Flying

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It takes energy to fly. How much?

Big Ideas: 
  • To fly, you have to move in a fluid, and move that fluid around, which takes energy.

 

Glider_final9.avi (right-click and choose "Save Link As..." to download to your computer)

Physics of Flight

The basic physics of flying has been well understood since about 19171. We do not intend to go into these details here; we are going to concentrate on the energy and environmental cost of air transport. The energy cost of transport is how much energy it takes to move a given mass a given distance2.

For an unconventional approach to the pedagogy of flight, see Ref.3

Why do we need energy to fly?

In order to fly, aircrafts or birds need to continually push air downwards so that the deflected air pushes back and provides a lift force. This continual addition of momentum to the air requires a continual input of mechanical energy. For straight and level flight, this energy comes from the burning of fossil fuels in the engines of an aircraft, or the conversion of food energy in the muscles of a bird. For gliding flight, the energy comes from the loss of potential energy as the glider descends.

 

Measuring the energy cost of flying.

Consider the forces acting on an aircraft in level flight at a constant velocity. All forces have to sum to zero, so we can be sure that the lift L equals the weight W, and that the thrust from the engines T equals the total drag D. The best measure of the quality of an aircraft, from the point of view of minimizing the energy cost, is the ratio of lift to drag, L/D. Plainly, the less drag you have for a given weight, the less thrust, and therefore energy, you need to get the aircraft from one place to another. The newest airliners like the Boeing 787 have lift to drag ratios of about 20. High performance sailplanes (i.e. gliders) have much higher ratios, but they are not configured to carry useful loads.

 

 

Now consider what happens if we turn off the engines and let the aircraft glide. For the sum of the forces to be zero, the aircraft pitches down and assumes a glide slope of angle θ. You can see from the diagram below that tan θ = D/L.         

 

So an aircraft with L/D = 20 will glide at an angle tan θ = 1/20, i.e. θ ≈ 3°. It is no coincidence that all airports require airliners to approach the runway for landing at an angle 3°, for at this angle airliners are almost gliding, and the engines are therefore fairly quiet.

Given L/D it is a simple step to calculate the energy cost of transport. If an airliner has a glide slope of 1 in 20, that means it loses potential energy mgh for every distance of d = 20h travelled. Hence the energy cost of transport, energy divided by (mass times distance) = mgh/(md) = gh/(20h) = g/20 ≈ 0.5 m/s2 = 0.5 MJ/(tonne.km). 

Now look up the fuel and range statistics for a Boeing 747-300 (the long-haul version, which we choose because short-haul versions spend a larger proportion of their time and energy taxiing around airports, accelerating and braking):4

Maximum mass5: 378 tonnes

Fuel capacity: 199,000 L

Range: 12,400 km

If we reckon on the aircraft carrying, on average, half its fuel capacity, and fuel has a density of 0.7 kg/L, then at the mid-point of a long flight, the mass of aircraft is about 300 tonnes. Jet fuel has a heat of combustion of around 34 MJ/L.

Energy cost = (127,000 L)(34 MJ/L)/((200 tonnes)(15,700 km)) = 1.8 MJ/(tonne.km)

Hmm. This nothing like the 0.5 MJ/(tonne.km) we estimated from the glide slope. We need to take account of the fact that the engines are not 100% efficient, in fact they are only about 35% efficient6, so for every 100 J of fuel burnt, only 35 J goes into pushing the aircraft forward; the rest just heats the environment (directly).

Hence a better calculation from the glide slope (which is 1 in 18 for the 7477) would be:

Energy cost = (g/18)/(0.35) ≈ 1.6 MJ/(tonne.km)

This is about 10% less than the in-service data - not bad for a simple calculation. No wading through tables of data. Just two numbers: the glide slope and the engine efficiency. This instantly tells us the two things that have to be improved to reduce the energy cost in terms of MJ/(tonne.km): the glide slope and the engine efficiency. However, each is a long slog. The glide slope is determined by the aerodynamic cleanliness of the aircraft, particularly the slenderness of the wings. However, the overall shape of a jet airliner has changed little since the Boeing 707 prototype first flew in 1954, and so there have not been enormous improvements in L/D

Lift-to-drag ratios for aircraft and birds (years shown are first-flight dates)
Type (L/D)max
$5 balsa glider8 4
Sparrow9 4
Gull8 11
Albatross8 20
Piper Warrior (shown in movie) 10
Boeing 707 (1954)7 18.5
Boeing 747 (1969, shown in movie)7 18
Boeing 787 (2009)10 21.5

Note how small things like sparrows and balsa gliders tend not to fly very well. Big birds like the Albatross fly much better than small ones. Ditto for aircraft. How to set up a simple experiment to measure the glide slope and L/D ratio for a balsa or paper glider can be found in our "Balsa Gliders and 747s" article11.

Engine efficiency is determined in part by how hot one can run the combustion chamber, and this is determined by the quality of materials used. A factor of two has been gained in the last 50 years; efficiencies have risen from about 17% to 35%6.

 

Energy per passenger-km or per tonne of freight

So far we have only considered the energy cost per tonne of aircraft, as that is the number basic physics tells us. Plainly a more telling quantity is the energy per passenger-km. Given that the energy per tonne-km is closely constrained by physics, the next most consideration in airliner design is the mass of the aircraft divided by the number of passengers. The more passengers you can get into a lighter aircraft the better. Here, materials science is having an effect. The Boeing 787 is largely built of composite materials while the 747 and earlier models were entirely aluminum. Comparing the long-haul 787-9 with the long-haul 747-300 we see that the former has a mass of 0.7 tonnes per seat, and the latter 0.8 tonnes per seat - a 14% improvement12.

 

The Jevons Paradox

Will these improvements in efficiency help the environment? Unlikely. As W. Stanley Jevons noticed when studying the British coal industry in the 1860s13, the more efficiently a resource is used, the more that resource gets used. In other words, improvements in the energy efficiency of flying are unlikely to reduce the carbon footprint of the global aviation business. The more efficiently planes fly, the cheaper flying will become, and the more people will fly, and the greater will be the GHG emissions. Too bad.

  

150 tonnes fuel x 44/14 = 470 tonnes CO2

  They still do it better than we do

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