Course:PHYS100/Fuel Economy in Air Travel

Phys 100 Sample Project

by G. Rieger, Nov. 07, 2007 (formatted for UbcWiki Oct. 27, 2010)

Introduction

Limiting your personal air travel is often mentioned as good measure to decrease your “carbon footprint”. Jet airplanes in particular have a bad reputation as being relatively inefficient and causing significant air pollution. I will present an estimate of the fuel economy of a Boeing 747, show where most of the energy is consumed and compare the fuel economy to other modes of travel.

Fuel Economy of Airplanes

For cars, the fuel economy is usually expressed in liters of fuel consumed per 100 km or as miles per gallon (mpg). We can use these definitions to calculate the fuel economy of an airplane. It is most meaningful to compare the fuel economy per passenger. A Boeing 747 has a maximum range of 13,450 km and a maximum fuel capacity of 216,840 L.

(http://www.boeing.com/commercial/747family/technical.html….)

The fuel economy is

${\tfrac {216,840L}{13450km}}=16L/km$

or 1600 L per 100 km. The number of passengers in a Boeing 747 is 400 - 500, so the consumption per passenger is 3.2 – 4.0 L per 100 km when the plane is full.

Comparison of Energy Efficiency

We can compare this now to other means of transportation. When talking about the fuel economy per passenger, often two numbers are given corresponding to average occupancy and maximum occupancy. In many situations, using the average occupancy is more meaningful, for instance when making political decisions. In North America, fuel economy is usually expressed in units of miles per gallon (mpg). 1.0 mpg (US) = 425 meter/liter On wikipedia, we find that the fuel economy for an airplane is 67 mpg = 28.5 km/L or 100 km/28.5 km/L = 3.51L per 100 km per passenger, which is compatible with our calculation for the Boeing 747.

(http://en.wikipedia.org/wiki/Fuel_efficiency_in_transportation)

Due to the typically high occupancy of an airplane its fuel economy is comparable to a relatively efficient car such as the Toyota Prius (46 mpg, multiplied by the average occupancy for cars = 1.57 = 72 mpg/passenger).

Why Do Jet Airplanes Fly at Such High Altitudes?

So the fuel economy of an airplane is roughly as good as that of a hybrid car. However, it has been argued that the pollution is more harmful at higher altitudes. In addition, climbing to high altitudes requires energy. So is there a benefit to flying at 10 km altitude? Yes, less fuel consumption.

How much fuel is needed for climbing to an altitude of ~10.5 km? The mass of the plane is 390 000 kg at take-off. (Source: www.boeing.com) The corresponding potential energy difference is

$mgh=4.0x10^{10}J$

How much energy is needed for the acceleration to the cruising speed of 915 km/h (v = 254 m/s)? Change in kinetic energy:

$\Delta K={\tfrac {1}{2}}mv^{2}=1.3x10^{10}J$

How much fuel is needed for accelerating and climbing? The energy efficiency of the jet engines is ~25% (estimated) so the equivalent energy from fuel should be approximately

$4x5.3x10^{10}J=21.2x10^{10}J$.

Energy in jet fuel: 37.6 MJ/L (Source: Wikipedia)

So we need

${\tfrac {21.2x10^{10}J)}{37.6MJ/L}}=5600L$

of jet fuel to accelerate to 915 km/h and go up to 10500 m. The rest of the fuel (~210,000L) is used to maintain speed against air drag. We see that minimizing the drag force is key in the fuel economy or air travel.

Air Drag Revisited

Air drag is usually given as

$D={\tfrac {1}{2}}\rho C_{D}Av^{2}$

Here shape and surface quality of the airplane are included in the drag coefficient $C_{D}$. A is the cross-sectional area of the airplane and v is the plane's speed. The density of air is $\rho =1.29kg/m^{3}$ at STP (standard temperature and pressure) but it depends on altitude and at an altitude of 10.5 km, it is only 40% of its value at sea-level. From the equation above we see that the air drag increases or decreases proportionally with the density of air. So flying at higher altitude decreases the drag force.

Conclusion

Airplanes have a fuel economy similar to hybrid cars. Planes fly at relatively high altitude to save fuel (and money). Their exhaust however is more harmful if it is emitted at high altitudes: The IPCC, for example, has estimated that the climate impact of aircraft is two to four times greater than the effect of their carbon dioxide emissions alone. (Source: www.davidsuzuki.org) Ecological impact: Due to the long distances, 4% of global annual $CO_{2}$ emissions from fossil fuels are due to air travel (Source: NASA Glenn).