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Typically the first turbine after the combustion chamber drives the compressor as shown in this picture.

turbine cross section
(wikimedia.org)

This first turbine spins the fastest and harvests the most energy from the gas flow. This seems inefficient when you want to maximize power output from the power turbine. So why not have the power turbine the first to take advantage of the gas flow and then the compressor/gas producer turbine second?

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You can only use for shaft horse power what is available after the energy to drive the compressor is subtracted. This subtraction is done by the high pressure turbine. Any energy left in the exit flow after that can be extracted to drive a propeller, a fan or a generator.

What happens in your arrangement if the power turbine extracts too much energy to keep the core engine running? No, clearly the priority has to go to keeping the core engine alive and to only use what is left.

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enter image description here

  • A slower compressor will lower the compression ratio, and therefore will hurt the fuel economy of the engine.
  • A faster free turbine will require a much stronger and bigger reduction gearbox (remember the prop rotates much slower than the turbine).
  • There is no easy way to have the concentric (or otherwise) shafts run the LPC (or prop) via the HPT (pictured above).
  • Assuming it can be done, adjusting the fuel flow will have a slower compressor response, resulting in overheating (not enough air) or choking of the turbine (too much air but not enough fuel).

In short, the gas flow is only strong because the compressor is run by the first stage(s) of the turbine.

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The compressor obviously requires a power source, in order to do the work of compressing the air before it is mixed with the fuel. The first turbine takes some of the energy from the combustion process and uses it to drive the compressor.

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