## Monday, August 2, 2010

### Does a heavier airplane descend quicker than a lighter one ?

It is not uncommon for air traffic control (ATC) to issue a descent clearance at a particular airspeed. A very interesting question here is, given two identical airplanes, one loaded to full capacity and the other almost empty, which one would have a greater rate of descent for a given true airspeed ?

Consider two identical Boeing 737s, one heavy and the other light.

For any given angle of attack, the heavier plane would have to fly at a greater speed in order to generate more lift to maintain level flight. This is evident from the lift equation, given below:

Total Lift L = 0.5 * Cl * p * S * v^2, where

Cl - coefficient of lift that corresponds to a particular angle of attack
p - density of the medium, in this case air
S - Effective area of the wing that contributes to generating lift
v - velocity of air flow over the wing, which can be approximated to True Air Speed (TAS)

The gradient of descent, or the horizontal distance gained per foot of altitude lost depends on the ratio of lift-to-drag, and is greatest (gives the maximum range for a given loss of altitude) when the lift-to-drag ratio is maximum. This condition (maximum lift-to-drag ratio) occurs at a particular angle of attack, for a given type of wing/airframe, and does not depend on the weight of the aircraft.

However, the AIRSPEED at which this given angle of attack occurs DOES vary with the weight of the aircraft, as it was seen that more lift is required for heavier weights, and this lift can be generated by flying at a higher airspeed. Consequently, the maximum "glide" range for the heavy 737 will be the same as for the light 737, PROVIDED that the heavy 737 is flown at a higher airspeed to give the same angle of attack which gives the maximum lift-to-drag ratio. The heavy 737, flying at its higher minimum drag speed, will come down faster than the light 737, flying at its lower minimum drag speed , but both planes will travel the same ground distance for a given altitude loss.

Now, to answer the question. Assume that the heavy 737 has a minimum drag speed of about 240 knots, and the light one, about 210 knots. If ATC issues a clearance to descend at 200 knots, then the light 737 will follow the green gradient line (above), and the heavy 737 will follow the red gradient line. The heavy 737 will not cover as much ground distance for a given altitude drop as the light 737 and it will come down much quicker than the light 737.

On the other hand, if ATC issues a descent clearance at 250 knots (as is more often the case), the heavy 737 will follow the green gradient line, and the light 737 will follow the red gradient line. This time, not only will the heavy 737 cover more distance for a given altitude drop (or drop less altitude for a given ground distance) than the light 737, it will also take longer to come down.

It can be concluded that, for airplanes with similar glide characteristics, but different weights, the airplane whose minimum drag speed most closely matches the descent clearance airspeed will be the one that has the most shallow descent gradient (green gradient line) and will take longer to come down; all the others will have a steeper descent gradient (red gradient line) and will come down faster.