Exponential Formula to Estimate
Airplane Emissions
Common Approach with Three Plateaus
Most Carbon Calculators use three levels of CO2e
per mile (or per kilometer), for "short," "medium," and
"long" Airplane flights.
They use the highest level of CO2e per
passenger mile for short flights, since takeoff burns a lot of fuel and is a
bigger fraction of short flights. However this approach
under-estimates emissions of the shortest flights in each level, where takeoffs
are a bigger fraction than average for the level.
They use a medium level of CO2e per
passenger mile for long flights, since long flights have to carry a huge amount
of fuel at the beginning of the flight, so they burn extra fuel to carry the
extra fuel. This approach under-estimates emissions of
the longest flights, where carrying fuel is an even bigger burden.
The dropping and rising levels also create anomalies
where a slightly longer flight is assigned much less or much more CO2e
than a slightly shorter flight. These anomalies show up in the comparative Graphs at the bottom of
the "Fuel" tab of xls.CO2List.org
Linear Formula
A linear equation with a constant per takeoffs can
give a very good approximation, especially for the large majority of short and
medium flights. However it slightly over-estimates medium flights and/or
under-estimates the longest.
Airplanes use above average fuel per mile for
takeoff and climb, while they use below average fuel per mile for descent and
landing. The constant per takeoff actually represents the net difference for
the flight as a whole, plus taxiing.
Exponential Formula
An exponential equation has the best theoretical basis, since
it reflects rising fuel burdens of longer flights, and it can also have a
constant per takeoff. The weight
constantly drops as fuel is burned, so less fuel is needed each mile.
Wm = Weight in last
mile, including plane, load and final reserves of fuel
Wm * R = Weight of fuel
burned in last mile "m"
Where R is a constant representing the ratio of
the weight of fuel being burned to total weight being flown at that moment
The weight of fuel for that last mile has to be carried to that
point, so weight in the preceding period is bigger by that amount:
Wm-1
= Wm + ( Wm *
R )
Simplifying:
Wm-1
= Wm * ( 1 + R )
In each previous period weight has to be yet bigger by the same
ratio:
Wm-2
= Wm *
( 1 + R )2
...
W0 = Wm
* ( 1 + R )m
Where W0 is starting
weight
The total fuel burned in a flight is the starting weight minus
ending weight, or
W0 - Wm = Wm
* [ (
1 + R )m
-1 ]
So we have this functional form:
CO2e per passenger during the whole
flight = A + B [ CMILES
- 1 ]
Where A is a constant
per takeoff, while B and C relate to fuel
consumption during the flight
A
More data would allow more confidence in
the coefficients, but this estimate is well-behaved, rising noticeably for very
short and long flights. From about 200 to 5,000 miles (300-8,000 km), the
exponential and linear formulas are close. The exponential is higher at both
ends.
The derivation lets A,
B, and
C
include other effects not discussed here. For example
A includes all effects which are independent of flight length, such as airport
usage and some probability of weather diversions at the destination. B and C
include all effects related to flight length, such as maintenance and size of
plane.