What is the maximum speed of all the stages of a geared bypass tubojet engine?

Question

What is the maximum speed of all the stages of a geared bypass tubojet engine before they start losing efficiency.

Answer 1

The maximum rotational speed of turbofan blades follows from compressibility effects, just like those for helicopter rotors and for propellers.

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On most high bypass turbofans the fan tips can get to slightly supersonic speeds. A Rolls Royce engineer at the Singapore Airshow mentioned Mach 1.05 for the fan tips, that is why the blades have swept back tip shapes.

All other stages are designed such that blade tip speed remains subsonic: compressibility effects cause a rise in blade drag before the speed of sound is reached. It then drops again, but:

• There is a large region of high drag that needs to be overcome, where torque will be waisted on compressibility for no gain in thrust.
• Blade root stress is proportional to rpm squared, and it is best to stay in the subsonic region.

For the blades in all stages of the turbofan, three main design considerations are:

• tip speed;
• blade mass;
• operating temperature.

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Blade rotational speed follows from blade radius and tip speed limit. For the fan, a slow turning long blade is best, since it works just like a rotor or propeller does. The compressor blades become smaller at each stage due to the smaller air volume to be compressed, the turbine blades reverse this trend. Rotational speed of the compressor and turbine stages is chosen such that tip speeds stay below the compressibility drag, as confirmed verbally by a Rolls Royce engineer.

Benefit of the gearing can be seen from the shape of the engine: the low pressure turbine blades have a smaller radius than the fan blades, an un-geared engine would turn the LP turbine too slow. With a geared turbofan, the fan tip speed will stay at around the same tip speed, but the RPM of the LP turbine will be brought to a higher value. Below drag divergence number for the hot exhaust gases.

Answer 2

A question with respect to efficiency is very difficult to answer.

Here is why: Its not entirely clear which efficiency you mean.
Let's assume you are talking about isentropic efficiency of turbine and compressor. Then the most efficient compressor will tend to turn slow and will have a high number of stages. The most efficient turbine will turn very fast, the stage count will also most likely be increase (no numbers here, this section is to get a general feeling for this).

But its not only efficiency which shapes today's engines, it's also Total Cost of Ownership or Life Cycle Cost. More stages will result in more material cost and also in increase maintenance efforts. The engine might be heavier, which influences the wing structure etc.

This is why in the design-phase of a jet engine the design-teams will make trade-studys to find a pareto-optimal design which satisfies as many requirements to the most possible way.

But low TCO is only one target, another one is to ensure a wide range of operating range. This means the engine is supposed to work with a similar efficiency under a wide range of ambient conditions.

This means in order to answer your question adequately you need to add some numbers, like pressure-ratio and/or thrust and/or weight and/or size and/or cost. These boundary conditions will then allow to answer your question.

A study by Lingen Chen et al. investigates the your question with respect to axial compressor design. But all the results are non-dimensional so in order to get a numerical value to your question, you have to plug in a numerical value first.