# Monte Carlo Simulations in Aviation Contrail Study: A Review
## 2 Contrails in Aviation: Generalities  
Contrails form in the mixing process of an expanding engine plume with the surrounding atmosphere. Exhaust from the propulsion system is generally warmer than air and contains a large amount of water vapor. For this reason, contrails have been mainly observed in the highest part of the troposphere when associated with the turbo-fan engine currently employed in medium and long-haul flights. The numerical criterion for evaluating the trail formation probability is dubbed the Schmidt-Appleman Criterion (SAC) [5] after the two scientists who developed pioneering studies in this field in the early phases of modern aviation. SAC is a global thermodynamic criterion based on the assumption of an adiabatic and water-conserving mix between the exhaust and the surrounding atmosphere, wherein heat and vapor diffuse at the same rate [6]. Under this condition, the state point representing the mixture in a Temperature-Partial vapor pressure graph evolves along a straight line, whose slope is given by  
\[G=\frac{P_{v}-P_{\infty}}{T-T_{\infty}}, \tag{1}\]  
where \(P_{v}=X_{H_{2}O}P\) represents the water's partial vapor pressure, \(X_{H_{2}O}\) represents the water's molar fraction, and the underscript \(\infty\) indicates the completely mixed system. For a turbo-fan engine, the \(G\) parameter in Equation (1) can be expressed as a function of the engine efficiency and the air pressure only [7]:  
\[G=\frac{c_{p}P_{\infty}}{\epsilon}\frac{EI_{H_{2}O}}{(1-\eta)Q}, \tag{2}\]  
in which \(c_{p}\) stands for the specific heat capacity of air at constant pressure, \(\epsilon=0.622=W_{H_{2}O}/W_{air}\) is the water/air molar mass ratio, \(EI_{H_{2}O}\) is the water emission index of the engine (whose test bench measurements in a laboratory are required by ICAO regulation for engines with an output rated greater than 26.7 kN [8; 9; 10]), \(Q\) is the total heat released per mass of fuel, and \(\eta\) = (thrust \(\times\) distance per mass of fuel)/\(Q\) is the engine's efficiency in cruise conditions, in such a way that \((1-\eta)Q\) represents the heat conveyed by the engine's exhaust. Once that the mixing line slope is known, it can be employed as a parameter for evaluating contrail formation and persistency. In particular, the threshold temperature for contrail formation can be derived by considering the temperature dependence of the partial pressure of the liquid water-steam phase transition \(P_{sat_{i}}^{Iiq}\); that is, the water liquid-vapour pressure saturation curve, described by the Clausius-Clapeyron relation [11; 12]  
\[\frac{dP_{sat_{i}}^{Iiq}}{dT}=\frac{1}{T}\,\frac{Q^{lat}(T)}{\alpha_{v}- \alpha_{l}}, \tag{3}\]  
in which \(Q^{lat}(T)\) is the latent heat of evaporation/condensation, and \(\alpha_{v}\), \(\alpha_{l}\), are the specific volume of the vapour and liquid phases, respectively. The threshold temperature, \(T_{thr}^{*}\), for condensation is trivially the temperature of the tangent point between the mixing line and the curve in Equation (5):  
\[\left.\frac{dP_{sat_{e}}^{liq}}{dT}\right|_{T_{th}^{n}}=G. \tag{4}\]  
Threshold temperature determination is graphically described in [6]. There, the condition for contrail formation and persistence is synthetically described: if the mixing line intersects the liquid-vapour saturation curve, the contrail is formed. The crucial questions for evaluating the environmental impact regards how long they persist in the atmosphere and which physicochemical processes are involved into their transmutation in clouds [13]. Sight cannot distinguish contrail from natural cirrus clouds [14]. For the condensed trail to persist, a second condition has to occur, which is that the atmosphere-exhaust mixing result has to lay above the vapour-ice saturation curve. Ice formation, however, and the following persistence of the contrail, also depend on the presence in the plume of soot and sulfuric acid aerosol, originated in the engine, acting as condensation nuclei for the microscopical water drops.