Contrail Mixing-Line Slope (Schmidt-Appleman Criterion)

When a jet engine burns fuel it puts hot, wet exhaust into cold air, and the Schmidt-Appleman criterion decides whether a contrail forms behind it. This page gives the first half: the slope $G$ of the mixing line on a diagram of vapour pressure against temperature, in pascals per kelvin, along which the exhaust mixes with the air around it. The steeper the line, the warmer the air can be and still make a contrail. The threshold temperature that goes with $G$ is on the next page.

This is not a fit to data. It is a relation among the properties of the air, the fuel and the engine, printed by Schumann (2012, equation 11), who cites his 1996 paper for these relations. In $G = \frac{c_p\,p\,EI_{H_2O}}{\varepsilon\,Q\,(1 - \eta)}$, $c_p$ is the specific heat of air (1004 J/(kg K)), $p$ the ambient pressure, $EI_{H_2O}$ the mass of water the engine emits per mass of fuel (1.23 kg/kg for kerosene), $\varepsilon = 0.622$ the ratio of the molar masses of water and air, $Q$ the heat of combustion of the fuel (43.2 MJ/kg for kerosene) and $\eta$ the overall propulsion efficiency at cruise.

More water and more pressure steepen the line, and so does a more efficient engine, which turns more of the fuel energy into thrust, so that $1 - \eta$ is smaller and less heat is left in the exhaust. For kerosene with $\eta = 0.3$, $G$ is 0.66 Pa/K at 100 hPa, 1.64 at 250 hPa and 3.28 at 500 hPa, inside the 0.6 to 4 that Schumann gives as usual for kerosene aircraft between 100 and 500 hPa. At 250 hPa, which is 25 000 Pa, the formula gives 1.641371 Pa/K.

The slope alone does not say whether a contrail forms. Whether a contrail that forms will last is a separate question, answered by whether the air is supersaturated with respect to ice. The relation works for any fuel once its emission index and heat of combustion are put in: hydrogen (8.94 kg/kg and 120 MJ/kg) at 250 hPa and $\eta = 0.3$ gives 4.2948 Pa/K.