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.

_{}
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

_{}

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}).

excess
power, visit NASA Glenn - GO! |

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 _{}

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

_{}

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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*

__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.

**Figure 4.7** Lockheed Martin
F-16 performing a vertical accelerated climb.

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