In this lecture we will make the connections between aircraft performance and propulsion system performance.
For a vehicle in steady, level flight, the thrust force is equal to the drag force, and lift is equal to weight. Any thrust available in excess of that required to overcome the drag can be applied to accelerate the vehicle (increasing kinetic energy) or to cause the vehicle to climb (increasing potential energy).
Figure 4.1 Force balance for aircraft in steady level flight.
Recall from fluids that drag takes the form shown below, being composed of a part termed parasitic drag that increases with the square of the flight velocity, and a part called induced drag, or drag due to lift, that decreases in proportion to the inverse of the flight velocity.
Figure 4.2 Components of vehicle drag.
The minimum drag is a condition of interest. We can see that for a given weight, it occurs at the condition of maximum lift-to-drag ratio
We can find a relationship for the maximum lift-to-drag ratio by setting
from which we find that
Now we can look at the propulsion system requirements to maintain steady level flight since
Thus the power required (for steady level flight) takes the form
Figure 4.3 Typical power required curve for an aircraft.
The velocity for minimum power is obtained by taking the derivative of the equation for Preq with respect to V and setting it equal to zero.
As we will see shortly, maximum endurance (time aloft) occurs when the minimum power is used to maintain steady level flight. Maximum range (distance traveled) is obtained when the aircraft is flown at the most aerodynamically efficient condition (maximum CL/CD).
Homework P4 (PDF)
To see the implications of excess power, visit NASA Glenn - GO!
Again, for steady, level flight,
The weight of the aircraft changes in response to the fuel burned
applying the initial conditions, at t = 0 W = Winitial \ const. = ln Winitial
the time the aircraft has flown corresponds to the amount of fuel burned, therefore
then multiplying by the flight velocity we arrive at the Breguet Range Equation which applies for situations where Isp and flight velocity are constant over the flight.
This can be re-written in other forms:
NASA Glenn has a great interactive Java program for aircraft range - GO!
For a given amount of available fuel energy (Joules), the maximum endurance (time aloft) is obtained at a flight condition corresponding to the minimum rate of energy expenditure (Joules/second), or Preqmin, as shown in Figure 4.3.
We can determine the aerodynamic configuration which provides the minimum energy expenditure:
So the minimum power required (maximum endurance) occurs when is a maximum.
With a little algebra we can arrive at an expression for the maximum endurance. Setting
we find that
Thus the minimum power (maximum endurance) condition occurs at a speed which is 3-1/4 = 76% of the minimum drag (maximum range) condition. The corresponding lift-to-drag ratio is 86.6% of the maximum lift-to-drag ratio.
Figure 4.4 Relationship between condition for maximum endurance and maximum range.
which can be substituted into
Such that, for maximum endurance
which can be integrated (assuming constant Isp) to yield
Have some interactive endurance fun at NASA Glenn - GO!
Any excess in power beyond that required to overcome drag will cause the vehicle increase kinetic or potential energy. We consider this case by resolving forces about the direction of flight and equating these with accelerations.
Figure 4.5 Force balance for an aircraft in climbing flight.
where is the accel. normal to the flight path
where is the accel. tangent to the flight path
So the change in height of the vehicle (the rate of climb, R/C) is:
which is instructive to rewrite in the form
excess power = change in potential energy + change in kinetic energy
For steady climbing flight,
and the time-to-climb is
for example, and
The power available is a function of the propulsion system, the flight velocity, altitude, etc. Typically it takes a form such as that shown in Figure 4.6. The shortest time-to-climb occurs at the flight velocity where Pavail Preq is a maximum.
Figure 4.6 Typical behavior of power available as a function of flight velocity.
Homework P5 (PDF)
To see more on climbing flight, visit NASA Glenn - GO!
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