Forming
The Schmidt–Appleman criterion: when a mixing exhaust passes water saturation.
The mixing line
An engine burns a kilogram of fuel and puts out its water, 1.23 kg of it for kerosene, and the share of its heat that did not push the aircraft. As the exhaust mixes with the air round it, both are diluted by the same air, so in a diagram of vapour pressure against temperature the mixture moves along a straight line from the exhaust down to the air. Its slope is
G = EI · cp · p / (ε · Q · (1 − η))with EI the water per kilogram of fuel, Q its heat, η the share of it
that pushes the aircraft, p the air's pressure, cp air's heat capacity
and ε water's molar mass over air's. At 11 km, for the docs' fighter, it is
1.4 Pa/K.
Crossing saturation
Saturation pressure curves upwards with temperature. A steep enough line, wet enough exhaust in cold enough air, crosses it on the way down: the plume passes water saturation, droplets form on its soot, and in air this cold they freeze almost at once. That is the Schmidt–Appleman criterion (Schmidt, 1941; Appleman, 1953), with Schumann's (1996) correction for the heat that pushes the aircraft and never warms the plume.
The line first nears the curve where the curve is as steep as it is. From
that tangent temperature the threshold follows: the warmest the air can be,
with the vapour it has, for the line to still reach saturation. schmidt_appleman
finds the tangent by bisection on Murphy and Koop's saturation curve, and
agrees with Schumann's fit to within a hundredth of a degree.
Efficiency
The more of the fuel's heat pushes the aircraft, the less is left to warm the plume, the steeper the line, and the warmer the air it trails in. It follows from the airframe: it is the thrust's power over the fuel's,
η = V · F / (ṁ · Q)for engines making F newtons on ṁ kilograms of fuel a second at airspeed
V. Dry, ṁ = TSFC · F and it is V / (TSFC · Q): a fighter's low-bypass
engine is about 25 % efficient at cruise, a modern fan about a third, and the
airliner's threshold is a degree warmer.
Reheat burns its fuel in the exhaust, at the pressure the turbine left it at, and turns far less of it into thrust: each newton costs 1.58 times the dry engine's fuel at Mach 0.85 (Mattingly, Heiser and Pratt, 2002, eq. 3.55). The fighter at full reheat is 16 % efficient, its mixing line a tenth shallower, and its threshold 0.9 °C colder. It trails in a little less of the sky, but with six times the water where it does, as Airframes shows.
Where along the plume
The criterion says whether; plume_formation says where. It follows the same
mixture out along the plume's measured dilution and finds the dilution where
it first saturates: for the fighter at 11 km, 0.14 s behind the nozzle, 36 m
at cruise. The trail starts there, the clear gap behind every engine.
Its droplets then freeze where the mixture cools past -38 °C, if it is still saturated over water there to keep them. In air warm enough to be near the threshold, they may evaporate first: a faint wisp that lasts only as long as the plume stays saturated over water.