The last season I spent in the 4Y, in four months I shut down five engines...over about a 50 hour period. Typical.
Then again, I've gone almost a hundred hours without shutting one down...once.
Radial engines have their place, but do not match turbines for power, reliability, maintenance frequency, vibration, etc.
The origional idea of the radial was to increase power in a piston engine with a minimal increase in weight, while retaining adequate cooling. The engine is short compared to an inline engine. Harmonic vibration issues created along longer crankshafts are not nearly the issue.
Someone mentioned vibration being a benifit of the radial...if you consider an increase in vibration to be a benifit, then yes, it is. Otherwise, you're stuck with a lot more airframe stress when running a radial engine. This is especially true if everything isn't running perfectly. Foul a plug, you have vibration. And so on. I've seen 3350's break their mounts, and 2600's crack of their prop governor housing assembly. I had two runaway props due to vibration breaking and shorting wires in the cannon plug to the prop governor.
Radials like to catch fire, some more than others. The 4360 seemed to like it a lot. So did the 1820, and 3350.
Radial engines are loud, and burn a lot of oil. I typically serviced each engine with 3-6 gallons of oil per flight hour.
I have lots of late-night experience changing cylinders from a ladder in the rain, being consumed by mosquitoes, and eating pizza that is wet and tastes like 60 weight oil and avgas. I also have a lot of clothing that has stains from 60 weight oil all over it.
You haven't lived until you've oiled your airplane in high winds using a 5 gallon bucket with a funnel and a ladder, and end up wearing half the bucket, and drinking the other half when you get blown off the ladder. Or burned your hands on hot exhaust stacks changing out gaskets and seals during a working day.
Radial engines are large, and heavy. With one only a couple of exceptions, they're old, and parts are no longer manufactured, but reconditioned.
With respect to piston engines, they're quite efficient in terms of what they can do for their size.
Folks will talk about how one could blow a jug and keep on flying, but those folks never owned one, or had to work on one. I've certainly never continued to operate one after it started banging; I shut them down right now. I've had jugs come off, pistons burn through, sucked valves, disintegrated superchargers and supercharger clutches, failed inductions, carburetor failures, propeller failures and governor failures, fires, etc. In each case, I shut them down, except one. In that case, I had a governor base rupture on #4 and a supercharger die on #1. I shut down the governor rupture due to excessive oil loss and a fire hazard, and kept #1. I flew to the next stop, replaced the governor base and assembly, and then departed on all four, shutting down #1 for the remainder of the flight.
The advantages are that you may be flying an airplane with radial engines, and those are what you've got. They keep you aloft.
That, and a radial engine certainly sounds better...unless you happen to live at the airport and need to get some sleep.
Radial engines do NOT take abuse well.
Radial engines are geared engines in most all cases, and demand a lot more respect. Cooling issues may be more critical in many cases, as well. Radials are not as tolerant of abusive power or operating techniques. A great deal more backlash is found in them, and these engines should be operated most all the time at power settings adequate to keep the slipstream from driving the prop. Radial engines are prone to hydraulic lock, something far less common in other types of engines, and rarely heard of in turbines

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