A Simple Approach to the Energy Cost of Flying

It takes energy to fly. How much?


 

Movie: filmed and produced by Oliver Millar for the C21 Project

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The basic physics of flying has been well understood since about 1917[1] J. D. Anderson, History of Aerodynamics (Cambridge 1997). 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 distance. For more on the aerodynamics see https://c21.phas.ubc.ca/article/air-transport/.

For an unconventional approach to the pedagogy of flight, see Ref.[2]C. E. Waltham “Flight without Bernoulli, Phys. Teach., 36 (1998) 457-462..

For a beautiful video debunking the standard textbook explanation of how airfoils work, see Ref.[3] Holger Babinsky, “How airplane wings really work” http://gizmodo.com/5878773/clever-1+minute-video-shows-how-airplane-wings-really-work [2019-10-02]..

Why do we need energy to fly?

In order to fly, aircraft or birds need continually to 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.

Fig.1. Balanced forces on an aircraft in level flight at constant velocity, resolved into lift, drag, weight and thrust (author diagram).

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.

Fig.2. Balanced forces on an aircraft gliding at constant velocity, resolved into lift, drag and weight (author diagram).

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.

Fig.3. Glide path (slope) of an aircraft with no power.

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 J/(kg.m) = 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]Boeing 747 http://en.wikipedia.org/wiki/Boeing_747

Maximum mass[5] I cannot bring myself to call it “weight”.: 378 tonnes

Fuel capacity: 200,000 L

Range: 12,400 km

If we reckon on the aircraft carrying, on average, half its fuel capacity, and fuel has a density of 0.8 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 35 MJ/L.

Energy cost = (127,000 L)(35 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% efficient[6] Vaclav Smil, Energy Transitions: Global and National Perspectives (Praeger, 2016), 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 747[7]Peter Wegener, What makes airplanes fly?, (Springer, NY, 1991) p.169.) would be:

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

This is about 10% less than that given by 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.

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. You can easily set up a simple experiment to measure the glide slope and thus the L/D ratio for a balsa or paper glider.

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%.

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 747-300 with the long-haul 787-9 we see that the former has a mass of 0.8 tonnes per seat, and the latter 0.7 tonnes per seat – a 14% improvement[8]We compare long-haul aircraft because short-haul ones carry much less fuel and consequently have much lower mass per seat; the short haul 787 is only 0.5 tonnes per seat.

Fig.4. Curtiss HS-2, c.1920. L/Dmax ~ 8 (author photo).

 

Fig.5. Boeing 247, c.1935. L/Dmax ~ 14 (author photo).

 

Fig.6. Boeing 767, c.1985. L/Dmax ~ 20 (author photo).

 

 

 

Updated (CEW) 2019-10-10

Footnotes   [ + ]

1. J. D. Anderson, History of Aerodynamics (Cambridge 1997)
2. C. E. Waltham “Flight without Bernoulli, Phys. Teach., 36 (1998) 457-462.
3. Holger Babinsky, “How airplane wings really work” http://gizmodo.com/5878773/clever-1+minute-video-shows-how-airplane-wings-really-work [2019-10-02].
4. Boeing 747 http://en.wikipedia.org/wiki/Boeing_747
5. I cannot bring myself to call it “weight”.
6. Vaclav Smil, Energy Transitions: Global and National Perspectives (Praeger, 2016)
7. Peter Wegener, What makes airplanes fly?, (Springer, NY, 1991) p.169.
8. We compare long-haul aircraft because short-haul ones carry much less fuel and consequently have much lower mass per seat; the short haul 787 is only 0.5 tonnes per seat

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