Why do propellers have less theoretical range (i.e. range ignoring impatient passengers wanting to go fast) than jets, despite greater efficiency? I was playing around with the breguet range equation recently, and came across a suprising result, that propellers have less range than jets, despite having greater efficiency.

The range equation for a prop is:

R = 550 * np * ((L/D)/(C / 3600)) * ln(Wi / Wi+1)

where C is the SFC, in lb/(hp * s), and R is the range, in feet. Note that the 550 here coincidentally is the same as the speed in the next calculation, but the 550 here is actually a conversion factor from hp to lb-f/s

According to wikipedia, the most efficient turboprop engine is the NK-12, used on the Tu-95 bomber (and Tu-114 airliner), and that engine gets a SFC of 0.36 lb/(hp*h). Assume a propeller efficiency of 0.8, which is typical.

R = 550 * 0.8 * ((L/D)/(0.36 / 3600)) * ln(Wi / Wi+1)

R = 550 * 0.8 * ((L/D)/0.0001) * ln(Wi / Wi+1)

R = 550 * 0.8 * ((L/D)/0.0001) * ln(Wi / Wi+1)

R = 4400000 * (L/D) * ln(Wi / Wi+1)

For jets:

R = (V / (C/3600)) * (L/D) * ln(Wi / Wi+1)

a SFC of 0.5 lb/(lbf*s) in cruise is considered pretty good for jets, and let's assume a 550 mph cruise speed (or 806.667 ft/s). Range is now:

R = (806.667 / (0.5/3600)) * (L/D) * ln(Wi / Wi+51)

R = 5808002.4 * (L/D) * ln(Wi / Wi+1)

The range expressions for jets and props is identical, except for the coefficient, and 5808002.4 is greater than 4400000, so therefore, jets have better range, even on paper, than props, assuming same L/D ratio and fuel ratio. This is kinda suprising, given that props are quoted to have higher efficiency than jets. I've heard people say props have less range, but I've always heard it backed up with the evidence that "props are slow, and people like to go fast". Are my calculations and data correct? If they're correct, why, intuitively, do props end up having less range?

Clarification: user3528438 has pointed out that some turboprop engines provide both a power specific fuel consumption number, as well as thrust specific. However, if you plug the thrust specific fuel consumption number for that turboprop into the jet range equation (essentially treating the prop as a jet), the resulting range is very different than the range you'll get if you plug the power specific fuel consumption number into the propeller equation -- so which one is right? I doubt my formula for propeller range is wrong (though I might be using it wrong - could someone please check) because the same formula (but without the conversion factor, because they're using metric) is found here - https://nptel.ac.in/content/storage2/courses/101104007/Module2/Lec8.pdf.

I also have my derivation of the formula on hand too, if anyone wants to check that.
For more background, this question is about the usage of Breguet Range Equation (I find this explanation better suited for this question)

There are many problems with your calculation:


Time unit of speed should be the same as the time unit of SFC. For your first calculation, since you used mph for speed and lb/(hp*h) for SFC, your SFC term shouldn't have 1/3600. For your second calculation, since your speed is fps and your SFC is also in lb/(lbf*s), there's no need to have 1/3600 either.
For both calculations, end result should be the same unit of length. Your first calculation uses mph and the second uses fps, so the the end result would be in miles and feet so not comparable. But since you made the mistake #1 you end with numbers still on the same order of magnitude. 
For the propeller case you are using PSFC (Power specific fuel consumption) rather than TSFC (Thrust-specific fuel consumption). Breguet Range Equation requires TSFC to work.




If you look at the Breguet Range Equation you see the only difference between prop case and jet case is TSFC. So to compare relative range differences, you only need to compare TSFC and avoid all other complications.

Take Cruise TSFC from some modern examples:


GE90: 0.545 lb/lbf/h
Europrop TP400: 0.38 lb/lbf/h


(1/0.38)/(1/0.545) - 1 = 43% so with everything else being the same, prop should have about 40% more range than jet.