The old Rover II gas turbine was used as an APU in the V bombers I think. It developed 100 bhp at the output shaft. A small and neat unit, one was fitted to an Auster and also to a Currie Wot single seater biplane (Hot Wot).
Unfortunately a turbine's specific fuel consumption is much higher than an equivalent output piston engine thus requiring bigger fuel tanks. Also the level of precision engineering required on the blades far exceeds that of a recip engine with the appropriate costs! Until there is a method of precision casting relatively cheap ceramic turbine blades which require no machining then cheap turbines are in the same level of reality as anti-gravity devices
