WARNING: This blog contains maths. It’s interesting! But it does contain maths. Sorry.
Updated on April 17 to use fuel consumption figures as suggested by Stephen Skinner in his comment.
A Boeing 747 fuelled for a journey from London to Buenos Aires – a flying distance around 6900 miles – might take off with just over 200 cubic metres of fuel – kerosene. Some of this is held in reserve – but let’s imagine a hypothetical flight in which all the fuel was burned.
With a density of 814 kg per cubic metre, this corresponds to a total fuel load of around 175 tonnes. Kerosene is a poorly specified mixture of alkanes – compounds of carbon and hydrogen – with between 6 and 16 carbon atoms per molecule. Here we will approximate the effect of this mixture of alkanes by considering an ‘average’ alkane – dodecane – with a chemical formula, C12H26. Each molecule of dodecane consists of 12 atoms of carbon, each with a relative molecular mass of 12, and 26 atoms of hydrogen, each with a relative molecular mass of 1. So the relative molecular mass of a dodecane molecule is 12 × 12 + 26 × 1 = 170
So 1 mole of kerosene – the Avogadro number of kerosene molecules – weighs 170 grams or 0.17 kg. The fuel load of the plane therefore consists of 175,000 kg ÷ 0.17 kg which is just over 1 million moles of kerosone. When burned in the jet engine,
- Each of the 12 carbon atoms within each kerosene molecule combines with oxygen from the air to make 12 molecules of CO2
- The 26 hydrogen atoms within each kerosene molecule combine with oxygen from the air to make 13 molecules of H2O
So the 1 million moles of kerosene in the original fuel load are turned into:
- 12 million moles of CO2 each weighing 12 + 2 × 16 = 44 grams or 0.044 kg. So the CO2 emitted weighs 12 million × 0.044 kg = 528,000 kg or 528 tonnes
- 13 million moles of H2O each weighing 1 × 2 + 16 = 18 grams or 0.018 kg. So the H2O emitted weighs 13 million × 0.018 kg = 234,000 kg or 234 tonnes
So even though the plane took off with only 175 tonnes of kerosene, by combining this carbon- and hydrogen-rich fuel with the oxygen from the atmosphere, en route to Buenos Aires, the aeroplane produces 234 tonnes of water (H2O) and 528 tonnes of carbon dioxide (CO2) and removing 587 tonnes of oxygen from the atmosphere.
- The effect of the carbon dioxide has been discussed at length. However, I still find these numbers shocking – something between 1 and 1.5 tonnes per passenger. For a shorter trip such as the 3,500 mile ‘hop’ to New York, these figures would represent the combined carbon emissions for the 7,000 mile return journey.
- The effect of the removal of oxygen is not widely discussed but you can see the data here. Don’t worry – we’re not running out.
- The effect of the water is to, sometimes, produce contrails.
Contrails: The air at cruising altitude (≈10 km) is typically close to – 60 °C and usually very dry. When the water vapour concentration exceeds ≈50 parts per million of the nitrogen and oxygen, then the water vapour condenses into ice crystals – and we see wispy ‘cirrus’ clouds. The plane typically releases its 234 tonnes of water over a flight distance of around 11,500 km, or around 20 kg per kilometre. if we guess that the exhaust gases form 4 tubes, each 10 metres in diameter, then the exhaust gases occupy a volume of 314,000 cubic metres per kilometre of flight.
- The water density is thus 20 kg ÷ 314,000 m3 = 0.000064 kg per cubic metre. One mole of water has a mass of 0.018 kg and so the molar density of water is 0.0035 moles per cubic metre.
- The density of air at this altitude is much less than at sea level, because the pressure is only around 30% of that at sea level. The low temperature slightly compensates yielding an air density of around 0.5 kg per cubic metre (compared with 1.2 kg per cubic metre at sea level). One mole of air has a molar mass of 0.029 kg and so the molar density of air is 17 moles per cubic metre.
Comparing these two figures we estimate that the water vapour concentration in the exhaust plume is around 0.0035/17 ≈ 200 parts per million – a factor 4 more than the 50 parts per million required to saturate the air. Thus the water vapour condenses into tiny ice crystals – the condensation trail – or contrail.
The effect of contrails is complex, but on a clear day we can see that they frequently drift across the sky, appearing to nucleate the growth of wispy white ‘cirrus’ clouds. I wrote about a particularly graphic example of this previously. This tends to reduce the amount of solar radiation reaching the Earth, somewhat compensating for the effect of the carbon dioxide emitted along with the water. But at night, the clouds will have a warming effect, and overall the balance is hard to estimate. A recent paper in Nature Climate Change, suggests the cooling effect of contrails actually exceeds the warming effect of the carbon dioxide, but considerable uncertainty still surrounds their net effect. But it would be nice to think that perhaps the solution to the global warming problem is more foreign holidays – not less! Sometimes things work out like that.