So in WW2, both radial and in-line engines are used, if both engines had the same cubic inch volume, which would be more fuel efficient?
(If there was a difference)
A simple question but the answer is not simple.
The Merlin (1650 cu in) is sometimes criticized for poor fuel consumption. It did use one of the lowest compression ratios of any high performance WW II aircraft engine (6:1) but that also allowed more boost to be used before detonation set in. The Allison (1710 cu in) used 6.7:1 compression and is generally credited with a bit better fuel consumption but a bit lower limit on boost (both engines went through a variety of models with upgraded parts so beginning of war and late war engines should not be compared to each other).
P & W never upgraded the R-1690 Hornet so we are left with the R-1820 Cyclone and the R-1830 Twin Wasp as our air cooled examples ( I want to try to stick to the US engines as we can assume the same fuel and the same or at least close technology in piston rings/bearings and carburetors/single point fuel injection). and both use compression ratios of 6.7:1 or 6.65:1 depending on exact model (or source).
At high power the advantage
may lie with liquid cooled engines. The air cooled ones start running really rich and use the extra fuel as an internal coolant. The Liquid cooled ones use the same trick but generally at even higher powers. When the engines are running lean they are pretty close to each other. If the engine cooling system is set up to the cool the engine when it is making 1200hp (give or take) then cooling it at cruise (around 750hp?) is no trick at all and any benefit of liquid cooling over air cooling is marginal.
In the early years the two radials often came with 2 speed superchargers and at low altitude when running in low gear didn't use up as much power driving the supercharger.
However a big wrench in the comparison is the actual installation of the engines, Until the FW 190 and later the air cooled engine was simply a higher drag installation and required more power to go the same speed as the liquid cooled powered fighter (a P-36A could fly 285mph/15.000ft using 750hp from it's P&W R-1830 while an early P-40 could fly at 286mph/15,000ft using 600hp from it's V-1710 Allison.
A later installation of the R-1830 in a P-40 test mule got the drag difference down to about 8%.
When cruising the American engines could get into the high 0.40s (lbs of fuel per HP hour) for either the radials or the Allison. The Merlin was usually rated around 0.50 or just above. At Military power (or combat power if allowed) fuel consumption could rise to
around 0.70lb per hp hour for any of the engines.