Nomenclature

${c}_{\textrm{p}}$

heat capacity, $\textrm{J}\,\textrm{k}{\textrm{g}}^{-1}\,{\textrm{K}}^{-1}$

$ \textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}}$

emission index of water vapour, $\textrm{k}\textrm{g}\,\textrm{k}{\textrm{g}}^{-1}$

$ F$

thrust of the aircraft, $ \textrm{N}$

$ G$

slope of the mixing line, $\textrm{P}\textrm{a}\,{\textrm{K}}^{-1}$

$ {G}_{\textrm{F}\textrm{C}}$

slope of the mixing line for fuel cell systems, $\textrm{P}\textrm{a}\,{\textrm{K}}^{-1}$

$ {h}_{\textrm{a}}$

ambient enthalpy, $\textrm{J}\,\textrm{k}{\textrm{g}}^{-1}\,{\textrm{K}}^{-1}$

$ {h}_{\textrm{E}}$

exhaust enthalpy, $\textrm{J}\,\textrm{k}{\textrm{g}}^{-1}\,{\textrm{K}}^{-1}$

$ {h}_{\textrm{m}\textrm{i}\textrm{x}}$

plume enthalpy, $\textrm{J}\,\textrm{k}{\textrm{g}}^{-1}\,{\textrm{K}}^{-1}$

$ {m}_{\textrm{a}}$

mass of ambient air, $ \textrm{k}\textrm{g}$

$ {m}_{\textrm{E}}$

exhaust mass, $ \textrm{k}\textrm{g}$

$ {m}_{\textrm{m}\textrm{i}\textrm{x}}$

plume mass, $ \textrm{k}\textrm{g}$

$ N$

dilution factor, 1

$ {N}_{\textrm{E}}$

exhaust dilution factor, 1

$ {p}_{\textrm{a}}$

ambient pressure, $ \textrm{P}\textrm{a}$

$ {p}_{\textrm{w},\textrm{s}\textrm{a}\textrm{t}}$

water saturation pressure, $ \textrm{P}\textrm{a}$

$ {\textrm{p}}_{\textrm{w}\textrm{v},\textrm{a}}$

ambient water vapour partial pressure, $ \textrm{P}\textrm{a}$

$ {p}_{\textrm{w}\textrm{v},\textrm{m}\textrm{i}\textrm{x}}$

plume water vapour partial pressure, $ \textrm{P}\textrm{a}$

$ Q$

lower heating value, $\textrm{J}\,\textrm{k}{\textrm{g}}^{-1}$

$ {Q}_{\textrm{a}\textrm{v}\textrm{a}\textrm{i}\textrm{l}}$

heat available for contrail formation, $ \textrm{J}$

$ {q}_{\textrm{a}}$

ambient specific humidity, $\textrm{k}\textrm{g}\,\textrm{k}{\textrm{g}}^{-1}$

$ {q}_{\textrm{E}}$

exhaust specific humidity, $\textrm{k}\textrm{g}\,\textrm{k}{\textrm{g}}^{-1}$

$ {q}_{\textrm{m}\textrm{i}\textrm{x}}$

plume specific humidity, $\textrm{k}\textrm{g}\,\textrm{k}{\textrm{g}}^{-1}$

$ \widehat{q}$

linearised specific humidity, mass mixing ratio, $\textrm{k}\textrm{g}\,\textrm{k}{\textrm{g}}^{-1}$

$ {T}_{\textrm{f}\textrm{r}\textrm{z}}$

homogeneous freezing temperature, $ \textrm{K}$

$ {T}_{\textrm{L}\textrm{M}}$

temperature at which the mixing line and saturation pressure curve touch, $ \textrm{K}$

$ {T}_{\textrm{m}\textrm{i}\textrm{x}}$

plume temperature, $ \textrm{K}$

$ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$

limiting temperature for contrail formation, $ \textrm{K}$

$ {u}_{\textrm{A}\textrm{C}}$

aircraft velocity, $\textrm{m}\,{\textrm{s}}^{-1}$

2.0 Generalised contrail formation theory

In this section, relative emission factors are introduced to describe the thermodynamic exhaust conditions of FC-propelled aircraft. Then, a detailed derivation of the classical mixing line theory [Reference Schumann8] is presented, which is used to decide whether or not transient supersaturation is reached in the diluting plume and a contrail forms. In particular, we explicitly state the underlying assumptions of this approach. Afterwards, a more generally valid theory that does not rely on these assumptions is developed. Instead of a straight mixing line with constant slope $ G$ , a non-linear relationship between plume temperature and WV partial pressure will be obtained. In analogy to the term ‘mixing line’, we refer to the new formulation as mixing curve theory.

2.1 Relative emission factors

For aircraft with combustion engines, $ {Q}_{\textrm{a}\textrm{v}\textrm{a}\textrm{i}\textrm{l}}$ is likely to be fully contained in the exhaust plume as thermal and kinetic energy, even though heat recovery systems might be included in future turbo-fan aircraft to increase the overall propulsion efficiency [Reference Krempus, Beltrame, Majer, de Servi and Vos39, Reference Xu, Tang, Chen and Duan40].

In common FC design concepts, however, only a fraction of the generated WV and energy is contained in the moist-air exhaust stream. To account for the potential reduction of emitted WV and heat in FC-propelled aircraft (cf. Fig. 1(b)), we introduce two relative emission factors. While

(2) \begin{align} \gamma =\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O},\textrm{e}\textrm{m}\textrm{i}\textrm{t}}/\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}} \end{align}

represents the actual amount of WV emitted compared to the unaltered amount,

(3) \begin{align} \delta ={Q}_{\textrm{e}\textrm{m}\textrm{i}\textrm{t}}/{Q}_{\textrm{a}\textrm{v}\textrm{a}\textrm{i}\textrm{l}} \end{align}

represents the actual amount of heat emitted compared to its unaltered amount. In future FC systems, $ \gamma $ and $ \delta $ might be well below 1 for several reasons. First of all, the exhaust and propulsive streams are spatially separated. Hence, the kinetic energy of the propulsive streams and associated friction losses at the propellers reduce the energy content of exhaust plumes. Moreover, Fig. 1(b) indicates the diversion of heat (via heat exchangers) and WV (via WV condenser devices) from the exhaust gas. The diverted heat can be exploited for secondary applications within the aircraft, such as cabin heating, or released in a separate ‘dry-air’ exhaust stream. Provided that the two exhaust streams are emitted at different locations and do not mix during plume dilution, the energy content of the moist-air exhaust remains reduced. Similarly, the condensed WV might be released as liquid droplets far away from the moist-air exhaust, such that its WV content remains reduced. As contrail formation can only occur inside the diluting ‘moist-air’ exhaust, we will simply refer to it as ‘exhaust’ in the remainder of this paper.

The adapted slope for the mixing line of FC systems, taking the relative heat and WV emission factors $ \gamma $ and $ \delta $ into account, reads

(4) \begin{align} {G}_{\textrm{F}\textrm{C}}=\frac{\gamma \, \textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}}\,{c}_{\textrm{p}}\,{p}_{\textrm{a}}}{\delta \, Q\left(1-\eta \right)\epsilon } \end{align}

If all available energy is used to heat the plume and all WV is contained in the exhaust, theoretically $ \gamma =\delta =1$ and the mixing line slope of the FC system is equal to the respective combustion propulsion system. We will discuss the effects of potential WV and heat recuperation systems on contrail formation in detail in Section 4.

2.2 Classical derivation of mixing line

First, we provide an overview of the derivation for the mixing line as presented in Ref. [Reference Schumann8]. As the plume dilutes, ambient air is added continuously to the exhaust. Isobaric mixing of the two air parcels is assumed, as the plume pressure relaxes quickly to ambient pressure. One air parcel represents the homogeneous exhaust plume, and the other parcel represents the infinite reservoir of surrounding atmospheric air. In this purely thermodynamic formulation, the effects of microphysical processes are disregarded. For consistency with the original derivation, which was developed for combustion propulsion systems, the relative emission factors are set to $ \gamma =\delta =1$ .

We determine the specific humidity, which is the ratio of the WV mass to the total air mass, and the enthalpy of the mixed plume. The plume’s specific humidity can be expressed by

(5) \begin{align} {\Delta }\widehat{q}=\frac{\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}}-{\widehat{q}}_{\textrm{a}}}{N}\approx \frac{\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}}}{N} \end{align}

where $ N$ represents the dilution, which is the air-to-fuel mass ratio of the diluting plume. The ambient specific humidity $ {\widehat{q}}_{\textrm{a}}$ is negligible compared to the exhaust’s specific humidity and is therefore set to zero.

The enthalpy of the plume is treated analogously:

(6) \begin{align} {\Delta }h=\frac{Q\left(1-\eta \right)-{h}_{\textrm{a}}}{N}\approx \frac{Q\left(1-\eta \right)}{N} \end{align}

Compared to the exhaust enthalpy, the enthalpy value $ {h}_{\textrm{a}}$ of the ambient air is small and therefore neglected. The excess enthalpy $ {\Delta }h$ can then be used to calculate the excess temperature for each plume condition during mixing

(7) \begin{align} \Delta h = \int_{T_a}^{T} c_p \, \textrm{dT} \approx \bar{c}_p (T - T_a) = \bar{c}_p \Delta T \end{align}

where ${\bar{c}}_{p}\approx 1004\,\textrm{J}{\left(\textrm{k}\textrm{g}\,\textrm{K}\right)}^{-1}$ is chosen as a constant whose value is close to the one of DA at ambient conditions. Equation (7) assumes that the plume kinetic energy contributes negligibly to the plume enthalpy (which is reasonable at least when the plume becomes supersaturated). By combining Equations (5)–(7), $N$ can be eliminated and the change of the plume specific humidity with plume temperature can be expressed as

(8) \begin{align} \frac{{\Delta }\widehat{q}}{{\Delta }T}=\frac{\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}}{\bar{c}}_{\textrm{p}}}{Q\left(1-\eta \right)} \end{align}

Following the usual assumption in meteorology, the WV partial pressure is expressed as

(9) \begin{align} \widehat{q}=\epsilon \, \frac{{p}_{\textrm{w}\textrm{v}}}{{p}_{\textrm{a}}} \end{align}

Reformulating $ {\Delta }\widehat{q}=\widehat{q}-{\widehat{q}}_{\textrm{a}}$ and $ {\Delta }T=T-{T}_{\textrm{a}}$ in Equation (8), using Equation (9) and rearranging gives a linear relation between the WV plume partial pressure and the plume temperature using $ G$ from Equation (1):

(10) \begin{align} {p}_{\textrm{w}\textrm{v}}={p}_{\textrm{w}\textrm{v},\textrm{a}}+G\left(T-{T}_{\textrm{a}}\right) \end{align}

It is advantageous for the following analysis to describe the plume moisture and thermal energy in terms of partial pressure and plume temperature. Unlike the saturation specific humidity, which depends on the ambient pressure, the WV saturation pressure $ {p}_{\textrm{w},\textrm{s}\textrm{a}\textrm{t}}$ depends only on temperature. This allows us to compute the limiting temperature $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ for a given ambient relative humidity by requiring that the slope of the mixing line matches the slope of $ {p}_{\textrm{w},\textrm{s}\textrm{a}\textrm{t}}$ . For this, we introduce the temperature $ {T}_{\textrm{L}\textrm{M}}$ at which both curves touch each other, implicitly given by:

(11) \begin{align} \frac{\textrm{d}{p}_{\textrm{w},\,\textrm{s}\textrm{a}\textrm{t}}\!\left({T}_{\textrm{L}\textrm{M}}\right)}{\textrm{d}T}=G \end{align}

An approximation for the evaluation of $ {T}_{\textrm{L}\textrm{M}}$ depending on $ G$ is given in the appendix of Refs [Reference Schumann8] or [Reference Gierens9]. The actual limiting temperature $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ depends on the ambient relative humidity $ \textrm{R}{\textrm{H}}_{\textrm{w},\textrm{a}}$ and can be found by solving the following equation:

(12) \begin{align} {p}_{\textrm{w},\textrm{s}\textrm{a}\textrm{t}}\!\left({T}_{\textrm{L}\textrm{M}}\right)+G\left({\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}-{T}_{\textrm{L}\textrm{M}}\right)=\textrm{R}{\textrm{H}}_{\textrm{w},\textrm{a}}\,{p}_{\textrm{w},\textrm{s}\textrm{a}\textrm{t}}\!\left({\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}\right) \end{align}

In general, this is an implicit equation that can be solved for $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ by a Newton iteration. A contrail forms, if the ambient temperature is both lower than $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ (i.e. the SAC criterion is fulfilled) and $ {T}_{\textrm{f}\textrm{r}\textrm{z}}$ of the water droplets. For ambient temperatures larger than $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ , no contrail forms because the cooling plume remains sub-saturated during the whole mixing process.

2.3 Review of assumptions in the classical formulation

In the past, the classical mixing line has been applied to scenarios of kerosene and hydrogen combustion. However, this formulation may not be applicable to the exhaust plumes generated by FC-propelled aircraft equipped with heat recuperation systems, which can produce moist and relatively cool plumes. Consequently, some assumptions underlying the mixing line derivation (Section 2.2) require re-evaluation.

The first approximation that may not be sufficiently accurate for moist plumes is the linear relation between the specific humidity $ q$ and the WV partial pressure $ {p}_{\textrm{w}\textrm{v}}$ , given by Equation (9). For this relation, which is used in traditional meteorological formulations, it is assumed that the WV mass within an air parcel is significantly smaller than the DA mass. Consequently, the contribution of $ {p}_{\textrm{w}\textrm{v}}$ to the total pressure is typically neglected, allowing the total pressure to be approximated as the DA pressure $ {p}_{\textrm{t}\textrm{o}\textrm{t}}={p}_{\textrm{d}\textrm{a}}$ . For very moist air parcels, this assumption is no longer valid; the pressure of the moist air $ {p}_{\textrm{t}\textrm{o}\textrm{t}}$ has to be described as the sum of the WV and DA partial pressures. Hence, the relationship needs to be extended to include the influence of WV on the properties of the moist air parcel:

(13) \begin{align} q\,:\!=\,\frac{{m}_{\textrm{w}\textrm{v}}}{{m}_{\textrm{t}\textrm{o}\textrm{t}}}=\frac{{\rho }_{\textrm{w}\textrm{v}}}{{\rho }_{\textrm{t}\textrm{o}\textrm{t}}}=\frac{{\rho }_{\textrm{w}\textrm{v}}}{{\rho }_{\textrm{d}\textrm{a}}+{\rho }_{\textrm{w}\textrm{v}}}=\frac{{p}_{\textrm{w}\textrm{v}}/{R}_{\textrm{w}\textrm{v}}}{{p}_{\textrm{d}\textrm{a}}/{R}_{\textrm{d}\textrm{a}}+{p}_{\textrm{w}\textrm{v}}/{R}_{\textrm{w}\textrm{v}}} =\frac{\epsilon \, {p}_{\textrm{w}\textrm{v}}}{{p}_{\textrm{d}\textrm{a}}+\epsilon {p}_{\textrm{w}\textrm{v}}}=\frac{\epsilon \, {p}_{\textrm{w}\textrm{v}}}{{p}_{\textrm{a}}+(\epsilon -1){p}_{\textrm{w}\textrm{v}}} \end{align}

$ m$ denotes the mass, $ \rho $ the density and $ {R}_{\textrm{w}\textrm{v}},{R}_{\textrm{d}\textrm{a}}$ the specific gas constants for WV and DA, respectively. Due to the isobaricity assumption, the total pressure of any air parcel considered equals $ {p}_{\textrm{a}}$ . The discrepancy between this formula and the simplified relation (Equation (9)) is illustrated in Fig. 3(a) and evaluated as a function of the ratio $ {p}_{\textrm{w}\textrm{v}}/{p}_{\textrm{a}}$ . The simplified formula provides a good approximation for $ {p}_{\textrm{w}\textrm{v}}/{p}_{\textrm{a}}\lt0.1$ , which encompasses the majority of applications in atmospheric science. However, for larger relative WV contents, the two lines diverge, indicating that the simplified formula becomes less accurate the moister the plume is.

Figure 3.

Exemplary simplifications for the mixing line theory. Black lines show the values used in classical derivation. a) Dependence of specific humidity on the normalised WV partial pressure. b) Dependence of heat capacities on temperature. c) Dependence of enthalpy on temperature.

Figure 3. Long description

The image contains three separate line graphs. The first graph (a) shows the dependence of specific humidity on the normalized water vapor partial pressure. The x-axis represents the normalized water vapor partial pressure, while the y-axis represents specific humidity. Two lines are plotted: a blue line representing the equation and a black line representing the estimated value. The second graph (b) illustrates the dependence of heat capacities on temperature. The x-axis represents temperature in Kelvin, and the y-axis represents heat capacity in joules per kilogram Kelvin. Three lines are shown: an orange line for heat capacity as a function of temperature, a green line for heat capacity at a specific condition, and a black line for constant heat capacity. The third graph (c) depicts the dependence of enthalpy on temperature. The x-axis represents temperature in Kelvin, and the y-axis represents enthalpy in joules per kilogram Kelvin. Three lines are plotted: an orange line for enthalpy as a function of temperature, a green line for enthalpy at a specific condition, and a black line for constant enthalpy. All values are approximated.

The second assumption to be re-evaluated is the use of a constant heat capacity $ {c}_{\textrm{p},\textrm{d}\textrm{a}}$ throughout the entire temperature range of the mixing process. To improve accuracy, it is more appropriate to employ a framework that accounts for the changes in heat capacity as a function of temperature and water content. Consequently, the enthalpy of the plume must be evaluated at each mixing state using the integral expression:

(14) \begin{align} h(T) = \int_{T_{\textrm{ref}}}^{T} c_p (q(\tilde{T}), \tilde{T}) \, \textrm{d }\!\tilde{T}\end{align}

where $ {T}_{\textrm{r}\textrm{e}\textrm{f}}$ is a reference temperature at which the enthalpy is set to zero. The heat capacity of moist air is a linear interpolation between the heat capacities $ {c}_{\textrm{p},\textrm{w}\textrm{v}}(T)$ and $ {c}_{\textrm{p},\textrm{d}\textrm{a}}(T)$ via the specific humidity $ q$ :

(15) \begin{align} {c}_{\textrm{p}}(q,T)=q\,{c}_{\textrm{p},\textrm{w}\textrm{v}}(T)+(1-q)\,{c}_{\textrm{p},\textrm{d}\textrm{a}}(T) \end{align}

To accurately represent the WV and DA heat capacities, we utilise the parametrisation provided by Ref. [Reference Bücker, Span and Wagner41]. For DA, a composition of $ 21{\%}$ oxygen and $ 79{\%}$ nitrogen is considered. Reference (Reference Bücker, Span and Wagner41) also gives a parametrisation of the enthalpy that is consistent with their WV and DA heat capacity parametrisation. We employ this parametrisation to calculate the enthalpy of the diluting plume analogously to the heat capacity:

(16) \begin{align} h(q,T)=q\,{h}_{\textrm{w}\textrm{v}}(T)+(1-q)\,{h}_{\textrm{d}\textrm{a}}(T) \end{align}

Figure 3(b) and (c) show the temperature dependence of the heat capacity and enthalpy of moist air, respectively. Both plots compare the value used in the classical derivation (black line) with the temperature-dependent values for DA and for moist air with fixed $ q=0.1$ . Even though the differences in the heat capacity diagram are exaggerated for the chosen y-axis range, the relative error between the constant and parametrised heat capacity for moist air exceeds $ 15{\%}$ for high temperature values, rendering the usage of a constant heat capacity questionable. The maximum difference for the enthalpy of the moist air is approximately $ 12{\%}$ , although it has to be noted that the relative difference depends on the chosen reference value for the enthalpy, which is $ 273.15$ K in the utilised parametrisation.

Moreover, the classical theory drops the ambient terms from Equations (5) and (6) leading to a slope of the mixing line in $ \widehat{q}$ – $ T$ space that is independent of ambient conditions. We deem the errors by this simplification negligible. Nevertheless, the ambient terms are retained in our derivation, as dropping the terms would not simplify the following derivations.

2.4 Generalised contrail formation criterion

In the following, we derive a formulation for the mixing process without the assumptions used in the classical formulation. We start from fundamental physical principles, specifically the conservation laws of mass and energy. As in Section 2.2, isobaric mixing of the plume and ambient air and a homogeneous plume described by a mean plume state is assumed. Note that this implies that the exhaust is instantaneously mixed, which may not be appropriate for modern turbo-fan engines with high bypass ratios. This effect on the mixing line has already been studied in Ref. [Reference Zink and Unterstrasser42]. In contrast to the classical mixing line, we allow the specific heat capacity and enthalpy to be functions of both temperature and specific humidity. Our objective is to determine the plume specific humidity $ {q}_{\textrm{m}\textrm{i}\textrm{x}}$ and its corresponding temperature $ {T}_{\textrm{m}\textrm{i}\textrm{x}}$ during the mixing process. To this end, the conservation laws governing the total mass, the WV mass and thermal energy of the plume are applied, respectively:

(17) \begin{align} \textrm{total}\,\textrm{mass}\,\textrm{conservation}\quad {m}_{\textrm{E}}+{m}_{\textrm{a}}(t)={m}_{\textrm{mix}}(t) \end{align}

(18) \begin{align} \qquad\qquad\,\, \textrm{water}\,\textrm{mass}\,\textrm{conservation} \quad {q}_{\textrm{E}}\,{m}_{\textrm{E}}+{q}_{\textrm{a}}\,{m}_{\textrm{a}}(t)=q_{\textrm{m}\textrm{i}\textrm{x}}(t)\,{m}_{\textrm{m}\textrm{i}\textrm{x}}(t) \end{align}

(19) \begin{align} \qquad\qquad\qquad \, \textrm{energy}\,\textrm{conservation} \quad {m}_{\textrm{E}}\,{h}_{\textrm{E}}+{m}_{\textrm{a}}(t)\,{h}_{\textrm{a}}={m}_{\textrm{m}\textrm{i}\textrm{x}}(t)\,{h}_{\textrm{m}\textrm{i}\textrm{x}}(t) \end{align}

The indices ‘E’ and ‘a’ correspond to the exhaust and ambient values, respectively. Note that $ {m}_{\textrm{a}}(t)$ does not describe the (infinite) mass reservoir of ambient air, but the increasing mass already entrained into the plume. In contrast, all other ambient variables correspond to time-constant properties of the ambient atmosphere. The three Equations (17)–(19) are used to calculate the three unknowns of the plume $ {m}_{\textrm{m}\textrm{i}\textrm{x}},{q}_{\textrm{m}\textrm{i}\textrm{x}},{h}_{\textrm{m}\textrm{i}\textrm{x}}$ . Using Equation (13), the specific humidity can be replaced by a function of partial pressure $ {p}_{\textrm{w}\textrm{v}}$ , while the temperature can be calculated from inverting the enthalpy parametrisation of moist air (see Equation (16)) with an iterative solution method. Following these steps yields a relationship with a form similar to the classical mixing line Equation (10).

To calculate the temporal evolution of the specific humidity $ {q}_{\textrm{m}\textrm{i}\textrm{x}}(t)$ , we use the temporal derivative of Equation (18), which results in the differential equation

(20) \begin{align} \frac{\textrm{d}{q}_{\textrm{m}\textrm{i}\textrm{x}}}{\textrm{d}t}=\frac{{\dot{m}}_{\textrm{m}\textrm{i}\textrm{x}}}{{m}_{\textrm{m}\textrm{i}\textrm{x}}}\left({q}_{\textrm{a}}-{q}_{\textrm{m}\textrm{i}\textrm{x}}\right) \end{align}

Accordingly, Equation (19) results in a differential equation for the enthalpy:

(21) \begin{align} \frac{\textrm{d}{h}_{\textrm{m}\textrm{i}\textrm{x}}}{\textrm{d}t}=\frac{{\dot{m}}_{\textrm{m}\textrm{i}\textrm{x}}}{{m}_{\textrm{m}\textrm{i}\textrm{x}}}\left({h}_{\textrm{a}}-{h}_{\textrm{m}\textrm{i}\textrm{x}}\right) \end{align}

Now, the combination of Equation (20) and Equation (21) leads to

(22) \begin{align} \frac{\textrm{d}{q}_{\textrm{m}\textrm{i}\textrm{x}}}{\textrm{d}{h}_{\textrm{m}\textrm{i}\textrm{x}}}=\frac{{q}_{\textrm{m}\textrm{i}\textrm{x}}-{q}_{\textrm{a}}}{{h}_{\textrm{m}\textrm{i}\textrm{x}}-{h}_{\textrm{a}}} \end{align}

This simple differential equation results in a linear function between the enthalpy and the specific humidity of the plume. With the initial condition that the mixing line should meet the exhaust condition $ q({h}_{\textrm{E}})={q}_{\textrm{E}}$ the linear specific humidity-enthalpy relation during the mixing process can be expressed as

(23) \begin{align} {q}_{\textrm{m}\textrm{i}\textrm{x}}={q}_{\textrm{a}}+({q}_{\textrm{E}}-{q}_{\textrm{a}})\frac{{h}_{\textrm{m}\textrm{i}\textrm{x}}-{h}_{\textrm{a}}}{{h}_{\textrm{E}}-{h}_{\textrm{a}}} \end{align}

This relation governs the whole mixing process in a generalised way. If we assumed linear functions $ {p}_{\textrm{w}\textrm{v},\textrm{m}\textrm{i}\textrm{x}}\left({q}_{\textrm{m}\textrm{i}\textrm{x}}\right)$ and $ {h}_{\textrm{m}\textrm{i}\textrm{x}}({T}_{\textrm{m}\textrm{i}\textrm{x}})$ (see Equations (7) and (9)), the classical mixing line (Equation (10)) would follow. With the full non-linear relations, we obtain the following implicit relationship between $ {p}_{\textrm{w}\textrm{v},\textrm{m}\textrm{i}\textrm{x}}$ and $ {T}_{\textrm{m}\textrm{i}\textrm{x}}$ :

(24) \begin{align} {q}_{\textrm{m}\textrm{i}\textrm{x}}\left(\frac{1}{{q}_{\textrm{E}}-{q}_{\textrm{a}}}-\frac{{h}_{\textrm{w}\textrm{v}}({T}_{\textrm{m}\textrm{i}\textrm{x}})-{h}_{\textrm{d}\textrm{a}}({T}_{\textrm{m}\textrm{i}\textrm{x}})}{{h}_{\textrm{E}}-{h}_{\textrm{a}}}\right)&=\frac{{q}_{\textrm{a}}}{{q}_{\textrm{E}}-{q}_{\textrm{a}}}+\frac{{h}_{\textrm{d}\textrm{a}}({T}_{\textrm{m}\textrm{i}\textrm{x}})-{h}_{\textrm{a}}}{{h}_{\textrm{E}}-{h}_{\textrm{a}}}\\\nonumber\textrm{with}\,{q}_{\textrm{i}}&=\frac{\epsilon \,{p}_{\textrm{w}\textrm{v},\textrm{i}}}{{p}_{\textrm{a}}+(\epsilon -1){p}_{\textrm{w}\textrm{v},\textrm{i}}}\ \textrm{for}\,\text{i}\in \{\textrm{m}\textrm{i}\textrm{x},\textrm{E},\textrm{a}\} \end{align}

In contrast to the classical mixing line, Equation (24) does not explicitly contain fuel properties and the overall propulsion efficiency $ \eta $ . To obtain a classical formulation of the mixing curve, these parameters can be calculated from the exit conditions

(25) \begin{align} {q}_{\textrm{E}}=\gamma \,{N}_{\textrm{E}}\,\textrm{E}{\textrm{I}}_{{\textrm{H}}_{2}\textrm{O}} \end{align}

(26) \begin{align} \quad\,\, {h}_{\textrm{E}}=\delta\, {N}_{\textrm{E}}\,Q(1-\eta ), \end{align}

where $ {N}_{\textrm{E}}$ is the dilution of the exhaust at the exit.

Using a parametrisation for $ {h}_{\textrm{w}\textrm{v}}$ and $ {h}_{\textrm{d}\textrm{a}}$ , e.g. Ref. [Reference Bücker, Span and Wagner41], this equation can be solved for $ {p}_{\textrm{w}\textrm{v},\textrm{m}\textrm{i}\textrm{x}}$ under specified ambient and initial conditions. The form and behaviour of the generalised mixing curve when changing the input parameters are less apparent than for the classical mixing line. Investigating the effects of fuel properties and ambient conditions on contrail formation is much more straightforward for the classical mixing line, because these are explicitly represented in Equation (10).

The calculation of $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ is straightforward with the classical mixing line theory. In particular, the independence of $ G$ from the ambient state simplifies the determination of $ {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}$ . The mixing curve (Equation (24)), however, depends implicitly on the ambient temperature, making it impossible to evaluate the limiting temperature condition (Equation (11)) explicitly. Therefore, we propose a numerical, iterative search for the maximum ambient temperature, at which the relative humidity with respect to water $\textrm{R}\textrm{H}_{\textrm{w}}$ is at least $ 100{\%}$ during mixing. This ambient temperature is then considered the limiting temperature:

(27) \begin{align} {\widehat{T}}_{\textrm{l}\textrm{i}\textrm{m}}=\textrm{m}\textrm{a}\textrm{x}\left({T}_{\textrm{a}}\right)\;|\;\textrm{R}\textrm{H}_{\textrm{w},\textrm{m}\textrm{a}\textrm{x}}\ge 100{\%}\,\textrm{f}\textrm{o}\textrm{r}\,\textrm{Equation}\,(24) \end{align}

While the solution of this formulation is computationally more demanding than the application of the classical theory, it provides a more generally valid representation of plume mixing with the ambient air.

Microphysical processes play an important role for the plume conditions once the plume becomes supersaturated with respect to water. While the thermodynamic formulation is well-suited to determine whether or not a contrail forms, it does not provide any information on the contrail’s properties. To include microphysical processes, the plume mixing has to be resolved in time, which is not necessary for the thermodynamical formulation. The current study focuses on the thermodynamical formulation, but Appendix A.1 provides a method to include the generalised equations in a numerical microphysical model.

As a side remark, we want to clarify the terminology used in existing literature. We introduced the specific humidity, which is defined as the ratio of WV mass over total air mass, while the mixing ratio $ \chi =\frac{{\rho }_{\textrm{w}\textrm{v}}}{{\rho }_{\textrm{d}\textrm{a}}}$ is defined as the ratio of WV mass over the mass of the DA. As mentioned before, for atmospheric applications this difference is mostly irrelevant. Nevertheless, it makes sense to apply some rigour and use the correct terminology in contrail formation studies. Using mixing ratio instead of specific humidity, the WV mass conservation equation (the equivalent equation for $ q$ is given in Equation (18)) and the prognostic equation (cf. Equation (20)) can be written as

(28) \begin{align} {\chi }_{\textrm{E}}\,{m}_{\textrm{E},\textrm{D}\textrm{A}}+{\chi }_{\textrm{a}}\,{m}_{\textrm{a},\textrm{D}\textrm{A}}(t)={\chi }_{\textrm{m}\textrm{i}\textrm{x}}(t){m}_{\textrm{m}\textrm{i}\textrm{x},\textrm{D}\textrm{A}}(t) \end{align}

and

(29) \begin{align} \frac{\textrm{d}{\chi }_{\textrm{m}\textrm{i}\textrm{x}}}{\textrm{d}t}=\frac{{\dot{m}}_{\textrm{D}\textrm{A}}}{{m}_{\textrm{D}\textrm{A}}}\left({\chi }_{\textrm{a}}-{\chi }_{\textrm{m}\textrm{i}\textrm{x}}\right), \end{align}

respectively. In the latter equation, the logarithmic derivative of the mass is called entrainment rate, which typically refers to the total plume mass (as it appears in the equation for $ q$ , Equation (20)) and not to the DA mass only (as in Equation (29)). While Ref. [Reference Schumann8] does neither use the term ‘specific humidity’ nor ‘mixing ratio’ and does not distinguish between the two, many other studies have used the term ‘mixing ratio’ [Reference Bier, Unterstrasser, Zink, Hillenbrand, Jurkat-Witschas and Lottermoser27, Reference Kärcher and Fabian43–Reference Bier, Unterstrasser and Vancassel46], where ‘specific humidity’ would have been more appropriate. Note that Ref. [Reference Richardson11] uses specific humidity and refers to it as ‘mass fraction’.

3.0 Application of the generalised contrail formation theory to future aircraft propulsion systems

This chapter applies the generalised mixing curve as defined by Equation (24) for various potential propulsion systems and fuels of future aircraft. We compare it with the classical mixing line (see Equation (10)) to assess the impact and validity of the assumptions outlined in Section 2.3.

Table 2.

Parameter settings for different applications for possible future aircraft propulsion systems

We analyse a wide range of possible future aircraft fuels and propulsion system configurations. The exhaust conditions are varied according to the specific propulsion system being considered. Specifically, four distinct fuel types are examined: kerosene, methane, ammonia and hydrogen. Additionally, the impact of WV and heat recuperation in hydrogen FC systems is investigated. Lastly, ambient conditions are varied for a specific hydrogen FC case to assess the influence of atmospheric conditions. In total, we consider eight cases that are summarised in Table 2. The heat of combustion and WV emission index for the different fuels are given in Table 1. We assume that the initial dilution factor $N_{\textrm{E}}$  is around 70 in the kerosene case [Reference Bier, Unterstrasser and Vancassel46]. For all other fuels, $N_{\textrm{E}}$ is implicitly adjusted such that ${Q_{{\textrm{avail}}}}$ is the same for all fuel types. When changing e.g. from kerosene to H ${_2}$ , the fuel consumption would be smaller due to the larger combustion heat. For comparable engine size, exhaust mass flow and ${T_{\textrm{E}}}$ , this implies an adjustment of ${N_{\textrm{E}}}$ . The resulting values for ${N_{\textrm{E}}}$ in Table 2 might differ from actual air-to-fuel exhaust ratios, because these also depend on the operation state of the engine and the aircraft design. Yet, the plume properties are not much affected by our choice of ${N_{\textrm{E}}}$ . A smaller ${N_{\textrm{E}}}$ leads to larger ${T_{\textrm{E}}}$ and ${q_{\textrm{E}}}$ values and would simply extend the plotted range, while higher values for ${N_{\textrm{E}}}$ lead to lower exhaust values. Both variants do not influence the shape of the mixing curves, leading to the same plume states during mixing as displayed in Fig. 4.

Figure 4.

Comparison between the mixing curve, given by Equation (24), and its respective mixing line (black dotted) for different propulsion systems and fuels (see legend for colours). All cases are outlined in Table 2. The right end points of the mixing curves are indicated by a filled circle, while that of the mixing lines by a black star. The additional pink symbols in panel c show the H ${_2}$ C case of panel b, the brown ones in panel d show the H ${_2}$ FC 1 case of panel c.

Figure 4. Long description

The image contains four line graphs labeled a, b, c, and d, each comparing the mixing curve and its respective mixing line for different propulsion systems and fuels. The x-axis represents temperature in Kelvin, while the y-axis represents the partial pressure of water vapor in hectopascals. Each graph includes a legend indicating the fuels or propulsion systems being compared. In graph a, the blue line represents kerosene, and the orange line represents LNG. Graph b features green and pink lines for NH3 and H2C, respectively. Graph c shows brown and purple lines for H2FC 1 and H2FC 2, with additional pink symbols indicating the HC case from graph b. Graph d includes red and yellow lines for H2FC 3 and H2FC 4, with additional brown symbols indicating the H2FC 1 case from graph c. The right endpoints of the mixing curves are marked by filled circles, while the endpoints of the mixing lines are marked by black stars. The graphs illustrate how different fuels and propulsion systems affect the mixing curve and line under varying temperature conditions.

We examine various propulsion systems that may be employed in the coming decades. As the efficiency of these systems is hard to project, a constant value of $\eta = 0.4$ is assumed for all cases. In reality, the propulsion efficiency may vary with fuel type due to differences in combustion behaviour and thermodynamic properties. However, a variation of $\eta $ over a reasonable range has a much smaller impact on plume thermodynamics than a variation of fuel type or the newly introduced factors $\gamma $ and $\delta $ . For simplicity, a constant overall propulsion efficiency is assumed. Although the technical realisation of a combustion turbine and a fuel cell are fundamentally different, for our study, only the emitted heat and WV are of interest. Hence, contrail formation is independent of whether the fuel is combusted in a turbine or converted to electric power in an FC. However, we do consider one potential technical advancement that may be feasible for FC systems: the recuperation of heat and WV before the gas is emitted. This could significantly reduce the emitted heat or WV and would clearly influence contrail formation. To account for this, we analyse an FC system, which emits only $25\;{\textrm{% }}$ of the available energy ${Q_{{\textrm{avail}}}}$ as heat while the remaining $75\;{\textrm{% }}$ are reused within the aircraft (H ${_2}$ FC 1). Additionally, an FC system is analysed, which only emits $25\;{\textrm{% }}$ of the produced WV (H ${_2}$ FC 2), and the remaining $75\;{\textrm{% }}$ are condensed and emitted as liquid droplets, separately from the heated exhaust. Note that this value is purely hypothetical, as no current aircraft systems employ WV condensation to mitigate exhaust humidity. The chosen values represent a range of potential future mitigation scenarios.

Ambient conditions at a relatively low flight level are considered, corresponding to an ambient pressure of ${p_{\textrm{a}}} = 400\;{\textrm{hPa}}$ and an ambient temperature of ${T_{\textrm{a}}} = 230\;{\textrm{K}}$ , which corresponds to a temperature that is $10\;{\textrm{K}}$ lower than the temperature defined by the ISA at this flight level. We choose a low flight level because the technical realisation of fuel-cell-propelled aircraft is currently expected to be implemented for small, regional aircraft, which operate at lower altitudes. Additionally, propeller-driven aircraft require higher air densities than combustion aircraft to operate efficiently and thus fly at lower altitudes. To investigate the impact of atmospheric conditions on the mixing process, two cases with independently reduced ambient temperature ${T_{\textrm{a}}} = 210\;{\textrm{K}}$ (H ${_2}$ FC 3) and increased flight altitude ${p_{\textrm{a}}} = 280\;{\textrm{hPa}}$ (H ${_2}$ FC 4) are added. For all cases, the ambient air is assumed to be saturated with respect to ice, i.e. ${\textrm{R}}{{\textrm{H}}_{{\textrm{i}},{\textrm{a}}}} = 100\;{\textrm{% }}$ . The exact value of ${\textrm{R}}{{\textrm{H}}_{{\textrm{i}},{\textrm{a}}}}$ has only a very small influence on contrail formation, e.g. [Reference Zink, Unterstrasser and Burkhardt47]. However, contrails can persist only when ${\textrm{R}}{{\textrm{H}}_{{\textrm{i}},{\textrm{a}}}} \gt 100\;\% $ . The occurrence frequency of ISSRs has primarily been studied in the context of current long-haul aircraft, which operate at higher cruising altitudes compared to regional aircraft. Based on findings from [Reference Kaufmann, Dischl and Voigt14] and [Reference Petzold, Neis, Rütimann, Rohs, Berkes, Smit, Krämer, Spelten, Spichtinger, Nédélec and Wahner48], the frequency of ISSRs may be slightly reduced at lower flight altitudes. Nevertheless, such reductions are not sufficient to eliminate the potential for persistent contrails, particularly for low temperatures during winter months.

Figure 4 depicts the mixing curve and its respective mixing line in a Schmidt-Appleman plot for each case listed in Table 2. While each mixing curve is represented by a different colour in Fig. 4, the corresponding mixing lines are all given as dotted black lines. The ambient conditions are represented by a black circle in the lower left corner of all panels, while the initial exhaust conditions are represented by a coloured circle for the general mixing curve and a black star for the classical mixing line. The grey lines represent the saturation pressure over water ${p_{{\textrm{w}},{\textrm{sat}}}}( T )$ . The zoomed-in windows highlight the temperature range relevant for contrail formation, i.e. the temperature range, where the mixing line and curve lie above the saturation pressure curve. The most notable observation across all panels is that the mixing curves closely align with their corresponding mixing lines. Figure 4 reveals that the mixing curves are only slightly curved and can be approximated by their corresponding mixing line.

Figure 4(a) and (b) depict the mixing curves and lines for different fuel types (first four cases in Table 2). Because all cases are constructed to emit the same amount of energy, the exhaust temperatures differ only slightly from one another in the top row. These small differences originate from the differences in heat capacities that depend on the specific humidity of the exhaust. The initial specific humidity ${q_{\textrm{E}}}$ varies with fuel type and is lowest for kerosene (blue line, case ‘kero’). This example represents a limiting case where the plume is only slightly supersaturated over a very narrow temperature range. For the LNG exhaust (orange line, case ‘LNG’), ${q_{\textrm{E}}}$ is larger than for kerosene due to the higher WV emission index, resulting in a plume that is clearly supersaturated during mixing. The difference in ${q_{\textrm{E}}}$ compared to kerosene is even more pronounced when ammonia or hydrogen is used, leading to a larger slope of the mixing curves and consequently to higher supersaturation values during plume mixing in Fig. 4(b). The mixing curves for ammonia (green line, case ‘ammo’) and hydrogen (pink line, case ‘H ${_2}$ C’) are very similar, which can be attributed to their similar energy-specific WV emission indices of $\zeta $ (see Table 1). Despite the higher moisture content present in the plumes when using ammonia or hydrogen, the mixing curves do not differ significantly from their respective mixing lines for temperatures where the plume is supersaturated. Figure 4(c) illustrates the effects of heat (‘H ${_2}$ FC 1’, brown line) and WV (‘H ${_2}$ FC 2’, magenta line) recuperation of a hydrogen FC-based propulsion system on the mixing process. As a reference, the initial exhaust conditions are shown for the unaltered ‘H ${_2}$ C’ exhaust (cf. Fig. 4(b)). Heat recuperation results in a significant reduction of ${T_{\textrm{E}}}$ while leaving ${q_{\textrm{E}}}$ unchanged. This leads to a substantial increase in the mixing curve’s slope and even larger supersaturation values than in the H ${_2}$ C case. In contrast, WV recuperation leads to a decrease in the mixing curve’s slope. For the specific case displayed, this results in a mixing curve that is only slightly supersaturated and lower than in the kerosene case shown in Fig. 4(a). This is expected as $\gamma $ is smaller than $\zeta _{{\textrm{norm}},{\textrm{H}}2}^{ - 1}$ . Figure 4(d) shows the dependence of the mixing curve on the ambient conditions. Note that for this panel, the limit of the temperature axis is much smaller than for the other panels, as both cases have a lower initial exhaust temperature. The brown symbols in this panel correspond to the initial exhaust conditions of the case with reduced exhaust heat at unaltered ambient conditions ‘H ${_2}$ FC 1’ as a reference. The red curve (‘H ${_2}$ FC 3’) illustrates that a decrease in ambient temperature leads to larger supersaturation values. Notably, for the displayed case, the exhaust itself is already nearly saturated. The yellow curve shows the mixing curve at a higher flight level (‘H ${_2}$ FC 4’). Due to the lower overall pressure, the WV partial pressure is reduced, resulting in lower supersaturation of the plume during mixing. Under real atmospheric conditions, ambient pressure and temperature do not vary independently, but a decrease in ambient pressure due to a higher flight level is, on average, associated with a decrease in ambient temperature. A simultaneous decrease of ambient pressure and temperature leads to counteracting effects on the mixing curve. Hence, varying only one of the two parameters may slightly overemphasise the sensitivity of the mixing curve slope on ambient conditions.

In general, the lower $q_\mathrm{E}$ and the higher $T_\mathrm{E}$ , the lower the slope of the mixing curve and its supersaturation values. Moreover, Fig. 4 discloses fairly small differences between the classical mixing line and the generalised mixing curve for all considered cases.

Figure 5 provides a detailed explanation of why the two formulations give basically identical results. It shows the mixing curves and lines for two scenarios, which are depicted for four different combinations of coordinate axes, namely $h$ or $T$ on the x-axis and $q$ or ${p_{{\textrm{wv}}}}$ on the y-axis. The ranges of the different axes are chosen such that the end and start points of the classical mixing line coincide on both ends for all axes combinations. Using the linear functions $\hat{q}(p_\mathrm{wv})$ (Equation (9)) and $h( T )$ (Equation (7)) in the classical formulation, the connecting straight line is the same for all four axes combinations, represented by the black line. For non-linear relations (see Equations (13), (15) and (16)), the initial exhaust values differ from those obtained by using their respective linearisation (Equations (9) and (7)). For better assessment of the plume’s saturation state, the grey line shows ${p_{{\textrm{w}},{\textrm{sat}}}}\left( T \right)$ .

Figure 5.

Comparison of mixing curve and line along different axes for two different cases for hydrogen-powered fuel cell: a) ‘H ${_2}$ FC 1’ case, see Tab. 2; b) extreme case: ${\eta _{extr}} = 0.5,\;{\gamma _{extr}} = 1,\,{\delta _{extr}} = 0.1,$ ${p_{a,extr}} = 400\;hPa,\;{T_{a,extr}} = 210\;K$ .

Figure 5. Long description

The line graph compares mixing curve and line along different axes for two different cases for hydrogen-powered fuel cell. The x-axis represents temperature in Kelvin, and the y-axis represents specific humidity and water vapor pressure. The graph includes four data lines: q(h), q(T), p(h), and p(T). The data lines show how specific humidity and water vapor pressure vary with temperature and specific enthalpy. All values are approximated.

Figure 5(b) illustrates the case ‘H ${_2}$ FC 1’, whose mixing line and curve have already been shown in Fig. 4(b). Both formulations (classical and generalised) are based on the linear $q - h$ relationship and share the same straight (black) line in these coordinate axes. In the $q - T$ diagram, the mixing curve is represented by the blue line, which is slightly convex near the upper initial point and ${T_{\textrm{E}}}$ is lower than for the mixing line. This is the effect of the heat capacity being a function of temperature and humidity, resulting in larger values compared to the DA value at constant temperature. Consequently, ${T_{\textrm{E}}}$ of the mixing curve is lower for the same ${h_{\textrm{E}}}$ . In the ${p_{{\textrm{wv}}}} - h$ diagram, the mixing curve is represented by the orange line. This curve is slightly concave near the upper point, where the partial pressure is lower compared to the mixing line. This effect originates from the employment of the non-linear Equation (13), which leads to a lower partial pressure for the same specific humidity value. In the ${p_{{\textrm{wv}}}} - T$ diagram (green line), both effects are present for the initial exhaust: the lower temperature due to the increased heat capacity and the lower WV partial pressure due to the usage of the correct specific humidity definition. The right end points of the mixing line and curve are not the same, but both effects basically cancel each other as the end point of the mixing curve lies almost on the mixing line. For contrail formation, however, not the exact exhaust conditions are relevant; the relationship between ${p_{{\textrm{wv}}}}$ and $T$ becomes crucial in the temperature range where the plume reaches supersaturation.

To confirm that the mixing line and mixing curve approaches give, in general, identical results regarding contrail formation, both formulations are compared for an extreme case. This extreme case should meet the following conditions: the initial exhaust should be very moist and its temperature as low as possible. With these conditions, the initial exhaust would have a large relative humidity and a high WV content, resulting in the largest expected differences between the line and the curve within the range where the plume is supersaturated. To meet these conditions, a hydrogen fuel cell case with a $90\;{\textrm{% }}$ heat recuperation ( ${\delta _{{\textrm{extr}}}} = 0.1$ ) is employed, while all of the generated WV is emitted. The efficiency is increased to ${\eta _{{\textrm{extr}}}} = 0.5$ . The ambient temperature is set to ${T_{{\textrm{a}},{\textrm{extr}}}} = 210\;{\textrm{K}}$ while the pressure is chosen to be ${p_{{\textrm{a}},{\textrm{extr}}}} = 400\;{\textrm{hPa}}$ in an ice-saturated atmosphere. The right end point corresponds to an initial dilution of ${N_{\textrm{E}}} = 50$ . We want to stress that this is an academic case, meaning that this is used to highlight possible differences between the formulations, and we do not expect that this case will be relevant in practice. Even for this extreme case where the initial exhaust is almost saturated with water, the mixing curve is very similar to the classical mixing line, cf. Fig. 5(b). The temperature range of the extreme case is smaller than for the ‘H ${_2}$ FC 1’ case, while the specific humidity range is larger. The trends are the same as described for Fig. 5(a), but due to the smaller temperature range, the differences are more pronounced. Nevertheless, the resulting green mixing curve aligns with the black mixing line, leading to no difference in the decision whether a contrail forms or not.

The cancellation of two linearisation errors occurs in all cases and leads to almost identical results for the classical mixing line and the mixing curve. This is a surprising, yet comforting, outcome, as it shows that the classical mixing line can be used to assess contrail formation for very moist plumes, despite being based on some potentially inaccurate assumptions.

As a direct consequence, we find that the limiting temperatures $\hat{T}_{\textrm{lim}} $ derived with both formulations are also close to each other. Specifically, $\hat{T}_{\textrm{lim}} $ differs by less than $0.2$ K between both formulations for all cases presented in Table 2. This similarity can be attributed to the close alignment of the mixing lines and the generalised mixing curves in the ${p_{{\textrm{wv}}}} - T$ diagram as exemplified in Fig. 5, especially in the temperature range where the plume is supersaturated. For the extreme case in Fig. 5(b), $\hat{T}_{\textrm{lim}} $ differs by approximately $0.2$ K between both formulations, highlighting that the similarity between both formulations holds for all cases.

We want to stress that the implementation of only one correction of the linear assumptions, without implementing the other correction, can lead to larger inaccuracies than using the classical mixing line. This is shown in Fig. 5(b), where the blue and orange lines differ from the aligned green and black curves. This difference is also reflected in the respective limiting temperatures. Due to the larger curvature of both the orange and the blue line, $\hat{T}_{\textrm{lim}} $ differs by several degrees from the limiting temperature evaluated with the generalised mixing curve. It is essential to note that $\hat{T}_{\textrm{lim}} $ calculated for this extreme case is very high, and discrepancies have likely no practical effect on contrail formation. $\hat{T}_{\textrm{lim}} $ is the critical factor for contrail formation only if $\hat{T}_{\textrm{lim}} \lt T_{\textrm{frz}} $ . If $\hat{T}_{\textrm{lim}} $ exceeds the freezing threshold, contrail formation is instead restricted by ${T_{{\textrm{frz}}}}$ . The latter occurs in all cases given in Table 2 except for kerosene combustion and the case ‘H ${_2}$ FC 2’. This suggests that contrail formation will occur as long as the ambient temperature is below the freezing temperature for future aircraft if fuels with high hydrogen content are used without reduction of the WV emissions. This holds only when future aircraft will not have a much lower overall propulsion efficiency than current aircraft (which is not expected).

Having also in mind the computational overhead of determining the mixing curve compared to the mixing line, we conclude that the mixing line is a suitable approximation for the mixing curve even for very moist plumes. The generalised mixing curve is almost identical to the mixing line for all considered cases, and the effect of the generalisation on the limiting temperature can also be neglected. Therefore, the mixing line can be used to investigate contrail formation for future aircraft propulsion systems.

4.0 Technical modifications of the FC exhaust and its effect on contrail formation

In this section, the possibility of contrail avoidance by potential heat and WV recuperation systems in FC-propelled aircraft is investigated. The goal of heat recuperation systems is to increase the efficiency of the propulsion system by using parts of the remaining energy of the gas before emission. In addition, heat exchangers that divert some of the generated heat away from the FC stack are necessary to ensure a constant operating temperature within the stack [Reference Kazula, de Graaf and Enghardt21]. While this internal heat management has no direct influence on the exhaust temperature, the usage of heat recovery results in lower exhaust temperatures, leading to plume conditions more favourable for contrail formation. In contrast, WV recuperation leads to less favourable conditions for contrail formation. The extracted WV could be liquefied into water droplets, which are then separated from the heated exhaust. We assume that the released latent heat by liquefying the WV only heats the stream of liquid water and does not increase the temperature of the exhaust. The exhaust heat and the water droplets should be emitted at different locations of the aircraft. Ideally, the droplets would be emitted at a location such that they will not be entrained into the expanding exhaust plume. Note that the following paragraphs do not discuss implicit changes of $\eta $ by technical modifications, but solely focus on changes of the emission factors $\gamma $ and $\delta $ .

For contrail avoidance by technical means, the exhaust should contain less water and/or have a higher temperature to decrease the mixing line’s slope and thus reduce the contrail formation propensity. Even if contrail formation is not suppressed completely, a drier or hotter plume results in smaller peak relative humidity values and a shorter supersaturation period during mixing. Both effects can reduce the number of ice crystals forming in the first seconds behind the aircraft [Reference Bier, Unterstrasser, Zink, Hillenbrand, Jurkat-Witschas and Lottermoser27, Reference Bier, Unterstrasser and Vancassel46] and the radiative impact of the mature contrail [Reference Unterstrasser and Gierens32–Reference Lottermoser and Unterstrasser34], which may evolve in suitable environmental conditions.

In the following, we assume that it is technically possible to install heat and WV recuperation systems in an FC-propelled aircraft to manipulate the humidity and temperature of the exhaust. The classical mixing line is used for the following analysis to reduce the computational effort relative to the mixing curve computations. To ensure that the use of hydrogen-based FC systems does not increase the contrail formation propensity, a first goal for contrail mitigation would be to reduce the water emission such that the mixing line of the FC-propelled aircraft’s plume aligns with that of an aircraft using kerosene combustion, i.e. we require ${G_{{{\textrm{H}}_2}}} = {G_{{\textrm{kero}}}}$ (first line in Table 3). Assuming equivalent efficiencies for both kerosene and hydrogen propulsion systems, the WV emission has to be reduced by the inverse value of ${\zeta _{{\textrm{norm}}}}$ given in Table 1, i.e. ${\gamma _{{{\textrm{H}}_2},{\textrm{ref kero}}}} = {\left( {{\zeta _{{\textrm{norm}}}}} \right)^{ - 1}} = 0.39$ . However, this value is based on the assumption that no heat is recuperated nor any energy losses occur around the propulsion units, which would reduce ${Q_{{\textrm{avail}}}}$ of the exhaust. To account for heat recuperation, a new variable $\lambda = \gamma /\delta $ is introduced, which represents the relative emission factor of the exhaust. This allows us to calculate the required WV reduction for a given value of heat recuperation. As an exemplary calculation, if $50\;{\textrm{% }}$ of ${Q_{{\textrm{avail}}}}$ for a ${{\textrm{H}}_2}$ propelled aircraft is recuperated, it would require a $78\;{\textrm{% }}$ reduction of the WV emission to avoid additional contrail formation compared to the kerosene propelled aircraft, because ${\lambda _{{{\textrm{H}}_2},{\textrm{ref kero}}}} = 0.39$ .

Table 3.

Overview of relative emission factors ${\lambda _i}$ to achieve different levels of contrail formation suppression by emission manipulation

While the requirement ${G_{{{\textrm{H}}_2}}} = {G_{{\textrm{kero}}}}$ keeps the contrail formation probability of H ${_2}$ FC aircraft at the level of kerosene aircraft (previous paragraph), emission manipulation can be used to further reduce the H ${_2}$ FC contrail formation probability and the associated climate impact of contrail-cirrus. It would be beneficial to completely suppress contrail formation by technical modification of the exhaust, by requiring that the plume ${\textrm{R}}{{\textrm{H}}_{\textrm{w}}}$ remains below $100\;{\textrm{% }}$ (second line in Table 3). Such a controlled exhaust modification, which may be associated with extra costs, is only required when flying in ISSRs. This proposition is similar to the idea of a targeted use of SAFs [Reference Teoh, Schumann, Voigt, Schripp, Shapiro, Engberg, Molloy, Koudis and Stettler49]. In both concepts, contrail formation is on-demand technically modified. The maximum $\lambda $ value is calculated, for which supersaturation with respect to water during mixing is avoided. This critical value ${\lambda _{100\;{\textrm{% }}}}$ depends on atmospheric conditions and is smaller (i.e. technically more demanding) than ${\lambda _{{\textrm{ref kero}}}}$ for cases where a kerosene contrail would form and less demanding when no kerosene contrail would form. In the limiting case for contrail formation, the relative humidity of the plume is at most ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} = 100\;{\textrm{% }}$ during the whole mixing process. We perform an iterative search of the mixing line slope necessary for the mixing line to be tangential to ${p_{{\textrm{w}},{\textrm{sat}}}}( T )$ . The resulting mixing line slope can then be used to calculate the corresponding relative emission factor $\lambda = \gamma /\delta $ via Equation (4).

Figure 6(a) shows ${\lambda _{100\;{\textrm{% }}}}$ as a function of ambient temperature for different fuels and two different flight levels at ${p_{\textrm{a}}} \in \left\{ {280,400} \right\}\,{\textrm{hPa}}$ . In addition to the two constant pressure levels, the circles in Fig. 6 use ${p_{\textrm{a}}}( {{T_{\textrm{a}}}} )$ as given by the ISA. The dotted horizontal lines indicate the relative emission fraction ${\lambda _{{\textrm{i}},{\textrm{ref kero}}}}$ necessary to restrict contrail formation to those scenarios where kerosene combustion would produce a contrail. For ${\lambda _{100\;{\textrm{% }}}} \gt 1$ , the unaltered exhaust would not form a contrail for the specified ambient conditions. Therefore, heat reduction can be applied up to $\delta = 1/{\lambda _{100\;{\textrm{% }}}}$ with the unaltered water emission without forming a contrail for these cases. Figure 6(a) reveals that this occurs only when kerosene is used as a fuel because $\hat{T}_\mathrm{lim} \gt 235 \: \rm{K}$ for hydrogen and ammonia. For all fuels, ${\lambda _{100\;{\textrm{% }}}}$ increases with increasing ambient temperatures and decreasing ambient pressure. For identical ambient conditions, the ${\lambda _{100\;{\textrm{% }}}}$ curves for different fuels differ simply by a factor identical to the ratio of ${\zeta ^{ - 1}}$ . This indicates that contrail avoidance requires more WV extraction for the same amount of recovered heat when ammonia is used instead of pure hydrogen. In general, the higher ${\lambda _{100\;{\textrm{% }}}}$ values for higher ambient temperatures make contrail avoidance easier for FC-propelled regional aircraft, as their lower flight altitude leads on average to higher ambient temperatures.

Figure 6.

Ratio between heat and WV recuperation $\lambda $ to limit the maximum relative humidity during plume mixing for different fuels and ambient. The lower part of the plots shows $\lambda $ , while the upper part shows $1/\lambda $ if $\lambda \gt 1$ . a) No contrail forms: ${\lambda _{100\;\% }}$ b) No HDN occurs: ${\lambda _{500\;\% }}$ .

Figure 6. Long description

The two line graphs illustrate the ratio between heat and water vapor recuperation to limit the maximum relative humidity during plume mixing for different fuels and ambient conditions. The x-axis represents the ambient temperature in Kelvin, ranging from 210 to 235 Kelvin. The y-axis on the lower part of the graphs shows lambda, ranging from 0.0 to 1.0, while the y-axis on the upper part shows the inverse of lambda, ranging from 0.0 to 1.0. The graphs compare the behavior of kerosene, hydrogen, and ammonia under two different pressure conditions: 280 hectopascals and 400 hectopascals. The solid lines represent the 400 hectopascals condition, while the dashed lines represent the 280 hectopascals condition. The circles on the lines indicate specific data points. The left graph (a) shows conditions where no contrail forms, and the right graph (b) shows conditions where no homogeneous nucleation occurs. All values are approximated.

Nonetheless, especially for low ${T_{\textrm{a}}}$ , the ${\lambda _{100\;{\textrm{% }}}}$ values are small, making it challenging to implement recuperation systems that can effectively meet this goal. If a complete suppression of contrail formation for FC-propelled aircraft is out of reach in a technical sense, it may still be beneficial to keep the peak humidity below a certain (technically less demanding) limit. As discussed above, heat recuperation leads to a decrease in exhaust temperature and consequently to an increase in the maximum relative humidity. Hence, heat recuperation without the use of WV reduction can lead to relative humidity values above $500\;{\textrm{% }}$ during plume mixing, if fuels with high hydrogen content like hydrogen or ammonia are used (third line in Table 3).

For these very high supersaturation values, an additional droplet-forming process can become significant: homogeneous droplet nucleation (HDN). This process condenses WV directly from the vapour phase into nanometer-sized droplets without the need for condensation particles. HDN commonly occurs in steam turbines [Reference Wittmann, Lück, Bode and Friedrichs50] and can also lead to the formation of aerodynamic contrails [Reference Jansen and Heymsfield51] over aircraft wings in cold conditions. Based on theoretical and experimental results, Ref. [Reference Wölk and Strey52] provides an empirical correlation of the nucleation rate with temperature and supersaturation. This correlation fits most experimental findings well, as shown in Fig. 3 of Ref. [Reference Jansen and Heymsfield51]. For typical plume temperatures at which supersaturation occurs, the nucleation rate of HDN is negligible for relative humidity values below $500\;{\textrm{% }}$ , but increases in a highly non-linear fashion with both supersaturation and temperature. For ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}}\gtrsim 500\;{\textrm{% }}$ , it cannot be ruled out that the droplet and consequently ice crystal numbers increase substantially compared to hydrogen plumes with ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} \lt 500\;\% $ . It cannot be excluded that ice crystal numbers might even be higher than in typical soot-rich kerosene combustion scenarios as studied in Ref. [Reference Kärcher and Yu53]. This effect would come along with increased contrail formation propensity in lower flight levels due to the higher limiting temperature of hydrogen compared to kerosene. The combination of both effects (higher contrail formation propensity and more ice crystals) would increase the climate impact of contrail-cirrus formed behind hydrogen FC-based aircraft.

Analogously to ${\lambda _{100\;{\textrm{% }}}}$ , ${\lambda _{500\;{\textrm{% }}}}$ is defined as the maximum $\lambda $ value for which the peak relative humidity within the plume is kept below $500\;{\textrm{% }}$ during mixing. Similar in style to panel 6(a), Figure 6(b) depicts ${\lambda _{500\;{\textrm{% }}}}$ . As expected, ${\lambda _{500\;{\textrm{% }}}}$ values are larger than the respective ${\lambda _{100\;{\textrm{% }}}}$ values, i.e. less WV has to extracted from the exhaust to remain below $500\;{\textrm{% }}$ . The trends in both panels are similar, with the curves in panel 6(b) shifted towards higher values of $\lambda $ . Nevertheless, for low ambient temperatures, it is still necessary to reduce WV emissions to suppress HDN, especially when hydrogen or ammonia is used. For kerosene with unaltered exhaust (i.e. $\lambda = 1$ ), we observe ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} \gt 500\;\% $ only for very low ambient temperature values. Additionally, for the kerosene case, the high number of emitted soot particles leads to a fast depletion of WV and the actual ${\textrm{R}}{{\textrm{H}}_{\textrm{w}}}$ values in the plume are much smaller than the theoretical peak value without considering microphysics (as done in the present study). For hydrogen plumes being void of soot particles and under the assumption of ice crystal formation on entrained ambient aerosol [Reference Bier, Unterstrasser, Zink, Hillenbrand, Jurkat-Witschas and Lottermoser27], WV is depleted more slowly and the ${\textrm{R}}{{\textrm{H}}_{\textrm{w}}}$ values with and without considering microphysics can be more similar. Reference [Reference Bier, Unterstrasser, Zink, Hillenbrand, Jurkat-Witschas and Lottermoser27] presented contrail formation simulations for hydrogen combustion, and the plume thermodynamic properties correspond to our hydrogen case with $\gamma = \delta = 1$ . Even though condensation/deposition depletes the WV, the peak relative humidity increases with decreasing ${T_{\textrm{a}}}$ and can reach values up to $500\;{\textrm{% }}$ . Hence, for hydrogen scenarios, ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}}$ could be a good indicator whether or not HDN may occur. The present study will not provide quantitative estimates on how many additional ice crystals may form by HDN. This work is deferred to an upcoming simulation study introducing the HDN process in the contrail formation model described in Ref. [Reference Bier, Unterstrasser, Zink, Hillenbrand, Jurkat-Witschas and Lottermoser27].

Figure 7 shows the maximum relative humidity ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}}$ during plume mixing for various fuels, considering $\lambda $ values between $0.2$ and $2$ and ambient temperatures below the freezing temperature. The ambient relative humidity is set to ${\textrm{R}}{{\textrm{H}}_{{\textrm{i}},{\textrm{a}}}} = 100\;{\textrm{% }}$ and the ambient pressure is set to ${p_{\textrm{a}}} = 400\;{\textrm{hPa}}$ , which relates to a relatively low flight level. It has to be noted that for this flight level, ambient temperatures ${T_{\textrm{a}}} \le 220\;{\textrm{K}}$ occur only rarely. The white area in each panel indicates the region where no contrail forms, i.e. ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} \lt 100\;\% $ . The white curve in each panel represents the condition where ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} = 500\;{\textrm{% }}$ . It is possible that HDN will be triggered if the plume conditions are located to the right of this curve. For kerosene, the maximum relative humidity remains below $500\;{\textrm{% }}$ except for very cold ambient conditions, which are unlikely at this pressure level. In contrast, the area where ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} \gt 500\;\% $ is significantly increased for hydrogen and ammonia. Additionally, the maximum relative humidity can reach ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}} = 4000\;{\textrm{% }}$ for ammonia if $\lambda \geq 2$ . Note that the leftmost panel on kerosene is not overly meaningful, as for soot-rich conditions, the actual ${\textrm{R}}{{\textrm{H}}_{{\textrm{w}},{\textrm{max}}}}$ values are much smaller when condensation/evaporation is considered. Nevertheless, this plot is included to showcase the general pattern and how it changes when switching to fuels with a larger energy-specific WV emission index. In general, this plot highlights the importance of water recuperation for FC systems to prevent the formation of contrails with potentially substantially increased ice crystal numbers due to HDN.

Figure 7.

Maximum relative humidity of the plume at various ambient temperature values and $\lambda $ values for an FC setup, when using (a) kerosene (b) hydrogen (c) ammonia. ${p_a} = 400\;\textrm{hPa}, \textrm{R}\textrm{H}_{i,a} = 100\;\%$ .

Figure 7. Long description

A heat map compares the maximum relative humidity of the plume at various ambient temperature values and values for an FC setup, when using kerosene, hydrogen, and ammonia. The heat map consists of three subplots, each representing a different fuel type: kerosene, hydrogen, and ammonia. The x-axis represents the lambda values, ranging from 1 to 2, while the y-axis represents the ambient temperature in Kelvin, ranging from 210 to 235. The color scale on the right indicates the maximum relative humidity percentage, ranging from 100 to 4000. Each subplot shows a gradient of colors, with darker colors indicating lower relative humidity and lighter colors indicating higher relative humidity. The 500 contour line is highlighted in white in each subplot, indicating a specific relative humidity value.