Burt Rutan's White Knight and SpaceShip One, Photo Courtesy of Scaled Composites

Thermodynamics and Propulsion

13.4 Aircraft Endurance

If the time spent in the air is of interest and not the distance traveled then one is concerned with endurance. The maximum endurance of an aircraft (or the time aloft) refers to a flight condition that requires the minimum fuel power. Assuming that the overall propulsion system efficiency, $ \eta_{overall}$ , defined as the propulsive power over the fuel power, is constant, maximum endurance can be obtained by minimizing the required propulsive power or rate of energy expenditure, $ P_R$ , as shown in Figure 13.3.

We can determine the aerodynamic configuration which provides the minimum energy expenditure:

so
where
Then
So the minimum power required (maximum endurance) occurs when [画像:$ C_L^{\frac{3}{2}}/C_D$] is a maximum. With a little algebra we can arrive at an expression for the maximum endurance. Setting
we find that

and
Thus the minimum power (maximum endurance) condition occurs at a speed which is [画像:$ 3^{-\frac{1}{4}} = 76\%$] of the minimum drag (maximum range) condition. The corresponding lift-to-drag ratio is 86.6% of the maximum lift-to-drag ratio, shown in Figure 13.5.

Continuing

which can be substituted into
Such that, for maximum endurance,
which can be integrated (assuming constant Isp) to yield

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