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Im trying to teach myself blade element theory for helicopters and propellers, but im getting quite confused.

Most of the papers and textbooks I read will reach an equation for differential thrust similar to equation 2.2 on this paper. They then define the differential lift, the local lift coefficient, the differential drag, and stop just short of defining the local drag coefficient. Instead, they claim it to be negligible, and disregard it in later equations.

I understand that this local drag coefficient is usually determined experimentally, but is there any way to determine it solely by the geometry of the blade? preferably in an algabreic/closed form solution? I found some conflicting answers from wikipedia

The wiki page for the Oswald efficiency number states that: $$C_{D}=C_{D0}+\frac{C_{L}^2}{\pi{e_{0}AR}}$$ The wiki page for the Lift to Drag ratio states that: $$(L/D)_{max}=\frac{1}{2}\sqrt{\frac{\pi\varepsilon{AR}}{C_{D,0}}}$$ This same page also provides another definition? I think these differ based on whether the propeller is moving at subsonic or supersonic speeds, not 100% sure. $$(L/D)_{max}=\frac{4(M+3)}{M}$$ (and yes, these describe the lift-to-drag ratio and not the local drag coefficient, but given that the local lift coefficient is defined, unless im mistaken you should be able to get the local drag coefficient from these ratios)

Finally, the wiki page for the Zero lift drag coefficient states that: $$ C_{D}=\frac{550\eta{P}}{\frac{1}{2}\rho_{0}[\sigma{S}(1.47V)^3]}$$

Which of these, if any, are applicable to the case of a helicopter?

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None of these are useful for you. They all pertain to a 3D full aircraft.

Please go back and cite specific equation numbers in the reference you cited.

The airfoil local drag coefficient appears in equation 2.6. It survives through to equations 2.13 and 2.14. What more do you want?

If you're looking for a reasonable source of data for that number, you'll want to use 2D airfoil data -- either from a wind tunnel reference like theory of Wing Sections data. Or from a computation like XFoil or CFD.

Usually, we use lower case $c_l$ and $c_d$ for airfoil coefficients. In the reference you cited, it appears they use mixed case $C_l$ and $C_d$ for airfoil coefficients -- but we pretty much universally use $C_L$ and $C_D$ for 3D aircraft-level coefficients.

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I'm trying to teach myself blade element theory for helicopters and propellers, but I'm getting quite confused.

Advanced aerodynamics is nothing that you can teach yourself unless you have a good understanding of calculus and fluid dynamics or aerodynamics.

They stop just short of defining the local drag coefficient. Instead, they claim it to be negligible, and disregard it in later equations.

You misread the paper. In blade analysis both drag and lift contribute in a substantial way to the thrust and to the torque and the figure 1b in the paper clearly shows it:

enter image description here

Lift $dL$ and drag $dD$ both contribute to thrust $dT$ as given in equation 2.2 and they both contribute to torque $dQ$ as given in equation 2.3. Note that in all of the following developments of those two equations both lift and drag survive, like for example in equation 2.13 and 2.14.

I understand that this local drag coefficient is usually determined experimentally, but is there any way to determine it solely by the geometry of the blade?

The drag coefficient on those equations depends only on the local airfoil. If you don't want to use experimental values you can interpolate the shape of $C_d$ vs. $\alpha$: normally a second order interpolation (parabolic) is enough.

I found some conflicting answers from wikipedia

They are not conflicting, they simply have nothing to do with the blade analysis you are investigating.

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