Figure of Merit

There are several difficulties in defining an efficiency factor for a helicopter rotor because many parameters are involved, such as disk area, solidity, blade aspect ratio, airfoil section characteristics, and tip speed. The power loading parameter discussed previously is one measure of rotor efficiency because a helicopter of a given weight should be designed to hover with the minimum power requirements; that is, the ratio T/P should be made as large as possible. However, the power loading is a dimensional quantity and so a standard nondimensional measure of hovering thrust efficiency called the figure of merit has been’ adopted. This quantity is calculated using the simple momentum theory as a reference. The ideas of a figure of merit were first introduced by Renard (1903) and Glauert (1935), but it was introduced in its present form during the 1940s by Richard H. Prewitt of Kellett Aircraft. The figure of merit is equivalent to a static thrust efficiency and defined as the ratio of the ideal power required to hover to the actual power required, that is,

Ideal power required to hover

FM = ——– f^—— ———————— <1- (2.44)

Actual power required to hover

The ideal power is given by the simple momentum result in Eq. 2.34. For the ideal case, the figure of merit must always be unity because the momentum theory assumes no viscous losses; hence the ideal power is entirely induced in origin. In reality, viscous effects manifest as both induced and profile contributions, and these are always present in actual power measurements. Therefore, for a real rotor the figure of merit must always be less than unity.

The figure of merit or FM can be used as a gauge as to how efficient a given hovering rotor is in terms of generating thrust for a given power. However, to qualify this, it should only be used as a comparative measure between two rotors when the rotors are also compared at the same disk loading, a point made again later. The figure of merit can also be written as

where the measured value of power coefficient, C/>meas, will include both induced effects and all of the other “nonideal” physical effects that have their origin from viscosity.

A representative plot of measured figure of merit versus rotor thrust is shown in Fig. 2.8. It will be apparent that the FM reaches a maximum and then remains constant or drops off slightly. This is because of the higher profile drag coefficients (> C^) obtained at higher rotor thrust and higher blade section AoA. For some rotors, especially those with less efficient airfoils, the curve can exhibit a peak in FM, followed by either a progressive or abrupt decrease thereafter. Therefore, the FM behavior in the high thrust range will, to some extent, be a function of airfoil shape and airfoil stall type (i. e., gradual or abrupt – see Section 7.9). In practice, FM values between 0.7 and 0.8 represent a good hovering performance for a helicopter rotor. State-of-the-art rotors may have figures of merit approaching 0.82 (see Fig. 6.2), although this probably represents the upper limit for a helicopter rotor with conventional technology.

Using the modified form of the momentum theory with the nonideal approximation for the power, the figure of merit can be written as

c3/2

kC3t/2 oCb

V2 8

(2.46)

with the results being shown in Fig. 2.8. Notice that at low operating thrusts the figure of merit is small. This is because in Eq. 2.46 the profile drag term in the denominator is large compared to the numerator. As the value of CT increases, however, the importance of the profile power term decreases relative to the induced term and FM increases. This continues until the induced power dominates the profile term and the figure of merit will begin to approach a value of 1/k, which is shown more clearly in Fig. 2.9. In practice, however, the profile drag contribution decreases this value somewhat. Therefore, the FM curve reaches a firm plateau region, which represents the maximum attainable FM of the rotor. In practice, at higher values of rotor thrust the profile drag (and power) increases quickly as the blade begin to stall, which will again cause a reduction in FM.

It has been mentioned that to be meaningful the figure of merit must only be used as a gauge of rotor efficiency when two or more rotors are compared at the same disk loading. This is because increasing the disk loading, DL (= T/ A) will increase the induced power relative to the profile power, producing a higher figure of merit and a potentially misleading comparison between two different rotors. This can be seen if the figure of merit is written dimensionally as

Therefore, it would be considered inappropriate to compare the values of the figure of merit of two rotors with substantially different disk loadings because the rotor with the higher disk loading will generally always give the higher figure of merit, all other factors being equal (i. e., it would not be meaningful to compare the values of FM of a helicopter rotor with its relatively low disk loading to a tilt-rotor with its much higher disk loading). Therefore, caution should always be exercised in the use of FM as a means of comparing the efficiency of different rotor systems.