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If I wanted to remove the piston engine from a cheap plane like a Cessna 152 and add small turbine engines with an equivalent thrust but weighing less, where would be the best place structurally to mount them? I'm thinking one small turbine on each side of the tail and two on the wings.
What problems may I expect?
Have a look at the Cri-Cri propeller and jet versions. Yes, jets can be mounted near the nose, but the plane must still balance at within its specified CG range. This is why removing the engine in front and placing jets in back is out (unless you want to really go crazy and mount a Gatling in the front). Remember, it must balance.
However, at Cessna (152) speeds, props are vastly more efficient, so a $turboprop$ is not out of the question and, indeed, these are found on crop dusters.
"Pocket jets" are also found on motor gliders, and I have often thought of putting 4 of these on a sailplane.
However, another key consideration is how much stress an airframe can take. With jets come higher speeds. These planes are built to flying within a certain speed "envelope" regardless of propulsion source.
The "Screamin' Sasquatch" may interest you. Here a jet is mounted on a sturdy bi-plane, and yes, it works.
You asked for it, you got it:
The x number of small non-certified engines will not add redundancy, they are in fact very noisy, and terribly uneconomical (as a reply to your comment).
The good old piston engine is extremely reliable compared to these mini jets. Certified GA pisyon engines are tried and tested, and you can bet your mamas behind that they will function properly the whole service interval. The mini (or micro) jets, not so much...
The mini jets are stupid loud. It is incredible how much noise they make compared to their size. Having multiple of those sitting in front of you screaming like nothing you could ever imagine, well... I think they actually are the most efficient way to convert fuel into noise. Which brings us to:
They will also be disappointingly unefficient (other than producing noise). As Robert already commented in his answer, they would be operating in a suboptimal speedrange. Sure, they are used in hoverboards and other doohickeys that are even slower than a C-152, but the reason they are used in such applications is their size and relative safety compared to piston engines with large propellers.
All that (and Roberts excellent answer) being said, if I had the time, I would certainly build what you have proposed in your question. Just for shits and giggles 😃
This is such an interesting question. I myself was thinking of adding triple microjets on both wings, for redundancy.
To add @Robert-DiGiovanni's good answer, the balance seems to be the tricky one. I'm thinking aloud here. Now, the plane must be able to be lift at the ends of both wings with
These and without stalling any of the wings, tail, flaps and control surfaces.
In my scenario
Adding fuel tank to the nose instead of the engine helps little as sooner or later the nose fuel tank is empty and again the forces need to be countered.
Now, what comes to fuel considerations, adding six JetCat P80SE (http://modelaircraftcompany.com/newshop/en/home/76-jetcat-p80-se.html) replaces the consumption with about 100 litres/hour. This is in the same ballpark as Jetranger 206B (100l) or Caravan C208 (150l). Don't expect to get much less for "cruise" instead of "peak", for turbine engines it's about the same (80-90%). For four hour endurance needs 400 litres (dm3) instead of the normal C152 70 litre tank. So the front space is needed for fuel.
As the @jpe61 put it, the noise would be incomprehensible so some kind of noise insulation around the engines is needed, adding to the drag and weight.
