I know the formula of figure of merit and blade loading coefficient individually but how the plot can be obtained I am not getting that. What would be the formula for figure of merit in terms of blade loading coefficient. Formulas I know are:

• Please edit your question to include which equations you have to start with. There is nothing we can do to help this question without simply doing your homework for you -- and that is not the purpose of this forum. Commented Apr 18, 2023 at 15:20
• @RobMcDonald when the FOM, coefficient of thrust, solidity of the rotor all are fixed which will be the variable in the plot?
– Eaza
Commented Apr 19, 2023 at 8:05
• My definition of blade loading coefficient is ($C_T/\sigma$). Plotting FOM vs. $C_T/\sigma$ is a common thing to do. Typically, the $C_T/\sigma$ variation represents a change in collective for a fixed rotor -- which means that $C_T$ is varying. Again -- please share the formulas you have and we can help show how to use them - or suggest the missing formula. Commented Apr 19, 2023 at 16:35

What you are missing is the rotor 'polar' equation. While not exactly the same as an aircraft's drag polar, it plays that role in performance equations.

$$C_P=\frac{\kappa {C_T}^{3/2}}{\sqrt{2}}+\frac{\sigma C_{d,0}}{8}$$

An ideal rotor has $$\kappa$$=1 and $$C_{d,0}$$=0. Such a rotor has $$FM$$=1.0.

The $$\kappa$$ is a non-ideal induced flow factor. It plays the same role as an Oswald efficiency factor, but is inverted in definition. A typical value might be $$\kappa$$=1.15.

The $$C_{d,0}$$ is the airfoil profile drag coefficient. It contributes to form the profile drag power. A typical value is $$C_{d,0}$$=0.01.

The figure of merit $$FM$$ is the ratio of the ideal power to actual power.

$$FM=\frac{\mathrm{ideal}}{\mathrm{actual}}$$

I.e. applying it to our definition above, we get...

$$FM=\frac{ \frac{{C_T}^{3/2}}{\sqrt{2}}}{\frac{\kappa {C_T}^{3/2}}{\sqrt{2}}+\frac{\sigma C_{d,0}}{8} }$$

Which you should be able to plot.

• How to get the value of k?
– Eaza
Commented May 5, 2023 at 19:50
• You perform complex analysis on your rotor design -- including planform, airfoils, twist, etc. Ideally under loaded conditions as rotors deflect substantially when operating. Or you measure performance of your rotor and choose k to match the data. Or you use historical data for similar rotors. Typical is 1.15 -- 1.0 is ideal, you can't go lower than that. Commented May 5, 2023 at 20:21