I've seen a discussion about this somewhere before. I'm aware that there's no real way to calculate this. So rather then comparing weight and volume for efficiency, let's compare prices. Which engine is more economical to operate? What price will get you further along?
(sorry in advance if the answers depend on weight or volume)
Piston engines are significantly less expensive to operate, as long as the aircraft's performance and reliability requirements can be satisfied by a piston engine.
It's not just the fuel, it's the maintenance. Turbines run longer between overhauls and rarely fail, but the overhaul costs tens of thousands, and more if repairs are required. There just aren't any cheap parts or cheap repairs to make.
Even though older carbureted designs have poor fuel economy, small turbines (under 1000 bhp) are even worse in that regard. As turbines get larger, they get more efficient, but also more expensive, even per kW, both to buy and to maintain. Modern fuel-injected piston engines are the cheapest to own, with tolerable fuel consumption (helped by supporting lead-free car gasoline) and tolerable maintenance cost. Not considering electric aircraft, which promise the lowest costs ever, but are too new to judge decisively.
The five cheapest turboprops will all cost over $100k/year just for owning them, no flying hours included.
Comparing apples to apples, this site cites USD 1100/hour for the Cessna 208, one of the cheapest turboprops to fly. Let's look at some comparable pistons and turboprops:
Turboprops simply cost more: to buy, to own, to fly. They offer better performance, reliability, and lighter engine weight. They also scale up well. After the initial premium for switching to turbines, cost per seat starts to go down.
There is a limit beyond which piston engines don't work well: they can't fly very high, they don't scale up well, and they aren't reliable enough for a common carrier airline. So there's no modern piston equivalent to commercial turboprops. But turboprops don't scale down well, since a smaller one still needs lots of expensive precision engineering.
As a result, within the size and performance envelope where piston-powered airplanes are still a thing, they are more economical. This is despite the higher cost of less energy-dense avgas. Diesels would've been the ultimate fuel-sippers, but they remain rare for now, with electrics taking up the efficiency niche.
NB: This answer applies to the kinds of aircraft that commonly fly today and where a choice between piston and turboprop engines is possible. Large aviation piston engines now survive only as historical items, like in warbirds.
From a fuel burn perspective, these ballpark numbers can be useful to remember:
Specific Fuel Consumption, Expressed as LBS/HP/HR:
Using .45 for a typical Lycoming carbureted engine, you can work out quickly that a 150 hp engine, cruising at 75% power or 112 hp, times .45, is burning 50.6 lb/hr, or just over 8 US Gal/hr, which is about what you normally see on those engines in the real world.
A particular class of aircraft gas engines, the turbocompound radials like the Wright R 3350, used supercharging plus Power Recovery Turbines to recover about 300 HP from the exhaust flow. These engines had SFC's in the high .3s, approaching diesels. The Canadair Argus used 3350s because of the incredible endurance they could provide, although unfortunately the choice became a problem later in life because of the engines' reliability issues.
Piston engines are much cheaper to run on direct operating costs, but when you factor in other variables, like reliability, weight, smoothness, TBO, obsolescence issues, availability of kerosene compared to gasoline, turboprops usually win out from a business case perspective once you exceed, say, 500 hp, because you can just add a little more kerosene to get the same range and avoid the large piston engine grief.
Up to about 450 horsepower, piston engines are more economical for the typical use model (short hops, slow speeds, unpressurized aircraft, low gross weights).
Above 450 horsepower, turbines are more effective (better power/weight ratio, longer TBO times, ready availability of spares) for the use model (longer stage lengths, higher speeds, pressurized planes, higher gross weights).
"Little" turbines are really uneconomical because turbine engines do not scale down as well as they scale up, so pistons still rule for Cessna 150s and 172s. "Big" piston engines are uneconomical because there's so much that can go wrong with the machinery. For example, the biggest commonly-used piston engine had 28 cylinders, 4360 cubic inch displacement, and 56 spark plugs. Fouling the plugs by running rich on the ground or an improper start sequence required a 6-hour maintenance gig to pull out and clean them all per engine. Those engines went away quickly once turboprops came on line.
One-for-one is a rather poor comparison here, the whole package must be considered for efficacy. It would be more appropriate to compare the economics of operating two 2000shaft hp turbines to four 1000hp piston engines.
This is because the turbines are far more than twice as reliable. So any flights over substantial amounts of water or other areas without suitable runways will require a plane with 4 piston engines but only 2 turbine engines for similar risk mitigation.
The reliability aspect is different from durability or TBO, reliability is chance of an engine failure in the middle of any given flight. Especially when you start to factor in reserve power, because the non-failed engine[s] will need to increase output after one fails.
This reliability is also why modern designs tend to progress one piston, two piston/oneturbine, two turbine; and you haven't seen 4 piston-engine planes since turbines moved out of the experimental stage. Even 2 piston-engine designs are slowly losing favor to single turbines.
Add to this the improved economics of high altitude flight. Especially regarding direct thrust turbines, which have even better high altitude performance, reliability, and simpler maintainence than turbo props.
