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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: Formula

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    $\begingroup$ 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. $\endgroup$ Commented Apr 18, 2023 at 15:20
  • $\begingroup$ @RobMcDonald when the FOM, coefficient of thrust, solidity of the rotor all are fixed which will be the variable in the plot? $\endgroup$
    – Eaza
    Commented Apr 19, 2023 at 8:05
  • $\begingroup$ 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. $\endgroup$ Commented Apr 19, 2023 at 16:35

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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.

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  • $\begingroup$ How to get the value of k? $\endgroup$
    – Eaza
    Commented May 5, 2023 at 19:50
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    $\begingroup$ 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. $\endgroup$ Commented May 5, 2023 at 20:21

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