Gaunty, I think that the only thing that we differ on is the definition of "work done"
You said
The turbines by definition are actually EXTRACTING energy from the gas path to use elsewhere in the engine. Not in themselves producing any thrust.
On their own they are incapable of doing anything.
I totally agree, the energy extracted is used to drive the compressors
You said
The energy in the accelerated higher energy air THATS LEFT is converted to thrust through that divergent duct or nozzle at the back end.
Agree, but the energy (by which I refer to as thrust) is producted by the combustion and subsequent expansion of the "exploded" air. This explosion accelerates the air backwards (through the turbine) at a high rate of knots. The "thrust" at the combustion chamber outlet, minus the energy extracted (work done) to turn the turbine blades leaves the engine thrust figure that is quoted by the engine manufacturers (simplistic summary I know!)
You said
In a straight no bypass turbojet the job of the turbines is simply to provide energy to the compressors to pump, or accelerate, add energy to the air thus produce thrust this requires an external energy input (fuel) to become self sustaining and hence my comment on the compressors actually doing the work.
Just our terminology is different!
You said
Its much simpler in a fan engine as almost all of the energy not required to directly drive the compressors is provided to the fan which produces around 80% or more of the actual thrust, sometimes refered as cold thrust as it does not pass through the hot part of the engine core.
Correct, a fact born out by the fact that the Rolls-Royce Trent family of engines (all quoted at different thrust) have the same core (intermediate and high pressure modules) but different fan diameters.
Your example is spot on but I think that
Usually if the engine is down on power or running hot the cause is usually found in the compressors.
is just a little bit too broad brush for my liking!
Halibut, you should be ready to write a thesis by now!!
------------------
Cheers,
Snigs