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The main efficiency comes from “downsizing” the turbo engine. This can be done because a turbocharger allows an engine to make more power for its size (displacement) than a naturally-aspirated one. The main reason why this is really effective for passenger cars, is that the average power required to drive is a small fraction (less than 10%) of the rated power of an engine. For example, a small engine delivering 10 HP to the driving wheels is going to be a lot more efficient than a large engine delivering the same 10 HP. It will be more efficient due to less friction and lower pumping (throttling) losses.
The exception is if the turbo engine is operated at high power levels, where it may be less efficient than the NA engine due to combustion reaching the knock limit and the exhaust temp reaching its limit. These limits force the engine calibration to retard ignition timing to avoid damaging knock, and the fuel mixture to go rich to avoid over-temping the the catalyst, exhaust valves, and the exhaust manifold. Both of these hurt engine efficiency.
Modern turbo matches don’t just kick-in a certain RPM. Boost builds from as low as 1200 rpm at full-throttle.
There is no single RPM to shift at. For a review of proper shifting strategy, see this: How to shift a manual transmission for both performance and economy (https://rothautomotivescience.com/how-to-shift-a-manual-transmission-for-both-performance-and-economy/)The energy used to run the compressor in a turbocharger comes partly from the wasted exhaust energy (part comes from the exhaust pumping work), so the energy needed for boosting is partly free. The turbo does create extra back-pressure, so that does have penalties (parasitic loss and worse knock resistance at high power). The energy used to run the compressor in a Supercharger comes 100% from the crankshaft, so it is all parasitic loss and all of it contributes to higher cylinder loading and worse knock resistance. The net result is that turbochargers can provide more boost more efficiently than Superchargers.
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