I have a few questions regarding constant speed props. As a pilot who just recently moved up to aircraft equipped with constant speed props (Cessna 182, Piper Dakota), I understand that a constant speed prop is meant to increase efficiency and decrease engine wear. However, the the performance tables in the POH of the C182T do not support the idea that a constant speed prop results in higher fuel efficiency. For example, at 3100 LBS, standard temperature, a constant 21 inches of manifold pressure, and a pressure altitude of 8000 ft, the ratio of true airspeed to gallons per hour (which is essentially nautical miles per gallon) barely increases with a decrease in engine RPM. From 2400 RPM to 2000 RPM, this ratio is 11.12, 11.42, 11.55, 11.69, 11.87, respectively. This does not seem like a very big increase in efficiency to me, considering the added complexity of a constant speed prop. A reduction in RPM with a constant manifold pressure also reduces true airspeed, so couldn't the same effect be produced with a fixed pitch propeller simply by reducing power? This would both reduce airspeed and increase fuel efficiency, so this raises the question, what is a constant speed prop actually achieving? What is the most efficient RPM in terms of nautical miles per gallon? Is it simply the lowest RPM? I feel like there's some knowledge I'm missing here.
With a constant speed prop you are paying for maximum performance across the operating spectrum.
A constant speed prop will have only a marginal efficiency improvement over a fixed pitch prop optimized for the cruise case (a so-called "cruise prop"). Both propellers will be operating fairly close to their optimum blade angle of attack and the overall performance will be similar.
What the constant speed prop gets you is efficiency across the speed range. If you want the best performance from a fixed pitch prop for the climb case, you have to pitch it for that speed range, and now you've lost efficiency for the cruise case. So you have to choose what to give up when you select a fixed pitch propeller.
This goes all the way down to the static (not moving) case. When stationary, a CS prop will produce substantially more thrust than a fixed pitch one pitched for climb, and way more thrust than a fixed pitch one pitched for cruise.
So in the end it depends on the performance objectives that you are willing to pay for. If a rate of climb of 1000 fpm vs, say, 1200 fpm is not a big deal, and acceleration from a stand still is not that big a deal (because you don't operate from short strips say) I might prefer to go with a fixed pitch prop and save a fortune on maintenance costs and maybe 80 lbs of weight.
On the other hand, if I want to accelerate from a standing start as hard as possible, get the highest climb rate possible, and cruise as fast as possible, all at the same time, then a CS prop is worth it.
The CS prop also gives power setting options you don't have with a fixed prop, like running "oversquare" to minimize RPM, since RPM is the primary determinant of engine wear (it comes down to how far the piston rings travel over a given number of hours), since low RPM is always less heat/friction than high RPM (subject to limitations of the engine itself, where for example, the engine may not be happy below a certain RPM in terms of roughness or vibration).
To answer your question directly, if the comparison is to a fixed pitch prop pitched for cruise, the difference in fuel efficiency will be negligible. If the comparison is to a climb prop, there should be a measurable difference because the climb prop will force you to run at a higher RPM at cruise to get the same percentage of power.