I say lower altitude, to ignore the effects of thinner air. I believe I know the answer, but the thermodynamics, in light of a turbo charger, aren’t trivial.
Reasons why TIO540 has lower efficiency at lower altitudes, compared to non-turbo charged engine:
Reasons why TIO540 has higher efficiency at lower altitudes, compared to non-turbo charged engine:
There are a lot of details that enter into the bottom line, and I’ll defer to others with experience running these engines to answer the overall question. However, its probably worth pointing out that a turbocharger increases efficiency, not the converse: it extracts energy from the exhaust flow and uses that energy to pump air into the engine, doing work that would otherwise be subtracted from crankshaft rotation. The exhaust is cooled across the turbo to reflect the energy extracted from the exhaust
A reduction in compression ratio (perhaps better expressed as expansion ratio) is sometimes required on turbo engines due to fuel limitations and that does leave more energy in the exhaust. So then it becomes a battle between the lost energy that wasn’t expanded out in the cylinders due to the lower CR, and whatever energy is subsequently recovered from the exhaust gas by the turbo.
If I look at the entire engine, including turbo charger as an adiabatic system, then only the difference between combustion temperature and exhaust gas temperature should matter, not the temperature at the outlet of the turbo charger. The work that the turbo charger extracts can only improve efficiency, if and only if either combustion temperature increases or EGT decreases (EGT of the gases wasted out of the exhaust stack). At least that’s how I understand it, but its 20 years ago when I last studied Carnot cycles. 
It’s even longer since I studied Carnot cycles I think I remember that turbo in a diesel increases power and efficiency. Turbo in a Otto engine only increases power.
Define efficiency?
Tio burns (a lot) more fuel.
A tc Commander has a lot longer groundroll than the equivalent io540.
But .. You will never be able to run full throttle in cruise with a TiO .. Even at altitude. Where as an IO540 will simply run out of air when you climb and at a certain point you will just floor it to get a reasonable MP.
My TIO typically has a max of 40/2575. Cruise usually is 29/2200 31/2200 29/2300 up to even 31/2400. These settings can be set even at FL200 and higher (allthough 31/2400 is too much power then for cruise). If I want to climb i can still easily add power at those altitudes.
Thr downside .. I calculate with 17gph average. On a good day it can be 15 .. On a bad day 18.
The discussion about temperatures is above my paygrade .. In your discussion I missed that a TC sends more air to the cylinders and compresses it?
I suspect Lucius is thinking particularly of the TIO540 and the IO540 fitted in the TB21 and TB20 respectively.
There is a big difference in the compression ratio, but also – reading the Socata user forum for years – most TB21 owners don’t fly them properly leaned. They don’t buy into the whole Peak-EGT or LOP / John Deakin / etc business, and they fly them well rich of peak, thus burning maybe an extra 10% fuel. I don’t know if there is a problem flying the TB21 at peak EGT (which is how I always cruise in my TB20, except above FL170 when best power is needed). Clearly there can’t be at low levels but maybe at say FL200 the air is too thin to cool the engine. But then one should be able to fly LOP…
The bottom line is that a TB21 has at least 10% less typical range than a TB20, but as I say this could simply be due to the way the owners (who write about it) choose to operate them.
If there is a difference in fuel consumption, it is going to be small.
What is going to be much more significant is the TIO’s consumption of exhaust stacks, cylinders and valves. I have a very benign turbo installation (normalized, manually enabled, engine flatrated, very good cooling with adjustable cowl flaps) and it definitely does make a difference in maintenance. I would say it is in the order of 2000 € / year compared to the very same aircraft and engine without the turbo. That includes the repair/replacement of turbo related parts, i.e. the additional cost can be 0 € in one year but also 5000 € in another. In the TB21 I would estimate it to be much more problematic due to sleek cowling with limited cooling.
let me add an observation from real life use: we operate the TIO-540-AK1A in a T182T. When leaning the mixture in lower levels the limiting factor to leaning (while ROP) is the TIT or turbine inlet temperature redline. When climbing into the flight levels it is possible to lean to LOP and TIT is no longer a factor. We are thus able to cruise at 65% with a LOP fuel burn of ~40l/h. Since the T182T climbs with almost 1000ft/min I gladly climb to FL80+ even for shorter trips (around 1h)
..just my $0.02
Yes – that is also my recollection from reading about the TB21.
On the TB20 there is no way to exceed any EGT limit and there basically isn’t an EGT limit AFAIK. Anyway, one should not be leaning to peak EGT above 75% power…
On the TB21, the TIT limit can be exceeded.
But that should not be an issue, because one would not be cruising at low levels above about (common consensus, I believe) 65% power – unless in a hurry to beat some airport closing time and then fuel doesn’t matter… and you won’t exceed the TIT at 65% power, I assume?
That’s why I never understood why the TB21 should be worse than the TB20.
Another problem is that the fuel burn figures in the TB20/21 POH are not real. They are pessimistic by at least 10%. I think they come out of some 1960s Lyco document, and when Socata went to the GT (“TB20 with Mod 151 applied” – you have to laugh) they missed the opportunity to revisit the topic and stayed with an old POH because changing the POH is more paperwork.
At 60% power, and too much leaning, I could easily exceed the 1650 F redline TIT on my TB21, so have to be careful.
