# IV.   Aircraft Performance

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. Q14 (PDF)

## A.   Vehicle Drag

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. where and Thus or 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 and  and ## B.   Power Required

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)

## C. Aircraft Range, the Breguet Range Equation The weight of the aircraft changes in response to the fuel burned or 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:  where  or ## D.  Aircraft Endurance

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 where Then 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 and  and 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.

Continuing which can be substituted into Such that, for maximum endurance which can be integrated (assuming constant Isp) to yield ## E.   Climbing Flight

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 or in words:

excess power = change in potential energy + change in kinetic energy

Q15 (PDF) and the time-to-climb is where 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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