Viewing page 67 of 79

This transcription has been completed. Contact us with corrections.

Fundamental limits to high-altitude flight

The case for compact instruments aboard unmanned aircraft can be summarized simply from first principles. We start with the definition that in sustained level flight, the lift L is equal to the weight W:
[[Mathematical Equation]]
(1)
L = W = (1/2)*p0*(V0^2)*Sw*CL
W/Sw = (1/2)*p0*(a^2)*(M^2)*CL

where p0 is the air density, V0 is the flight speed, Sw is the wing area, and CL is the non-dimensional lift coefficient. The speed of sound a is related to the pressure and density by a2=yp0/p0 and thus we obtain the following relation:
[[Mathematical Equation]]
(2)
Mu^2 = (M^2)*CL = (2/gamma)*((W/Sw)/P0)

This clearly separates the effects of wing loading W/S, air pressure p0 (or equivalently, altitude), and the "reduced" Mach number
[[Mathematical Equation]]
(3)
Mu = M*sqrt(CL)

The maximum Mu achievable by an air foil is therefore a ceiling parameter which determines the minimum tolerable ambient pressure, and hence the maximum altitude:
[[Mathematical Equation]]
(4)
(gamma/2)*p_min = ((Wt/o)/S_Omega)*(1/Mu_max)

Once Mu_max is set, the ceiling can be raised only be reducing wing loading or sweeping the wing (in which case p_min ~cosA, very roughly, A being the sweep angle). Since Mach numbers are limited to approximately 0.7 (to avoid passing chemical constituents through a shock prior to sampling) and CL is limited to values of approximately 1, Mu_max is roughly 0.8. Figure 1 plots achievable altitude as a function of wing loading. 

For a given payload weight, an aircraft's size is determined by its wing loading and its payload fraction (the ratio of payload weight to gross weight). Achievable payload fractions depend on many things, including required range or duration, level of structural technology employed, and desired safety factors. In general, however, it is extremely difficult to achieve payload fractions greater than about 15% in any practical aircraft (the ER-2, for example, has a payload fraction of about 6% at take-off). Aircraft with low wing loadings quickly become physically very large: carrying a 50 kg payload to 35 km with the maximum allowable wing loading of 250N/m2 (about 5 lbs/ft2) requires an area of about 20 m2, whereas a 1000 kg payload requires about 400 m2 (for comparison, the wing area of the ER-2 is 93 m2 with a wing loading of about 1000 N/m2, that of a 747 about 510 m2 with a wing loading of about 6,000 N/m2). Large aircraft stressed to the same levels as smaller aircraft have higher structural weight fractions due to the square-cube effect. [[footnote 1]] It must also be considered that the low payload fractions mean that every unit mass

[[footnote 1]] The square-cube effect states that the stress in similar structures increases with linear dimensions if the imposed load is proportional to the structural weight, since the latter grows as the cube of the linear dimensions while the material cross-section carrying the load grows only as the square. Thus, for constant structural technology, wing loading, and design conditions, larger aircraft will have smaller useful payload fractions. See F.A. Cleveland, "Size Effects in Conventional Aircraft Design", Journal ofAircraft, Vol. 7, No. 6, Nov-Dec 1970.

II-2

Transcription Notes:
Attempted to formulate the equations in a usable format, inserting the names of Greek letters where appropriate.