What is the range of ratios between optimal propeller efficiency and true propeller efficiency?

Question

In https://aviation.stackexchange.com/a/15383/20394, the optimal prop efficiency is given as:

$$\eta^{opt}_{Prop} = \frac{1 - \frac{\Delta v\cdot \left({v_{\infty} + \frac{\Delta v}{2}}\right)}{d_P^2\cdot\Omega^2}}{1 + \frac{\Delta v}{2\cdot v_{\infty}}}$$

If we think about $\eta_{Prop}$ as $k\eta^{opt}_{Prop}$, where $0\leq k \leq 1$, what would be the typical range of $k$?

Furthermore:

• Can we draw general conclusions about $k$ for various props, based on material and the modernity of its construction/materials/airfoil?
• How good of a fit is the $ \eta_{Prop} = k\eta^{opt}_{Prop}$ model for the true prop efficiency?

---

_{P.S. I am making a distinction between propeller efficiency and propulsive efficiency. This question is only about the former. C.f. Propeller Efficiency calculation different by two methods}

Answer 1

My XROTOR.DOC file from October 1988 reads:

<--- snip --->

Like CL, profile CD is defined piecewise in alpha such that in the unstalled    
region, CD has a quadratic dependence on CL (after Prandtl-Glauert scaling)
and a power-law dependence on Reynolds number as follows.
        |                   2|           f
 CD  =  |CD  +  b (CL  - CL) | (Re/Re   )
        |  o         o       |       ref

where
CD    =  minimum drag coefficient        ; Fortran name:  ( CDMIN )
  o
CL    =  CL at which CD = CD                              ( CLDMIN )
  o                         o
b     =  CL weighting coefficient  d(CD)/d(CL**2)         ( DCDCL2 )
Re    =  Reynolds Number at which CD formula applies      ( REREF )
  ref
f     =  Reynolds Number scaling exponent.                ( REXP )
         Typically:
          f = -0.1 to -0.2  for high-Re turbulent flow
          f = -0.3 to -0.5  for low-Re airfoils
Only the above parameters are supplied by the user, and the program
figures out the CD(alpha) function in SUBROUTINE CDCALC.

<--- snip --->

So we run XROTOR once with realistic drag data and once again with the same propeller but all drag coefficients set to zero. Since I last used XROTOR in the last millennium, I have only my old MacOS version which I run in Sheepshaver. It took me a while to get familiar again, and I took Mark Drela's Daedalus propeller file as the input. Copy-Paste from the old to the new OS doesn't work, so I have pasted screenshots here:

Next, I use the AERO submenu to set the drag parameters and redo the calculation at the same RPM:

What a surprise, without airfoil drag our efficiency equals the ideal efficiency. Looks like I did everything right to isolate viscous effects. The difference is a jump in efficiency from 0.886 to 0.94, a bit more than 5%. This is for a very lowly loaded prop at a very low Reynolds number, so the viscous effect is rather large. Expect less for larger props at higher speed, so your factor k will range between 0.943 and 0.984. The last number is valid for a 6-bladed prop driven by an AE2100 engine, the input file of which I also had at hand.