engine’s have opted to use a separate cool section and hot section diffusers. Rolls Royce seems to prefer a single diffuser which combines both the fan and gas core exhaust prior to exiting the nozzle for noise reduction reasons. Both designs are very efficient for their appropriate applications.
High bypass turbofans have exhaust flow speeds in the high subsonic and transonic regions but do not require a fixed or variable convergent-divergent diffuser for supersonic gas flow. The reasons are primarily that it is 1) less fuel efficient for a subsonic application and 2) supersonic exhaust flow is extremely noisy and limits the airports where the jet could fly into due to noise restrictions.
Your exact design for your particular diffuser is going to depend on the configuration of the engine as well as the installation constraints. You’re also going to have to calculate the impulse the engine needs to provide and this is going to determine the exit speed of the gas through the diffuser. It’s going to be a little more involved than simply copying an existing design and making something that looks similar.
