In this chapter we will make the connections between aircraft performance and propulsion system performance. For a vehicle in steady, level flight, as in Figure 13.1, 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 13.1: A schematic of the forces on an aircraft in steady level flight

13.1 Vehicle Drag

Recall from fluids that drag takes the form shown in Figure 13.2, 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 13.2: 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

13.2 Power Required

Now we can look at the propulsion system requirements to maintain steady level flight since

and

Thus the power required (for steady level flight) takes the form of Figure 13.3.

Figure 13.3: Typical power required curve for an aircraft.

The velocity for minimum power is obtained by taking the derivative of the equation for with respect to 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 ). 13.3 Aircraft Range: the Breguet Range Equation

Consider an aircraft in steady, level flight, with weight , as shown in Figure 13.1. The rate of change of the gross weight of the vehicle is equal to the fuel weight flow:

For steady, level flight, , , or

The rate of change of aircraft gross weight is thus

Suppose and remain constant along the flight path:

We can integrate this equation for the change in aircraft weight to yield a relation between the weight change and the time of flight:

where is the initial weight. If is the final weight of vehicle and , the relation between vehicle parameters and flight time, , is

The range is the flight time multiplied by the flight speed, or,

The above equation is known as the Breguet range equation. It shows the influence of aircraft, propulsion system, and structural design parameters. 13.3.1 Relation of overall efficiency, , and thermal efficiency Suppose is the heating value (``heat of combustion'') of the fuel (i.e., the energy per unit of fuel mass), in J/kg. The rate of energy release is , so

and

Thus

and

or

Keep in mind that, in general,

13.3.2 The Propulsion Energy Conversion Chain

The above concepts are depicted in Figure 13.4 as parts of the propulsion energy conversion train mentioned in Part I, which shows the process from chemical energy contained in the fuel to energy useful to the vehicle.

Figure 13.4: The propulsion energy conversion chain from Part I

The combustion efficiency is near unity unless conditions are far off design. We can therefore regard the two main drivers as the thermal and propulsive13.1 efficiencies. The evolution of the overall efficiency of aircraft engines in terms of these quantities was shown in Figure 11.8. 13.4 Aircraft Endurance

If the time spent in the air is of interest and not the distance traveled then one is concerned with...