I am sure this has been discussed before but it just came to my mind when I got into my car.
My car currently has 1213 hours on the engine. That is over a lifespan of 24,000 miles. With that average we are looking at a lot of hours before the vehicle even reaches 100,000 miles.
Why can we not get the same reliability in airplane engines?
Is it the RPM that we run them?
If that is the case, why then can we get thousands of hours out of a turbine engine that has an N1 speed of 53,000 RPM?
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First, TBO is only advisory for privately-owned aircraft. Lots of people operate to 3,000 hours and beyond. Even commercially-registered aircraft can go past TBO "on condition". It's just a guideline.
Second, as others have mentioned, we operate our engines at high power continuously in flight, while a car engine runs mostly at what we'd consider to be taxi power (except merging onto the highway, accelerating to pass, pulling a trailer up a hill, etc).
Third, our engines have to be as light as possible, so we can't overbuild them for longer life the way you could with a diesel engine in a truck or tractor—they're light, fragile things, like everything in an airplane. For the same reason, most small piston engines have to rely on air cooling rather than (more-effective but heavier) liquid cooling.
Fourth, we can't afford gradual power degradation like we could in a lawnmower or generator. It's fine if your lawnmower engine is putting out 25% less power after a few years, but with an airplane, that could mean flying into the trees at the far end of the runway. We need to overhaul before there's significant power loss.
Fifth, airplanes operate in extreme environments, sometimes going from 20c to -10c in a matter of minutes, accompanied by major air pressure changes. Cars, lawnmowers, tractors, and generators rarely deal with that.
Sixth, many planes spend a long time just sitting around unused. Corrosion seriously shortens the life of engines. Rentals usually go well past TBO, despite the rough treatment from student pilots and renters, just because they're always flying.
Seventh, you're right that our piston engines are old tech. The industry could probably do better if there were 10× as many of us with 20× as much money to spend, but there's not (we're getting older on average and gradually dying off), so everyone who tries to introduce new engine tech just loses their money. My prediction is that we'll have improved battery tech (e.g. 4–5 hours endurance) for electric planes before we have modern gas engines in most piston planes, because we'll benefit from all the R&D on electric cars.