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 P_req 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 C_L/C_D).

Homework P4 (PDF)

To see the implications of excess power, visit NASA Glenn - GO!

 

C. Aircraft Range, the Breguet Range Equation

Again, for steady, level flight,

                       

                       

The weight of the aircraft changes in response to the fuel burned

                       

or

                       

applying the initial conditions,    at    t = 0         W = W_initial        \  const. = ln W_initial

                       

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        

 

NASA Glenn has a great interactive Java program for aircraft range - GO!

 

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 Preq_min, 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

                       

 

Have some interactive endurance fun at NASA Glenn - GO!

 

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)

For steady climbing flight,

                       

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 P_avail ­ P_req 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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