### AIRCRAFT PERFORMANCE

### Fixed Wing Aircraft

### RANGE AND ENDURANCE

The range of the aircraft is a function of the rate at which fuel is being burnt, the duration of cruise, the flight speed and the aerodynamic performance during the flight. The mode of flight can also have a measurable effect on the distance traveled. Of major importance is the fact that the fuel used to supply the engines must be carried onboard the aircraft. This fuel weight must be supported by the aerodynamic lift along with the aircraft structural weight and the payload. While the structure and payload weights are fixed, the fuel weight changes significantly during the flight, leading to changing performance during long range cruise.

For cruise performance in level flight the excess power goes to zero. It is then a problem of analyzing the aircraft internal energy balance. The balance of energy being created by the combustion of fuel and the energy dissipated in overcoming the resistance to motion through the air.

For a short period of time, **dt**, while the aircraft moves
a distance, **ds**, at a velocity **V**,

so that, **ds = V.dt**, if the energy is balanced then,

Energy(combustion of fuel) = Energy(dissipated by drag)

that is

_{in}=Energy

_{out}

where

**dW**_{f}
is weight of fuel burnt during time **dt**,

**Hc** is energy
content (calorific value) of fuel and

**η** is the
overall efficiency of the propulsion system.

This is also equivalent to the Energy produced by the propulsion system over the time step

where,

**T** is the thrust
produced,

**V** is the flight
velocity, hence

**T.V** is the supplied
power over time **dt**.

Thus $ TV.dt = dW_f H_c η $

Rearranging this term into parameters that are readily able to be measured gives

where, **TSFC** is thrust specific fuel consumption, and is
normally available from engine manufacturer tests.

where

**D** is drag on the
vehicle,

**V** is flight velocity,

hence **D.V** is required
power to be overcome during time **dt**.

This balance of energy components **E**_{in} **=
E**_{out} gives,

Ifthere is excess thrust over drag then an excess of energy will be availble to change the motion of the aircraft.

In terms of specific energy, $E\/W$,
normally called energy height, **h**_{e}_{ },

Rearranging gives,

Based on the assumption of level flight, **L = W**, the
assumption of roughly constant altitude and velocity cruise, Δhe ≈
0 . If the change in weight of the vehicle is directly related to the
weight of fuel burnt, dW_{f} = -dW , then

or

Assuming approximate steady conditions where velocity and
propulsion performance is kept constant and that the aerodynamic
parameter **L/D** is maintained at a constant value then
integration of this equation gives a prediction of range,

This is the classic Breuget Range equation. It can be used to
give reasonable estimates of range in still air. However the
assumption of constant **L/D** and constant **V/TSFC**
may not be too accurate in practice. To maintain constant **L/D**
with the changing weight the aircraft would need to drift up in
altitude so that a constant angle of attack was maintained. It may
also be required that the aircraft change speed to maintain a
constant** V/TSFC**. So to get detailed estimates over
long range cruise conditions it may be necessary to do a numerical
summation over short segments where the assumptions are accurate.

### Range with Headwind or Tailwind

The above calculations are based on still air conditions. If the aircraft is flying into a headwind or with a tailwind, then the distance travelled through the air is the same but the ground distance covered will change.

The energy dissipated in overcoming drag is unchanged but the
effective energy output relative to the ground will be based on the
relative velocity, ( **V - V**_{w}_{ })
where **V**_{w} is the velocity of the wind,
positive meaning a headwind.

The energy balance becomes,

and the final range equation becomes,

### Endurance

This is the total time taken during flight. It is directly
related to the rate of fuel consumption. Assuming level flight with **L=W** and **T=D**

then endurance can be found from,

### Optimum Range or Endurance

In order to maximise range for a given load of fuel, a balance
of **V/TSFC** and **L/D** must be found.
Increasing **V** will increase range up to a point
where the increasing drag will start to reduce **L/D**
so that the range, which is the product of these two terms, again
starts to reduce.

In order to maximise endurance, fuel flow rate must be reduced.
For engines with roughly constant **TSFC** this means
reducing thrust by flying slowly at minimum drag speed or minimum
power speed.

The optimum flight speed for maximum endurance will be quite low and may be approaching the stall speed, whereas the optimum flight speed for range will be high.