Analytical Models of Flight Fuel Consumption and Non-CO₂ Emissions as a Function of Aircraft Capacity

Abstract

The sustained growth of air traffic over the past decades has increased the aviation’s contribution to anthropogenic radiative forcing through both ${CO}_2$ and non-${CO}_2$ emissions. Although the industry has committed to achieving net-zero emissions by 2050, this goal appears unrealistic without curbing, or at least stopping, the continued rise in traffic. To assess the potential of alternative travel options and quantify their environmental benefits, simple and flexible emission models are needed. In this work, we present a set of analytical models for estimating fuel consumption and associated emissions, including ${CO}_2$, SO_x, water vapour, and other key non-${CO}_2$ emissions such as NO_x and carbon monoxide. We also examine the emissions of non-volatile particulate matter. These models require only flight distance and aircraft seat numbers, enabling broad applicability across traffic scenarios. The models are openly available via a GitHub repository, and their practical use is demonstrated through a case study of a representative day of Spanish air traffic.

1. Introduction

In recent decades, with the exception of a huge decrease during the COVID-19 crisis, demand for air travel has been growing every year, raising environmental concerns since it contributes to the increase in greenhouse gas emissions and thus to anthropogenic radiative forcing, with both ${CO}_2$ and non-${CO}_2$ emissions. While other industries, including those within the transport sector, have been better able to partially decarbonise, aviation faces serious challenges. The pledge to achieve net-zero emissions by 2050 is based on technological solutions [1,2], which are not currently available, sustainable aviation fuels [3], whose production is still too low for the demand, and carbon compensation mechanisms or offsets [4,5], which have received increasing criticism [6]. The goal thus appears unrealistic without curbing, or at least stopping, the continued rise in air traffic.

This concern has become more evident recently, even at a political level. For example, in 2023, during the negotiations for the appointment of the Prime Minister of Spain, Pedro Sánchez, it was agreed to include the provision to eliminate short flights if a train alternative of no more than 2.5 h is available in an attempt to reduce aviation-related greenhouse gas emissions [7]. There was no mention of how this 2.5 h threshold was determined, but it corresponds to that applied in France since 2023 [8]. Even within the industry, major airlines are now offering integrated tickets to circumvent short-haul flights, replacing them with rail when stopovers are required [9].

Over the past decades, research and industry efforts have mainly concentrated on quantifying aviation’s carbon footprint, while non-${CO}_2$ emissions have received comparatively little attention despite their substantial environmental impact. The climate effects of aviation’s non-${CO}_2$ emissions, particularly contrails, regarded as the most significant non-${CO}_2$ contributor, are at least as relevant as those of ${CO}_2$ itself and may even triple the overall climate impact of a single flight [10]. These pollutants are released at cruising altitudes of up to 13 km, where they directly influence the atmosphere, with their effects being highly dependent on dynamic meteorological conditions [11]. In addition to their climate implications, non-${CO}_2$ emissions also have serious consequences for air quality and human health. While decarbonization strategies in aviation have largely targeted ${CO}_2$ reduction, addressing non-${CO}_2$ pollutants should also be prioritized, particularly to protect the health of the population living close to airports [10].

In 2020, the European Commission requested the European Union Aviation Safety Agency (EASA) to deliver an updated assessment of aviation’s non-${CO}_2$ impacts on climate change, in line with the requirements of the EU’s Emissions Trading System (ETS) directive [12]. The report examined potential financial measures, such as imposing monetary charges on aircraft NO_x emissions and/or including these emissions within the EU ETS. A key finding was the urgent need for a reliable, internationally harmonised methodology to estimate cruise NO_x emissions. By the end of 2027, the European Commission is expected to publish the results of its monitoring, reporting, and verification system for non-${CO}_2$ aviation impacts, and if deemed appropriate, to introduce a legislative proposal addressing these effects [12]. Nevertheless, practical tools for assessing non-${CO}_2$ emissions remain scarce or highly complex. For example, Lee et al. employed an extensive multi-sheet spreadsheet, covering contrail cirrus, ${CO}_2$, NO_x, water vapour, sulfate and soot aerosols, and corresponding ${CO}_2$-equivalent metrics, to estimate the contribution of global aviation to anthropogenic climate forcing [13].

Some available models to assess aviation emissions were reviewed in [14]. Most of these models use constant average seat numbers, depending on the flight distance, or only a few aircraft types to build their models, such as the International Civil Aviation Organisation (ICAO) Carbon Emissions Calculator [15].

Many alternatives can be found in the literature for the estimation, with high precision, of fuel consumption (and hence emissions) for individual flights. These estimators, based on a set of operational parameters, rely on complex and often closed models. For example, many employ simulators that use EUROCONTROL Base of Aircraft Data (BADA) models, which require licensing and limit their use in terms of potential comparison of aircraft models [16]; others are based on actual fuel consumption recorded by the flights, stored in their Quick Access Recorder (QAR) [17], and reported by the Aircraft Communications Addressing and Reporting System (ACARS) [18]. The models usually need, among other things, information on aircraft performance (aircraft type) and path profiles [19], real track data [20,21], or detailed trajectories, incorporating weather conditions [22,23]. Due to their complexity, they might even only focus on a part of the flight, such as the climbing phase, as in [17]. In general, high accuracy can be obtained, even considering uncertainties, but with the need for many parameters that might not be available at the strategic level when details on the operations are limited (e.g., only the distance between airports considered, and not the actual route (or 4D trajectory), is available). Very importantly, these are models that are closed and therefore not readily available to the community.

EUROCONTROL released a web-based modelling platform, IMPACT [24], for the calculation of detailed 4D trajectories for user-defined aircraft operations, providing information on engine thrust and fuel flow estimated using BADA [25]. This system enables the estimation of the environmental impact of aviation (in terms of emissions) but still requires a small set of operational details, such as distance flown and aircraft type. IMPACT has many advantages: it requires a limited set of parameters, circumventing some of the limitations of the previously described models and enabling the evaluation of use cases even for planning and strategic analysis; it relies on state-of-the-art BADA performance models; and it is backed up by an internationally recognised institution (EUROCONTROL). These considerations mean that the obtained estimations can be broadly considered as acceptable by relevant stakeholders. However, being a web-based platform, it cannot be integrated into external modelling tools, and as it relies on BADA, some of their licensing limitations still apply. IMPACT was used in previous research to generate generic models for the estimation of emissions as a function of distance and seats in the aircraft for short- and medium-haul flights, as reported in [14].

In recent years, other models have been presented, such as the Fuel Estimation in Air Transportation (FEAT), a reduced-order fuel consumption approximation with the origin-destination airport pair and aircraft type as inputs, enabling the estimation of fuel consumption for global scheduled aircraft movements described in [26]. This is a simplified model that enables quick computation of fuel usage per route, in a similar manner to how it is developed in this article, but it still requires the consideration of actual individual aircraft types. As shown in this article, we generalise the emissions by implicitly considering the aircraft type using the number of seats available as a proxy, bringing some significant benefits.

Finally, it is worth noticing how other alternatives for the estimation of emissions are also available, such as airline emission calculators. These might, however, present some limitations on their integration with other models and on the analysis of their underlying assumptions. In any case, a recent study shows the importance of uncertainty estimation among existing models, since they use different degrees of approximation and assumption [27].

The consideration of the number of seats and the type of flight, based on its distance, is essential to comprehensively evaluate the environmental impact of different modes of transportation. This can be used for the estimation of the emissions associated with providing an amount of supply between a given origin and destination [28]. This enables the evaluation of policies and multimodal networks [29].

With all these considerations, this article provides a set of analytical models to estimate emissions (${CO}_2$ and non-${CO}_2$) for the total trip (gate-to-gate) using as input only the flight distance(great circle distance) and number of seats. This allows for the estimation of fuel consumption even when the aircraft type is unknown, for example, in applications of the substitution of air travel by rail. The models also support sensitivity studies concerning the number of seats and can be used for analysing different transportation modes, considering carbon emissions costs, as studied in [30]. Other applications include the strategic consideration of new operations, such as the splitting of flights with an intermediate stop for refuelling to increase fuel efficiency by shortening the stage length of a mission, as shorter stage lengths allow for a reduction in the amount of fuel and therefore the weight of the aircraft and the kerosene burnt [31]. The small number of inputs supports the analysis of these aspects even when limited information is available. To consider uncertainties associated with the operations, corrections are applied, e.g., considering flown distances longer than the great circle distance, or the actual aircraft performance considered to fit the model. The selection of modern aircraft models, as reported by their historic usage, means that the emissions models reported in this article can be considered valid for current and future operations.

This article does not aim to create a new set of models to estimate fuel consumption based on detailed trajectories or to validate the model with reported fuel usage. Instead, it relies on the community-accepted estimations from IMPACT. It develops a metamodel that generalises the results from IMPACT to require only two variables: the great circle distance between origin and destination and the number of seats in the aircraft. The fitted models are therefore validated against the goodness of fit with respect to IMPACT. These models are parsimonious in that they rely on few variables [32] and effective because they are analytically tractable, physically realistic, and conceptually insightful [33]. As such, they are particularly well suited for strategic analysis in air transportation, providing system-level insights that might be obscured by details while requiring only limited input information.

A first model estimates the total fuel consumption from which greenhouse gases (GHG) proportional to this fuel can be computed (namely carbon dioxide (${CO}_2$), sulphur oxide (SO_x), and water vapour). Then, additional models are generated fitting the results from IMPACT on other non-${CO}_2$ emissions that are not proportional to fuel (nitrogen oxide (NO_x), carbon monoxide (CO), and non-volatile particulate matter (nvPM)).

Note that the models provide an estimate of the ${CO}_2$ and non-${CO}_2$ emissions but do not provide an estimate of the impact of aviation on climate (for example, as equivalent radiating force), as that is beyond the scope of these models. For that, the consideration of dynamic weather conditions should be included [11].

In this work, we thus propose indicators that can easily be used to evaluate the totality of flight emissions. Specifically, we develop the theoretical framework for the models and showcase their applicability. The article is structured as follows. First, the data and methodology used to estimate the fuel and emissions from IMPACT are described in Section 2. This section includes all assumptions and correction factors used to generate full trip fuel and emission estimations from only the great circle distance. Then, the fitting of the analytical models for fuel (and emissions proportional to fuel) and for other non-fuel proportional emissions is described in Section 3. This section also includes a comparison of the results obtained with the goodness of fit with respect to IMPACT and the comparison with other models. Finally, the article presents a case study demonstrating the capabilities of the produced models to analyse all scheduled commercial flights departing or arriving in Spain on a given day (in Section 4) and closes with conclusions in Section 5.

2. Data and Methodology

This section describes the data and methodology used to build the analytical models of fuel consumption and non-${CO}_2$ emissions.

2.1. Aircraft Models

The focus of our previous study was on short- to medium-haul flights [14]; therefore, commercial aircraft operated in those distances were used (medium-sized categories). In this article, we develop comprehensive models for all flight ranges and available aircraft seats; therefore, a wider range of aircraft models is considered.

A preliminary study showed that when trying to include many aircraft of different sizes, the high dispersion of the results makes it impossible to develop a single model that fits all distances. Here, we thus decided to use a general model divided into two sub-models, depending on the available seats (as).

First, we identify the aircraft most used according to their Available Seat Kilometres (ASK) in 2022 [34], which includes 10 models: Airbus A320, Boeing 737-800 Passenger, Boeing 737-800 (winglets) Passenger, Boeing 777-300ER Passenger, Airbus A321, Boeing 787-9, Airbus A350-900, Airbus A330-300, Airbus A320neo, and Boeing 787-8. These account for $2.4793\times {10}^{12}$ ASK, 33% of the total ASK of 2022 [35]. They all belong to category B (upper heavy) and category D (upper medium) of the RECAT-EU wake turbulence categories [36]. They cover an available range of seats from 172 to 365, which defines the seat range of the first sub-model.

Considering newer and current engine versions of the same aircraft (e.g., A320ceo and A320neo) leads to a high dispersion of results. Therefore, to obtain analytical models that can be used for the next few years, we opt to base our models only on the newest generation of aircraft. This means that the models slightly underestimate the current emissions, as the new engines can reduce emissions by around 15% [37]. With time, this underestimation will decrease as this new generation of aircraft progressively replaces the old one; for instance, the Airbus neo family made up 31% of the total Airbus narrow-body flights in 2024 [38]. This could be considered a fair assumption when analysing the impact of operating new routes. With all these considerations, eight aircraft are selected, since in the ten most used ones, both Airbus A320 and A320neo and two versions of the Boeing 737-800 (substituted by the max version) are present.

Then, to cover the range of low available seats and obtain the second sub-model, we look at data from the Official Airline Guide (OAG) in 2019. The most used aircraft are Bombardier CRJ200 and CRJ1000, Airbus 319, Embraer 190, and ATR 72. The latter is not considered, being the only turboprop aircraft. Furthermore, to ensure the continuity between both sub-models, we use, again, Boeing 737-800 max. All aircraft belong to categories D and E (lower medium) and have a seat range from 50 to 172.

Table 1. List of aircraft types considered per category.

RECAT-EU Category	Model D, E: $50\le \mathit{as}\le 172$	Model B, D: $172\le \mathit{as}\le 350$
CAT B	-	A359, A339, B77W, B788, B789,
CAT D	B38M, A319	A21N, A20N, B38M
CAT E	CRJ2, CRJ9, E190	-

summarises the list of aircraft types selected per category for each sub-model.

2.2. Fuel Consumption Calculation

First, the average available seats for each aircraft type in Table 1 were extracted from [39,40]. The aircraft models considered ranged from 50 to 365 available seats.

Then historical data following the methodology presented in [14] were used to estimate the flight level used, as a function of the distance, and the fuel consumption and emissions. For each aircraft type, their usage domain (minimum and maximum distance) was sampled every 50 km; and for each distance, the mean maximum operated FLs were computed from historical data. Finally, for each possible sampled distance and FL, fuel and emissions were calculated using EUROCONTROL’s web-based modelling platform IMPACT [24]. Built upon a reference data warehouse, IMPACT incorporates a common input data processor that generates detailed 4D trajectories for user-defined aircraft operations, together with corresponding engine thrust and fuel flow characteristics [24]. At altitudes below 3000 feet, fuel consumption is modelled according to the landing and take-off (LTO) cycle specified in the ICAO Engine Certification standards, whereas for operations above 3000 feet, corresponding to the climb, cruise, and descent (CCD) and phases, fuel burn is estimated using EUROCONTROL’s Base of Aircraft Data (BADA) [25].

Nominal speed and vertical profile as provided by IMPACT using EUROCONTROL’s BADA performance model were used. Finally, the calculation of the trajectory requires an assumption on the aircraft weight, based on the notion of weight-to-trip length relationships. The distance from the departure airport to the arrival airport determines a stage/trip length value. For example, a trip range of less than 500 nm is assigned a Stage Length 1, between 500 and 1000 nm a Stage Length 2, etc. [41]. IMPACT also uses the stage length to select an associated aircraft default weight from the aircraft noise and performance (ANP) database. As a consequence, and to account for the additional fuel to carry on, the aircraft’s default weight increases, in steps, with the flight distance.

2.3. Emission Calculation

Water vapour and ${CO}_2$ emissions arise directly from the oxidation of the carbon and hydrogen present in the fuel with atmospheric oxygen. As for the SO_x emissions, they are determined by the sulphur content of the fuel. Consequently, all three pollutants are directly proportional to the quantity of fuel consumed and can be estimated using non–engine-specific emission indices. Namely, 0.84 g of SO_x and 1237 g of water vapour are emitted for each kilogram of jet fuel [25]. For the ${CO}_2$ emissions, the proportional factor of 3160 g/kg is the one usually used [42], and it corresponds to the tank-to-wheel emissions. This leaves aside the well-to-tank emissions, which include crude oil production and refining emissions [43], and can also be considered, leading to a 3780 g/kg factor for well-to-wheel ${CO}_2$ emissions [44].

The ICAO aircraft engine emissions databank (AEED) provides emission indices and fuel flow parameters for turbofan and turbojet engines. Within IMPACT, the Advanced Emission Model (AEM) uses this databank to match each aircraft in the flight input sample with an appropriate engine. Although the underlying emission calculations are based on the AEED, both emission factors and fuel flow are adjusted to altitude-specific atmospheric conditions using the Boeing Fuel Flow Method 2 (BFFM2), a methodology originally developed by Boeing. This approach enables the estimation of NO_x and CO emissions along each trajectory segment, accounting for atmospheric conditions, aircraft speed, and fuel flow as computed by IMPACT [41,45]. BFFM2 leverages the correlation between fuel flow and the combustor inlet temperature (T3). At sea level, higher fuel flow leads to higher T3. By replicating this relationship at altitude, the model estimates the fuel flow needed to maintain the same T3 as at sea level. This is achieved through an energy balance across the combustor, allowing altitude emissions to be derived from sea level data [46].

Non-volatile particulate matter emissions (nvPM), resulting from the incomplete combustion of fuel, are estimated using the First-Order Approximation Version 3 (FOA3). This method is adapted to full-flight conditions using BFFM2 interpolation and altitude adjustments. For each trajectory segment, the AEM applies particulate matter emission indices in combination with segment-specific aircraft speed, IMPACT-derived fuel flow, and atmospheric conditions [41,47]. It is important to note that these estimates carry uncertainty, particularly when converting measured values to emissions in the engine exit plane. This is primarily due to challenges in quantifying system particle loss corrections, which remain difficult to assess accurately.

2.4. Correction Factors

IMPACT computes fuel consumption from take-off to landing, considering GCD between airports. Consequently, for each performance computed, correcting factors are implemented here to include taxi fuel consumption and flight distance inefficiency.

First, a distance correction factor is added to account for different air traffic procedures, such as air traffic congestion or routing deviations. To obtain the corrected flight path distance $d_{corrected}$, we use the same correction as in [26,48], which adds a fixed distance to the GCD and a percentage of inefficiency:

$$d_{corrected}=1.0387d_{GCD}+40.5\left(\mathrm{km}\right)$$

(1)

where $d_{GCD}$ is the great circle distance between the origin and destination airports. Note that the additional distance flight with respect to the GCD is higher than the horizontal en-route flight inefficiency of 3.16% reported in the Performance Review Report of 2024 [49], which does not include vertical inefficiency. The constant added term accounts for extra distance flown during the approximation phases.

Although taxiing contributes to the total fuel usage, IMPACT does not account for fuel consumption during this phase, since engines are in idle mode while taxiing before take-off and after landing. Taxi-in and taxi-out times vary by route, influenced by factors such as gate location, airport layout, and configuration, which differ at each airport. To include this fuel consumption, we use the average taxi duration (18.3 min) for flights operating to and from European airports from EUROCONTROL in 2022 [50]. Considering then that taxi fuel consumption can be computed based on the fuel flow in idle conditions, we identify the idle fuel flow for each aircraft based on its engine model [51]. Note that all chosen aircraft are twin-engine, and some airlines use single-engine taxi operations, which could potentially reduce this taxi fuel consumption.

Table 2. Idle condition fuel flow and taxi fuel consumption for the considered aircraft.

Aircraft Type	Engine Identification	Idle Fuel Flow (kg/s)	Taxi Fuel (kg)
A359	Trent XWB-79	0.560	614.9
A339	Trent7000-72	0.518	568.8
B77W	GE90-115B	0.682	748.8
B788	GEnx-1B64	0.386	423.8
B789	Trent 1000-G3	0.500	549.0
A21N	LEAP-1A35A/33/33B2/32/30	0.196	215.2
A20N	PW1122G-JM	0.160	175.7
B38M	LEAP-1B27	0.190	208.6
A319	CFM56-5B5/P	0.184	202.0
E190	11GE143	0.166	182.3
CRJ9	CF34-8C5	0.126	138.3
CRJ2	CF34-3B	0.096	105.4

presents the idle fuel flow and taxi fuel for the aircraft considered. Taxi time represents around 15% of the total block-time for intra-European flights [50], and its fuel usage is approximately 5 to 6% of the total fuel consumption for a 1000 km flight with a Cat D aircraft.

Similarly, we obtain the CO, NO_x, and nvPM emission indices at idle condition from [51] and display them in

Table 3. CO, NO_x, and nvPM emission index at idle condition for the considered aircraft.

Aircraft Type	CO (g/kg)	NOx (g/kg)	nvPM (mg/kg)
A359	22.51	4.55	25.1
A339	6.45	6.1	18.8
B77W	39.11	5.19	5.3
B788	21.62	4.24	2.3
B789	6.54	6.04	19
A21N	18.69	4.85	0.6
A20N	29.78	4.72	10.6
B38M	15.29	4.74	0.2
A319	30.00	3.8	1.8
E190	46.1	3.55	13.4
CRJ9	18.25	4.6	1.5
CRJ2	47.59	3.72	1.8

. Note that the emission indices are highly dependent on the engine model, especially in the case of nvPM, which shows extremely high variations between aircraft (and therefore between engine) models.

3. Fuel and Emission Analytical Models

In this section, we develop models of gate-to-gate fuel consumption per Available Seat Kilometres (ASK) which enable the direct computation of emissions proportional to fuel consumption, namely ${CO}_2$, SO_x and water vapour. Non-${CO}_2$ emissions, NO_x, and CO are then computed and fitted into analytical models. Finally, non-volatile particulate matter (nvPM) is also computed, but it is shown that no analytical model can adjust to their values.

3.1. Fuel and Proportional to Fuel Emission Models

3.1.1. Fuel Model Fitting

Figure 1 presents the ratio of fuel consumed per passenger per 100 km as a function of flight distance for all aircraft considered. For this figure only, emissions are expressed on a per-PAX basis rather than per ASK; thus a load factor of $83.5$% (value of 2024 [52]) and a PAX-to-freight factor of $96.1$% (following the ICAO Carbon Emissions Calculator Methodology [53]) were applied.

It can be seen that fuel consumption per PAX per 100 km first strongly decreases with distance. As expected, short flights present lower performances; for flights under 1000 km, the fuel per passenger can go as low as around 3 L per 100 km for the most efficient aircraft at this distance, but also up to around 7 L per 100 km for the largest aircraft, and for some of the smallest with a worse value of fuel consumption per passenger since they do not carry many passengers.

Vertical discontinuities are observed for the fuel consumption corresponding to the change in the flight stage length defined in IMPACT and to the fact that the take-off weight increases discretely due to the extra fuel needed for longer distances. For example, flights between 2660 and 4400 km are assigned a stage length of 3 [41], and the same aircraft weight is considered for this range of distances. This leads to a decreasing fuel per passenger per 100 km ratio. Then, from 4448 to 6188 km, the stage length is 4, leading to an increase on fuel consumption between 4400 and 4448 km. Overall, the fuel per passenger per 100 km ratio has a decreasing tendency until around 4000 km and then stabilises or even increases for the heaviest aircraft model due to the increase in take-off weight.

EUROCONTROL reported in 2023 an average of $3.4$ L of fuel per passenger per 100 km, a value that has shown steady improvement over the last 15 years and is expected to decrease down to $2.1$ L/100 km by 2050 [54]. Nevertheless, this average value hides important differences between very short flights, for which the fuel burned during the taxiing, take-off, and climb cycles is the major component of fuel consumption, and more efficient medium-haul flights. Differences are also observed between medium and heavy category aircraft, the latter having to carry more fuel for their extended flight distance, and thus showing a larger fuel consumption.

Figure 1 also shows that a single analytical model cannot be easily found if fuel consumption is solely related to flight distance. In previous studies, fuel consumption as a function of the GCD was fitted by a polynomial function of order two for each aircraft model [26]. In this article, the dependency of the analytical model on the aircraft type is removed by substituting it with its available seat number, creating a more generic model.

Figure 2 shows the ratio of fuel in grams per available seat and per kilometre (Fuel (g)/ASK) as a function of available seats and stage distance. The discrete values obtained using IMPACT are displayed, as well as the corresponding analytical fitting, for two ranges of seats available.

For the smallest aircraft (category E), increasing the number of seats leads to an improvement in the fuel consumption intensity: the extra fuel due to the higher weight of the aircraft is compensated by the additional available seats, improving the overall efficiency. For category B aircraft, the opposite effect is observed; the fuel intensity increases with the number of available seats, as the increase in seats requires a significantly higher weight. Other factors, such as larger aircraft potentially carrying larger cargo payload, might also affect these fuel efficiencies. These patterns reflect a structural characteristic of the market: narrow-bodies on short-haul routes can densify seating, whereas wide-bodies on the long haul cannot and require stronger fuselage sections, raising weight per seat. The result is consistent with Wei and Hansen [55]. This leads to an overall optimal fuel intensity usually obtained for category D aircraft, and this also shows that it is complicated to fit a global analytical model for all aircraft models, justifying having an analytical model divided into two sub-models depending on the available seat range.

Various polynomial fittings, including high-order models, were tested, but increasing the polynomial order did not improve the goodness-of-fit. Thus, the relationship between fuel in grams per ASK and flight distance ($d_{GCD}$) and available seats ($as$) could be expressed as:

$$\begin{array}{c}\mathrm{for}50\le as\le 172,\mathrm{and}\mathrm{with}100\le d_{GCD}\le 5000\left(\mathrm{km}\right),\\ \mathrm{Fuel}\left(\mathrm{g}\right)/\mathrm{ASK}=34.67+{\displaystyle \frac{6608}{d_{GCD}}}-1.196\times {10}^{-3}{d}_{GCD}-0.1354as+1.338\times {10}^{-5}{d}_{GCD}\times as,\end{array}$$

(2)

$$\begin{array}{c}\mathrm{for}172\le as\le 365,\mathrm{and}\mathrm{with}200\le d_{GCD}\le 12,000\left(\mathrm{km}\right),\\ \mathrm{Fuel}\left(\mathrm{g}\right)/\mathrm{ASK}=0.7361+{\displaystyle \frac{6651}{d_{GCD}}}+5.989\times {10}^{-4}{d}_{GCD}+6.152\times {10}^{-2}as-1.014\times {10}^{-6}{d}_{GCD}\times as.\end{array}$$

(3)

The analytical expression obtained is consistent with previous models [26,48,56], which relate emissions per passenger versus distance through a second-order polynomial in the distance variable; here, it corresponds to a first-order term since we work with fuel intensity. In the present work, the number of seats is additionally explicitly included in the analytical model as a term of the same order as the distance, decreasing for a low number of seats and increasing for larger ones. Note that the fuel consumption per ASK is inversely proportional to the flight distance, as a result of the inefficiency of short flights.

3.1.2. Accuracy of the Fuel Model

We next assess the accuracy of the analytical fitting functions given in Equations (2) and (3).

Table 4. Goodness-of-fit statistics for fuel models: Sum of square due to error (SSE), R-square, and Root mean square error (RMSE).

Model	SSE	R-Square	RMSE
Fuel per ASK, model D, E	$3.156\times {10}^3$ g	$0.951$	$2.778$ g
Fuel per ASK, model B, D	$1.862\times {10}^3$ g	$0.934$	$1.113$ g

presents the model’s goodness-of-fit statistics. The RMSE ranges from approximately 1.1 g to 2.8 g, while typical values of fuel (g) per ASK are between 15 g and 60 g, which is a reasonable level of error. Note that the accuracy is as high as that found in our previous model that focused only on short-to-medium-haul flights [14]. This is due to the fact that, as previously commented, we select aircraft types of the newest generation to avoid too much dispersion in the fuel intensity values and that we use two sub-models.

Table 5. Fuel (g)/ASK for the “B, D” and “D, E” models, the model of [57], and the average obtained for all aircraft models simulated by IMPACT for several distances and as.

Distance (km)	Seats	Model B, D	Model D, E	Model [57]	Avg IMPACT
490	-	-	-	38.53	31.47 (28.86 CAT DE)
490	172	25.10	25.41	-	-
1020	-	-	-	28.72	24.36 (22.39 CAT DE)
1020	172	18.27	18.99	-	-
1984	-	-	-	21.99	20.95 (18.91 CAT DE)
1984	172	15.51	16.91	-	-
4979	-	-	-	15.21	19.97 (22.41 CAT B)
4979	200	16.35	-	-	-
4979	290	21.43	-	-	-
8022	290	-	-	12.56	23.03 (CAT B only)
8022	290	21.85	-	-	-

compares the fuel intensity for the models “B, D”, “D, E”, that of [57], which uses an analytical model where the fuel intensity decreases exponentially with distance, and with the average obtained for all aircraft models simulated by IMPACT for a given distance. For the comparison with the IMPACT values, the average value is computed for all simulated models of aircraft (when the range allows), and also for the most suitable category for the given distance.

When comparing our current model with that of [57], it can be seen that they agree for a distance close to 5000 km. For shorter distances, our model reaches lower fuel intensity, and for larger distances, higher ones, which is consistent with their observation that they overestimate short-distance air trips and underestimate the long-distance ones. Also, recall that our model accounts for the extra fuel that needs to be carried on for longer flights, leading to an increase in fuel intensity observed from around 4000 km (see Figure 1).

When comparing the analytical model with the discrete values obtained with IMPACT, it can be seen that the model agrees well with the data for medium-to-long-haul flights but shows more discrepancy for short distances. The global average values obtained for a given distance of the IMPACT simulations can be a bit misleading, since, for instance, CAT B aircraft would not be adequate for short-haul flights, and averaging with all models thus overestimates the fuel intensity. This is why the average using only the most suitable aircraft category is also indicated. Note that only CAT B aircraft can reach the longest distance considered. Finally, the continuity between both sub-models is well obtained, since for the range of distances where both can be used, they agree very well.

We also compare our current model with the FEAT model from [26]. To that end, we sample the distance every 100 km for all the aircraft models of the FEAT model, even for those that fall outside the range of application of our model (aircraft type with a number of seats at a distance not supported by our model). We compute the fuel per ASK with both models and show their values in Figure 3.

Our model generally leads to lower fuel intensity, but the agreement is quite good except for a few exceptions. For one model of aircraft, our model sub-estimates the fuel, and for an aircraft with fewer than 172 seats, when it falls outside the sub-model distance range, a jump is observed in the fuel intensity, suggesting that the sub-model for small aircraft should not be used above its distance limit. On the contrary, the sub-model for larger aircraft behaves quite correctly above 12,000 km, proving its robustness. For all flights sampled, the average difference of our model with FEAT is of −$2.3$ g/ASK and the relative one −9%, while the relative standard deviation of the difference is 17%. Note that this difference was expected since, as previously commented, we use many new-generation aircraft, leading to a decrease in fuel intensity.

Thus, these validations show that, despite our model being more generic and, as a consequence, more flexible than others, such as FEAT, it is competitive from an accuracy point of view and shows good goodness of fit with respect to the values obtained from IMPACT.

3.1.3. ${CO}_2$, SO_x, and Water Vapour Models

As commented on in Section 2.3, the fuel intensity model proposed here enables the estimation of ${CO}_2$, SO_x, and water vapour, as these are directly proportional to the fuel intensity:

• 3160 g of ${CO}_2$ per kg of jet fuel [42],
• 0.84 g of SO_x per kg of jet fuel [25], and
• 1237 g of water vapour per kg of jet fuel [25].

3.2. Non-${CO}_2$ Models

Since IMPACT also allows for computation of other engine-dependent emissions, we also studied NO_x, CO, and nvPM emission intensities as a function of distance and seats, with the objective of seeing if these can also be analytically modelled.

3.2.1. NO_x Emissions Model

Starting with NO_x emissions, Figure 4 shows the NO_x (g) per ASK against flight distance and available seats values and the corresponding fitting surface. It can be seen that, as previously observed for fuel intensity, NO_x intensity is at first strongly decreasing with distance, and it also increases with available seats, especially for category B aircraft, not so much for smaller aircraft.

Although the behaviour of NO_x is slightly different for the two cases of available seat ranges, we chose to use the same terms for the analytical fittings presented in the following equations:

$$\begin{array}{c}\mathrm{for}50\le as\le 172,\mathrm{and}\mathrm{with}100\le d_{GCD}\le 5000\left(\mathrm{km}\right),\\ {\mathrm{NO}}_{\mathrm{x}}\left(\mathrm{g}\right)/\mathrm{ASK}=0.1512+{\displaystyle \frac{63.34}{d_{GCD}}}+{\displaystyle \frac{0.2954}{as}}-2.214\times {10}^{-6}{d}_{GCD}+6.217\times {10}^{-4}as,\end{array}$$

(4)

$$\begin{array}{c}\mathrm{for}172\le as\le 365,\mathrm{and}\mathrm{with}200\le d_{GCD}\le 12,000\left(\mathrm{km}\right),\\ {\mathrm{NO}}_{\mathrm{x}}\left(\mathrm{g}\right)/\mathrm{ASK}=-1.427+{\displaystyle \frac{152.1}{d_{GCD}}}+{\displaystyle \frac{143.5}{as}}+3.625\times {10}^{-6}{d}_{GCD}+4.18\times {10}^{-3}as.\end{array}$$

(5)

The analytical model of NO_x emission intensity includes a term inversely proportional to the available seats, which was not present in the fuel intensity model. This can be explained by the fact that there is not much difference in NO_x emissions between aircraft types, as can be seen, for instance, in the taxi phase, see Table 3. This penalises more the smaller aircraft when considering emission intensity per ASK. Also, note that the coefficient of the term proportional to the available seats is smaller in the smallest category case, as previously commented on. Finally, the term multiplying available seats by the distance present in the fuel model is not necessary here; it does not improve the accuracy of the model.

Table 6. Goodness-of-fit statistics for NO_x emission model: Sum of square due to error (SSE), R-square, and Root mean square error (RMSE).

Model	SSE	R-Square	RMSE
NOx emissions per ASK, model D, E	$0.74$ g	$0.854$	$0.043$ g
NOx emissions per ASK, model B, D	$6.68$ g	$0.792$	$0.067$ g

shows the goodness-of-fit statistics for the NO_x models. The R-square error is, in general, larger than that of the fuel models. This might be partially because, contrary to fuel consumption that is directly related to aircraft size and engine, the emission index of NO_x is related to the engine, as previously commented on; see Table 3.

3.2.2. CO Emissions Model

Figure 5 shows the CO (g) per ASK againts flight distance (km) and available seat values and the fitting surface.

Again, several analytical models were tested to fit the discrete data, and the most accurate one is described in the following equations:

$$\begin{array}{c}\mathrm{for}50\le as\le 172,\mathrm{and}\mathrm{with}100\le d_{GCD}\le 5000\left(\mathrm{km}\right),\\ \mathrm{CO}\left(\mathrm{g}\right)/\mathrm{ASK}=0.08338+{\displaystyle \frac{96.54}{d_{GCD}}}+{\displaystyle \frac{2.184}{as}}+2.433\times {10}^{-6}{d}_{GCD}-8.602\times {10}^{-4}as\\ +6.053\times {10}^{-8}{d}_{GCD}\times as,\end{array}$$

(6)

$$\begin{array}{c}\mathrm{for}172\le as\le 365,\mathrm{and}\mathrm{with}200\le d_{GCD}\le 12,000\left(\mathrm{km}\right),\\ \mathrm{CO}\left(\mathrm{g}\right)/\mathrm{ASK}=-0.5736+{\displaystyle \frac{65.11}{d_{GCD}}}+{\displaystyle \frac{51.85}{as}}+2.489\times {10}^{-5}{d}_{GCD}+1.411\times {10}^{-3}as\\ -8.39\times {10}^{-8}{d}_{GCD}\times as.\end{array}$$

(7)

Note that the CO emission intensity model has the same terms as the NO_x model, plus the term multiplying seats and distance.

Table 7. Goodness-of-fit statistics for CO emission model: Sum of square due to error (SSE), R-square, and Root mean square error (RMSE).

Model	SSE	R-Square	RMSE
CO emissions per ASK, model D, E	$1.214$ g	$0.902$	$0.055$ g
CO emissions per ASK, model B, D	$0.488$ g	$0.778$	$0.018$ g

shows the goodness-of-fit statistics for the CO models. For the category D, E model, the R-Square value is quite good, while it is lower for model B, D. Nevertheless, Figure 5 shows that the error of the analytical fitting is higher for long-haul distances with a few seats, which is a configuration that does not exist. So when applying the model to realistic combinations of distance and seats, the error is lower. Also, note that, as seen in Table 3, the dispersion in the CO emission index is very high and does not directly depend on the aircraft size, contrary to fuel consumption.

3.2.3. nvPM Emissions Model

Finally, non-volatile particulate matter values were computed for all considered aircraft models, and they are represented in Figure 6. Here again, and as already seen to a lesser extent for the NO_x and the CO emissions, the dispersion of the results is very large due to the differences in aircraft engines. It is thus not possible to model nvPM with analytical functions. However, it can be observed that in general, the nvPM intensity decreases with distance, although in some cases it is quite constant. No clear tendency can be found for the number of available seats. This dispersion was also observed for the nvPM emission index in idle condition shown in Table 3.

4. Case Study

All analytical models described in Section 3 can be found for direct use in a GitHub repository [58]. They are coded both in Python 3.10 and in MATLAB R2024b, with the inclusion of the models’ ranges of application.

To show the applicability of these models, we analyse a case study considering all scheduled commercial flights departing from or arriving in Spain on 6 September 2019, based on data from the Official Airline Guide (OAG). The objective is to estimate total ${CO}_2$, NO_x, and CO emissions. This approach allows for the evaluation of emissions in relation to commercial variables that influence airlines’ strategic decisions on aircraft acquisition and network design, particularly regarding aircraft size, flight frequency, and route distance.

On the selected date, 5038 flights were operated from or to Spain between 278 airports, consuming 34,160 tonnes of fuel and emitting, considering only tank-to-wheel emissions, a total of 107,945 tonnes of ${CO}_2$, 528 tonnes of NO_x, and 68 tonnes of CO.

Figure 7 presents the Cumulative Distribution Function (CDF) of ${CO}_2$, NO_x, and CO, and compares then with the CDF of flights and ASK. The analysis shows that 82% of flights to or from Spain occur on routes shorter than 2000 km. These account for 45% of total ASK and 43% of ${CO}_2$ emissions. When the threshold is extended to 4000 km, the cumulative shares rise to 95% of flights, 66% of ASK, and 60% of ${CO}_2$ emissions. These figures indicate that only 5% of flights realised long-haul routes that day (i.e., longer than 4000 km), yet they generated 34% of total ASK and 40% of ${CO}_2$ emissions. This value is slightly lower than what was observed in [59], considering passenger flights from 31 European countries, but still implies a relatively concentrated share of emissions from a small fraction of flights, despite their limited frequency.

For other pollutants, the distribution patterns are somewhat different. NO_x emissions closely mirror ${CO}_2$ emissions for flights under 2000 km (40% vs. 43%, respectively). However, they diverge at longer distances. Between 2000 and 4000 km, NO_x accounts for 14% of emissions (compared to 17% for ${CO}_2$), while flights over 4000 km contribute a larger share of NO_x (46%) than ${CO}_2$ (40%). This reflects the fact that cruise phases, longer in long-haul operations, are key contributors to NO_x formation. CO emissions are even more concentrated in short-haul segments: $53.4$% occur on flights under 2000 km and $65.4$% under 4000 km. This underlines the strong impact from take-off and climb phases on CO formation and the inefficiency of short sectors in terms of local air quality.

Although flights between 2000 and 4000 km exhibit relatively higher emission efficiency, short-haul flights remain a major contributor to overall emissions due to their high volume. In this range, operational efficiency has led to a more competitive service and an increased demand, as Jevons’ paradox states. In the range above 4000 km, we can observe that a small share of long-haul operations contributes disproportionately to total emissions, which is a structural feature of the air transport system with implications for policy and network design. It is definite that the cumulative evolution of emissions aligns more closely with ASK than with the number of flights, across all pollutants and fuel consumption.

The same case study is represented by an origin–destination (OD) matrix of flight frequencies, associated distances, and aircraft sizes. Fuel consumption is estimated as a function of aircraft size and distance, and shows the pattern in which fuel per ASK decreases up to approximately 100–120 seats and increases thereafter, as already seen in Section 3. Ignoring airline identity, the analysis now focuses exclusively on the aircraft mix per route to evaluate the potential for emission reductions through fleet reshaping.

Using a continuous approximation model to optimise seat allocations across routes, we propose a new aircraft assignment under ideal regulatory conditions that allows for cross-carrier coordination and the hypothetical adoption of optimal aircraft sizes. Although this scenario is not operationally realistic, as it is implemented across all airlines, it provides a useful upper bound on the potential emission reductions achievable through aircraft resizing. This policy-oriented analysis is facilitated by the use of the continuous models proposed in this article, which avoid the time-consuming task of matching specific aircraft types, often constrained by limited or proprietary data, while allowing for capturing systemic trends and boundary conditions.

For each OD pair, a more efficient aircraft assignment is identified following these rules:

1.

For aircraft with fewer than 100 seats, we explore up-gauging opportunities, replacing multiple small aircraft with a smaller number of larger ones to consolidate capacity and reduce fuel consumption. The algorithm sums the total number of seats from the subset of aircraft with fewer than 100 seats (considering only subsets with more than one aircraft) and searches for a minimum fleet of 100-seat (or smaller) aircraft such that the resulting average size is less than or equal to 100 seats.

2.

For aircraft with more than 350 seats, typically long-haul wide-bodies or jumbos, we explore down-gauging, splitting high-capacity flights into multiple smaller ones, in line with industry trends favouring fuel-efficient mid-size jets. In this case, the algorithm scans a subset of aircraft with more than 350 seats and explores how to split them into the minimum number of flights with aircraft seating of at most 350 passengers, such that the resulting average seat capacity remains below 350.

Figure 8 shows the distribution of flights on the day of study as a function of their seats and distance. Note that here each dot can represent one or several flights; the flight density is thus further studied with the lateral histograms, showing the number of flights as a function of their distance and of their seats. Figure 8a shows the original scenario, while Figure 8b reflects the new scenario after applying the fleet reshaping.

To properly interpret Figure 8, it is essential to consider the structure of Spanish air mobility, which can be understood by examining the distribution of flights across domestic and international routes. Spain’s national airport network is categorised into four groups: (i) airports on the Spanish Peninsula, (ii) airports serving the Balearic and Canary Islands, (iii) European airports, and (iv) intercontinental airports. The extensive Spanish coastline and the significance of tourism, accounting for $12.3$% of national GDP in 2023 [60], underscore the strategic relevance of air services to and from island regions. It is also relevant to note that most of Spanish flights (67%) are intra-European. These routes are primarily served by aircraft with seating capacities between 100 and 250 seats (average: 180 seats) and with typical stage lengths exceeding 1000 km (average: 1528 km). A small number of these flights are operated with wide-body aircraft, likely reflecting airline-level resource optimisation and fleet utilisation strategies.

On the reference day, $13.7$% of flights were operated by aircraft with fewer than 100 seats. These were predominantly deployed on intra-island routes in the Balearic and Canary Islands (51%), followed by short-haul flights within the Peninsula (18%). This operating pattern results in a distinct cluster of flights below 500 km and 100 seats in the seat-distance scatter plot, clearly observable in Figure 8a. After applying the reshaping strategy, this cluster is substantially reduced in Figure 8b, as also reflected in the marginal histograms for aircraft under 100 seats.

Intercontinental services accounted for only 9% of total flights, and aircraft with more than 350 seats represented just 1% of operations. These large-capacity flights form a sparse upper band in the scatter plot of Figure 8a, which disappears after fleet reassignment. Most wide-body aircraft with 250–350 seats operate on routes exceeding 4000 km, contributing to the long right tail observed in both the seat and distance distributions. Despite representing a small fraction of flights, this segment accounts for approximately 11% of total fuel consumption.

Following the proposed aircraft reassignment, the total number of flights is reduced from 5038 to 4986 (−$1.03\%$), using aircraft with larger seat capacities, as observed on the seat histogram of Figure 8b, which is shifted up with respect to Figure 8a. This leads to a decrease in the total fuel consumption from 34,160 tonnes to 33,008 tonnes ($-3.37\%$), with equivalent reductions in ${CO}_2$ emissions. NO_x emissions fall by $-10.06\%$, and CO by $-5.76\%$, reflecting the increased benefit of optimised aircraft sizing for these pollutants.

5. Conclusions

We develop analytical models to estimate fuel consumption and emissions in the aviation sector. These analytical models are a fit to EUROCONTROL’s IMPACT model with a set of operational assumptions. This set of parsimonious models, which rely on very few variables (distance and seats), provide a simplified but accurate estimation of fuel, ${CO}_2$ and key non-${CO}_2$ emissions. This approach is particularly effective for strategic assessments involving large datasets, as it enables researchers and policymakers to understand the fundamental structure of emissions without requiring detailed flight-level performance data. By focusing on key variables, namely aircraft size and flight distance, the model isolates the core mechanisms of fuel and pollutant output in a scalable and tractable manner. However, it necessarily simplifies the technological heterogeneity of the fleet. Moreover, our analysis focuses on fuel per seat-kilometre and does not capture the role of cargo, a limitation that should be kept in mind when interpreting the results.

While fuel consumption and ${CO}_2$ emissions remain central concerns due to their contribution to global warming, non-${CO}_2$ pollutants such as NO_x and CO are also highly relevant, particularly in relation to public health and local air quality. Applying the same modelling framework to non-${CO}_2$ pollutants allows comparative assessments across aircraft types and operational profiles and highlights the need for pollutant-specific strategies within broader decarbonization initiatives.

When validating against other aircraft-dependent models, the proposed models show a good level of accuracy, with a slight underestimation since the newest generation of aircraft has been considered to develop these analytical models. The accuracy will thus increase over the years, as the share of these aircraft increases, corresponding to the usual trend of decreasing fuel intensity (emissions per seat-km).

The models are suitable for current and future operations (as new models are assumed when several alternatives are possible) and consider corrections for operational aspects, e.g., horizontal and vertical inefficiencies with respect to great circle distance, which are particularly suited for the European context. The models, however, assume current fuel usage, as the relationship between fuel consumption and emissions is based on conventional jet fuel. If new engines and fuel mix are used (e.g., sustainable aviation fuels or hydrogen aircraft), the model would need to be adjusted to reflect the new relationship between fuel usage and emissions. Other factors, such as corrections due to average winds, could be included in future versions of the model.

We apply the models to a one-day snapshot of Spanish air traffic and identify several significant patterns. Although long-haul flights represent a small share of total operations, they account for a disproportionately large share of fuel consumption and especially NO_x emissions. Conversely, short-haul flights operated by small regional aircraft tend to produce higher CO emissions per unit of output. These distinct profiles underscore the importance of size-aware and pollutant-specific analysis.

To explore mitigation potential, we implement a reshaping strategy under idealised conditions, reallocating inefficient aircraft types. The strategy involves consolidating short-haul flights operated by small aircraft into fewer, larger flights (up-gauging), and replacing oversized aircraft on long-haul routes with smaller, more efficient models (down-gauging). This reassignment yields a $3.37\%$ reduction in total fuel consumption, primarily due to the removal of large wide-bodies, with only a $1.03\%$ reduction in total flights. NO_x and CO reductions are even more pronounced, demonstrating the disproportionate emissions impact of aircraft size.

Finally, while this study emphasises emission intensity, total emissions must also be considered. This can be easily achieved using the analytical models presented here, since they provide system-level insights from both perspectives. The results reveal a persistent structural challenge in aviation: traffic growth continues to outpace efficiency gains, leading to rising aggregate emissions. This highlights the necessity of systemic interventions in addition to technological improvements.