In Part 2 of this series on aircraft performance, we will be discussing some of the basic calculations that go into aircraft performance evaluations, with an eye towards gaining a basic appreciation for the mathematics involved.
The cornerstone of any performance or mission analysis calculation, will be the airplane's drag polar. For the purposes of this presentation, we will be looking at a classical drag polar representation. There are modified versions of the drag polar in use, but for our purposes, the classical representation should serve just fine.
The drag polar relates the airplane's total drag to a combination of the zero-lift, or parasitic drag, and the induced drag - or the drag due to lift. The induced drag, in turn, can be related to the lift coefficient squared, divided by the airplane's aspect ratio and the aerodynamic efficiency, or Oswald's efficiency. It should be noted as well that the aspect ratio is itself related to the airplane's wing span squared, divided by the wing reference area.
In order to be effective, the drag polar needs to take into account a variety of real world effects. The first element to note is that the wing reference area is itself a somewhat arbitrary quantity. It's what we would get if we were to extend the wing leading edge, and the wing trailing edge all the way to the airplane's centerline. What this means in practice is that the aerodynamic efficiency must be adjusted to take into account such real world effects as the wing cross section, wing-to-fuselage integration, whether the design has a canard or a horizontal tail, and so forth.
Similarly, the parasitic drag will be a function of the airplane's wetted area, the coefficient of friction (book keeping the effects of rivets and similar protrusions), and the Mach number. Particularly in transonic and supersonic flight, the parasitic drag can be expected to rise significantly due to wave drag.
Finally, in addition to being affected by fuselage integration, wing cross section, and similar effects, the aerodynamic efficiency will also be a function of the airplane's Mach number, and aerodynamic loading (or g-loading).
The mission analysis in turn is calculated by breaking the mission down into small segments, which can then be analyzed for fuel burn. Effectively, the airplane is analytically "flown" through the mission. In its most simple form this principal can be expressed by the Breguet range equation - which also provides insight into what these equations can tell us.
The Breguet range equation relates the range of an airplane to its initial weight, at the beginning of the flight, and its final weight, at the end of the flight - in a logarithmic manner. In other words, the relationship will be highly nonlinear. Range is also affected in direct proportion to the cruise velocity, as well as the lift-to-drag ratio, and is inversely proportional to the thrust specific fuel consumption (TSFC). The Breguet range equation, however, is also accurate only if the velocity, fuel consumption, and lift-to-drag ratio remain constant. It should also be pointed out that the lift-to-drag ratio is itself an output of the drag polar.
More detailed mission analyses will break a mission down into a series of segments, each small enough that variations in speed, altitude and specific fuel consumption become negligible across that particular segment. As an example, an equation for a constant altitude and speed cruise segment is portrayed here. For a complete mission, each segment of flight would need to be evaluated separately, with longer segments - such as cruise - further subdivided.
Part of the real beauty of the relationships described here, is that they allow us to draw comparisons and conclusions from different design trades without necessarily exercising a detailed mission calculation. Consider for example the Boeing 777X currently under development as a more fuel efficient, longer range development of the original 777 design. The 777X is expected to increase the wingspan of the airplane from the 212-ft 7-in found on the 777-200LR, to 235-ft 6-in for the 777-8X - a 10.8% increase.
From the drag polar, the maximum lift-to-drag can be calculated from the relationship shown below. Note that the lift-to-drag ratio is related to the square root of the wing aspect ratio, which in turn is related to the wingspan squared. Assuming a constant wing area, therefore, we would conclude that the 777X should offer a 10.8% increase in range over existing 777 models - just by virtue of the increased aspect ratio alone.
This of course, might lead us to wonder why Boeing did not take advantage of an extended aspect ratio design much earlier. The answer to this lies in the structural limitation for an all-metal wing. The existing 777 design represents the practical limit for airline aspect ratio for an all-metal structure. What makes the higher wing span and aspect ratio of the 777X possible is the introduction of a composite structure.
Again, a great deal of insight can be gained just by virtue of examining the underlying relations for range and the drag polar.
No comments:
Post a Comment