Aircraft performance

Air density

The four forces acting on an airplane are trust, lift, drag and weight. All factors except weight are influenced by the surrounding air density. Air density is mainly directed by temperature and pressure, to be exact vapour pressure. Higher temperatures makes the air less dense (thinner) and at a lower temperature the air will be more dense (thicker). Warmer air causes to generate more molecule energy then colder air, molecules need more space to move. Moisture (water vapour) displaces air also it makes the air also thinner. So humidity is also a factor which needs to be considered.


molecule enegry

Trust

Engine performance is influenced in multiple way. Less dense air degrades engine performance. The fuel air ratio will change in a bad manner and causes the engine to generate lesser power. When we fly at higher altitudes (normal aspirated engine) we have to lean the fuel air mixture, this is done by reducing the amount of fuel entering the cylinder to the best air / fuel ratio (1:15). Leaning decreases the total mass of fuel entering the cylinder which results in less brake horse power generated at the crankshaft. Turbo or super charged engines can counteract the effect of thinner air to a certain degree by ‘pushing’ or ‘pulling’ in more air into the engine and making the air artificially less dense. A second effect degrading engine performance is generated by the propellor. Caused by having less ‘grip’ in lesser dense air. The propellor will simply generate less lift in thinner air.

Lift

A very important formula in aviation is the lift formula. We can conclude lesser air density (r) will generate lesser lift. But a lower airspeed caused by the decreased engine trust will be effected by square root of airspeed. So less trust generates also less lift!

L = (1/2) * r * v^2 * a * CL

L = lift (which must equal the airplane’s weight in pounds)
r = density of the air
v = velocity
a = wing area of an aircraft in square feet
CL = coefficient of lift , which is determined by the type of airfoil and angle of attack

Drag

Drag is also influenced by air density. The less dense air generates less drag. The drag formula is almost the same as the lift formula but opposes trust.

D = (1/2) * r * V^2 * a * CD

D = drag
r = density of the air
v = velocity
a = reference area
CD = coefficient of drag

Weight

Weight is also an important factor, the less weight the less lift needed. A full plane, high temperatures, high humidity and high elevation airfield can be a dangerous combination.

Putting pieces together

We can conclude, the most significant factor influencing the performance of aircraft is air density, which is basically influenced by temperature and pressure. Take-off and landing performance calculations take these elements in account. This is the reason why we have to calculate pressure altitude (PA) and density altitude (DA) and combine these with performance tables in our manuals. Also the serious important factor weight needs to be addressed.

Pressure Laps Rate

This formula: PLR = 96 × ( T in kelvin) / QNH (in hPa) is used to calculate the exact pressure lapse rate. For example at 15°C and with ISA QNH the lapse rate will be:

Pressure Lapse Rate = 96 × (15 + 273) / 1013,25 = 27,3 ft/hPa

Pressure Altitude, PA

This is altitude or elevation corrected for non standard pressure.

Pressure Altitude = Altitude + (1013 - QNH) × 27

Density Altitude, DA

Density altitude is pressure altitude (PA) in the standard atmosphere corrected for non standard temperature. DA can be calculated by taking PA and adding (or subtracting) 120 feet for each 1°C temperature difference above (or below) the standard atmosphere temperature at that ISA altitude. ISATpa = 15°C.

Density Altitude = Pressure Altitude + (OAT - ISATpa) × 120

In practice

Mostly it is enough to calculate the pressure altitude (PA) to find the performance data in the aircraft’s manual / POH. In such performance tables all important factors are gathered. Have a look at the Take-off performance table of the Pilatus PC-12/45 beneath.

Pilatus pc12 take off performace table

Importance of calculating performance data