# How to find trim condition of a sectional airfoil without knowing the angle of attack?

I am using a reflexed airfoil designed for positive pitching moments to achieve a pitching angle matched to the changing of free-stream velocity (Aerodynamically tailored). I ended up with a pitching moment equation $$dM = c*Cm_{a.c} - (C_{l_0}+C_{l_α}*α)*Χ_{a.c}$$ where $c$ the chord length, $Cm_{a.c}$ the pitching moment coefficient about the aerodynamic center of the airfoil $Cl_0$ the zero lift pitching coefficient, $Cl_a$ the lift coefficient at a certain angle of attack and and $Xa.c$ an offset distance about aerodynamic center.

I want to design a propeller and the data i have is free stream velocity, RPM, and geometry of the blade. I assume a thin airfoil so the lift coefficient ($(C_{l_0}+C_{l_α}*α)$) is roughly $2π*α$ and $α$ varies with free stream velocity and rotational speed: $α = Δβ - tan^{-1}(V_{inf}/V_r)$ and $Δβ$ the pitch angle.

The idea is to find the ideal angle of attack $α$ that satisfies the pitching moment equilibrium condition $dM = 0$. However, pitching moment coefficient $Cm_{a.c}$ is also unknown.

*Let's say $X_{ac}$ value is given.

So, my question is how this equation can be solved? Can i get any other data from the airfoil's profile that help to find the solution? Are reflexed airfoils have any characteristics that i should consider in such situation?

• I don't understand the chord with the $C_M$, that would make the dimensions not right. Jul 27 '17 at 11:11
• @Koyovis This comes from the equation $$dM = dM_{ac} - dL*X_{ac} = 1/2ρV^{2}c^{2}Cm_{a.c}dr - 1/2ρV^{2}cC_LX_{ac}dr = 0$$ $C_L = (C_{l_0}+C_{l_α}∗α)$ Jul 27 '17 at 11:27
• Yes indeed. I would expect a length divided by a length, a dimensionless entity, to accompany the $C_M$ Jul 27 '17 at 11:39
• @Koyovis Imagine schematic shown in this link: aviation.stackexchange.com/questions/40910/… Jul 27 '17 at 11:44
• Do you want to solve the equation for $Cm_{ac}$ or for $\alpha$? If you have one equation you cannot solve for both. From the title of your question it appears that $\alpha$ is an unknown, however if free stream velocity, rotational speed and twist are known, then $\alpha$ is known and you can solve for $Cm_{ac}$ Jul 28 '17 at 7:03

If you have access to a model of your airfoil, a moment balance, and a wind tunnel, you may determine $C_{M_{ac}}$ experimentally: the moment the wing experience may be normalized to calculate $C_{M_{ac}}$
Then solving for alpha is trivial. Otherwise a tool like XFLR5 may be useful to find $C_{M_{ac}}$ for your airfoil.
• @user74671 Yes. It is published, but the only thing i can find is a relation between $Cm$ and $α$. airfoiltools.com/airfoil/details?airfoil=e328-il (last plot) *Pitching moment has to be zero for static stability Jul 28 '17 at 11:23