Mate, are you serious?
This is my serious face.
You can argue all you like, but I have a physics degree and the principles of thermodynamics have been unchallenged for a few hundred years.
I'm not challenging the laws of thermodyamics. What I'm disagreeing with is how you've explained them as they apply to a real engine. You may have a degree in physics but your post will have to stand on its own merit.
"Sorry no-one has answered the question you originally answered. I'm sure you realise the answers given about volumetric efficiency have nothing to do with your question about thermodynamic efficiency of an engine."
His original question was 'why does a turbine engine need a compressor' and that was answered in a number of ways including one direct reference to the Brayton cycle, which is correct.
Volumetric efficiency impacts on thermal efficiency though. If the VE is poor then you are suffering pumping losses and using more energy just to keep the engine running. Idealised heat engine models are all well and good up to a point but you have to move beyond a PV diagram to understand the constraints on the efficiency of a real engine.
"When the ignition occurs, a lower compression ratio engine will have a cooler air/fuel charge in the cylinder - and so it will absorb more energy (which is wasted as exhaust gas heat)."
I don't think so. You want the working fluid to absorb all available heat from combustion so as to raise the pressure. That's where the umph comes from. Waste heat in the exhaust is simply that which you are unable to turn into work before the power stroke ends.
"A high compression ratio engine will ignite a hotter air/fuel charge which will absorb less heat. Less energy wasted as heat = more energy transferred to the crank."
Again, the method of transferring energy to the crank is to heat the working fluid so it expands against the piston. The fluid needs the heat if it is to do any work. For this reason modern engines are run a little hotter than they used to be so that less of the combustion heat is taken out of the working fluid.
You are barking up the wrong tree. The significance of running the compression ratio high is to get more heat into the charge (without going too far) in order to promote better combustion. It is NOT to get more heat into the charge so the charge then absorbs less heat from combustion because that is not going to work:
The losses incurred by compressing the air have nothing to do with it, because that energy is regained...Compressing an air charge with a piston is like a squashing a spring - you get back what you put in (ignoring friction losses)
As well as friction you have blow-by and heat loss to the coolant (which both increase if CR increases I believe). Therefore, you need to get more out of the power stroke than you took from the crank to achieve the extra compression. That 'more' comes from the improved combustion, not from the heat of compression itself. It is to do with better mixing of the fuel and increased flame speed.