The original question appeared to call for an overall gear ratio of 1:12; resulting, in speed being reduced by a factor of 12, and theoretically, in an increase of torque by a factor of 12. Based on the 10:36 first stage, my calculations agree with everyone else who cited a 12T pinion to drive the 40T gear in the second stage.
If the turbine is required to produce 1 Mw then doubling the diameter of the rotor, would, presumably still result in an output of 1Mw, but at a quarter of the original speed? I can only share the fears of those who believe that maintaining the original speed would result in failure from overspeed. If failure did not occur, and the pressure drop across the turbine was the same in both cases, one would expect that, theoretically, the output should become 4Mw?
Turbochargers, IF the turbine and compressor wheels are small diameter, can run at very high speeds (over 100,000 rpm for the sizes currently employed on small petrol (gasolene) or diesel engines.
How do you measure these speeds?
By magnetising the nut retaining the compressor wheel, and having a sensing coil, with pole pieces surrounding the compressor casing, with resulting voltage displayed on an osilloscope. A signal generator trace displayed on the same screen, and synchronised will allow the speed to be calculated from the frequency. Alternatively, the compressor nut is painted black on one half, and white on the other, to allow the speed to be sensed optically, and the pulses counted electronically.
The original gas turbines originally used in jet aircraft were limited by the creep properties, at high temperature, of the materials available, for the turbine rotors. Initially, this was circa 1,000'C Hence the development, by Henry Wiggin, of Nickel alloys of the Nimonic type which increased the ability to withstand high temperature. We now have Rolls- Royce growing a turbine blade as a single crystal, which has no interstices, (and with internal cooling ducts) for the current ultimate in high temperature creep resistance.
At various times compound engines have been developed which geared an exhaust turbine to an internal combustion engine. The engine was acting as a gas producer to feed the turbine. Indeed experiments have been carried out where a reciprocating engine, or a free piston engine, was purely used as a gas producer to drive the turbine, to which it was not coupled.
Currently, these alternatives do not seem to have not proved to be viable for whatever reason, (weight, fuel economy, complication, cost?)
Howard