Mariem Ben Fadhl

Gil Utard

An Anonymous Data Control Access Protocol in Distributed Storage Systems

Keywords: Access rights, Anonymity, Cloud Storage System

In this paper, we describe a new access control mechanism for distributed data on cloud or peer-to-peer storage systems. The purpose of this protocol is to ensure that only the data owner or a user with rights are able to recover, modify or even delete it anonymously, and be able to delegate access to other users with revocation capability.

I. INTRODUCTION

Today, thanks to the introduction of the blockchain technology, a new business model is arising in the market for Cloud Storage Service. New actors, such as Sia [START_REF] Vorick | Sia: Simple decentralized storage[END_REF], Filecoin [START_REF] Lab | Filecoin: A decentralized storage network[END_REF] or Storj [START_REF] Wilkinson | Storj: A peerto-peer cloud storage network[END_REF] are proposing new open architectures to small data centre players, allowing the provision of their space storage to new clients for a specified time. In this new storage marketplace, when a client puts a file to store, he buys storage space with a specific smart-contract crypto-currency. Usually, the data is encrypted by the client before storing it. To ensure the integrity of the data stored, the smart contract is such that the storer agent is paid only after the retention period (after a check of the data integrity). To avoid data retention by the storer, another smart contract may be introduced to pay the storer when it sends back requested data. This incentive mechanism allows a high level of trust of the storers: another mechanism may be introduced to ban malicious storers, and some data redondancy is used to face it.

Moreover, anonymity must be provided by this system to ensure some privacy. Some anonymous routing protocols may be introduced (like onion routing) for this goal. The present paper focuses on how to restrict access to data to a client who is anonymous. Other concerns, such as smart-contract management, or anonymous routing, are outside the scope of this paper. The interested reader can consult previously cited papers.

These new architectures are based on the peer-to-peer (P2P) paradigm. Despite their increased use, the lack of control due to the absence of any centralized authority makes the security services difficult to implement, and has limited the adoption of these systems, especially by industry. Cryptography can solve the confidentiality problems, but does not restrict access to sensitive data. The use of keys to encrypt data protect the data and avoid disclosure of the data, but may induce user privacy issues. Moreover, the keys have to be managed in a distributed environment. Data access control and user privacy becomes a necessity.

The access to the shared resources in a storage system can include two main aspects: the ability to store data and the ability to access to data anonymously, to preserve privacy.

In the new market model, storing the data is performed through a transaction between the user and the storer. This transaction is supported by the underlying crypto-currency. The use of an anonymous crypto-currency, such as CoinJoin [START_REF] Maurer | Anonymous coinjoin transactions with arbitrary values[END_REF], ZCash [START_REF] Peck | A blockchain currency that beat s bitcoin on privacy [news[END_REF] or Monero [START_REF] Van Saberhagen | Cryptonote v 2.0[END_REF], allows preserving the user privacy. The stored data is then identified by the anonymous transaction ID, and not by an user ID.

In the present paper, we focus on access to data while preserving users' privacy. We present a new mechanism derived from a remote integrity protocol, allowing the control of anonymous data access rights.

The remainder of this paper is organized as follows. Section II discusses some related work. Then we present some previous work on integrity from which we extracted the cryptographic material of our solution. We propose then the new mechanism for realizing anonymous access control. The we show how to use it to check the integrity of any data modification by the owner. We then introduce an Access Control List allowing to delegate access to some designed users. Finally we present some avenues for future research.

II. RELATED WORKS

Traditional access control models, such as ACL or RBAC [START_REF] Ferraiolo | Role-Based Access Control (RBAC): Features and Motivations[END_REF], generally rely on central servers for authorization operations. Access control policies are defined on a single authority. This simplifies the control of access rights. This scheme is not suitable for the specificities of a distributed environment.

One trivial solution is to use an asymmetric public key protocol, where each file is stored with the public key of the data owner. On access demand, the storer create a challenge based on the associated public key, the owner answers thanks to the private key. The main drawback of this solution is that to insure anonymity, one pair of private/public keys must be generated for each stored file and managed to a dedicated wallet. Otherwise all files stored by a specific user may be tracked by a malicious peer. Moreover, in the case where the memory is a scarce resource, it may be problematic to store one key for each stored file: think, for instance, of an IoT infrastructure, where Internet objects store the generated data in the cloud.

Access policies for personal data can be stored in a blockchain [START_REF] Zyskind | Decentralizing Privacy: Using Blockchain to Protect Personal Data[END_REF]. That protocol describes the interactions between a user and a service (fore example, a mobile application). This service wants to collect data about the user. The blockchain is used as an intermediary between these two parts. It manages the access rights defined by the user, stores the encrypted data in a distributed hash table (DHT), and redistribute the data to authorized services. This allows the user to control data access. However using a DHT will slow data access times, which makes its implementation difficult. Moreover, the authors don't mention user anonymity but rather data privacy.

The decentralized access control scheme in clouds of [START_REF] Ruj | Decentralized access control with anonymous authentication of data stored in clouds[END_REF] addresses user revocation and ensures the authenticity and the privacy of the stored data while preserving user privacy. The authors use attribute-based signature scheme [START_REF] Maji | Attribute-Based Signatures: Achieving Attribute-Privacy and Collusion-Resistance[END_REF] to ensure anonymous authentication. Key distribution is done by using many key distribution centres (KDCs). Unlike the existing research about the cloud which only allows read permission, this scheme provides authentication and other rights for users.

Merkle's puzzles [START_REF] Merkle | Secure communications over insecure channels[END_REF] and zero-knowledge proofs are used to ensure authentication with controlled anonymity in P2P systems [START_REF] Wierzbicki | Authentication with controlled anonymity in P2p systems[END_REF]. This protocol allows a peer to securely store data on another peer and to change it anonymously. This scheme requires a superpeer to generate the necessary keys for the calculation of a zero-knowledge proof. Only the owner is able to modify the previously stored file. Intermediate nodes ensure the communication between peers to provide anonymity. The authors present a novel authentication method successfully ensuring anonymity and accountability. However, the use of a superpeer, a trusted authority is similar to a trusted third party scheme. Thus the superpeer should be honest. Our contribution consists in removing the role of a superpeer, while keeping the same security goal. In the present paper, we extend this mechanism using a flat model without any hierarchies.

III. BACKGROUND: REMOTE INTEGRITY CHECKING

Our new mechanism is based on a remote integrity protocol presented in [START_REF] Deswarte | Remote integrity checking[END_REF]. In this protocol, a verifier, typically the data owner, generates a challenge to the data storer. The storer is obliged to use the entirety of the data to compute the proof. Thus this protocol is deterministic and allows an unlimited number of verification.

The security of the protocol is derived from the discrete logarithm problem proposed by Diffie-Hellman algorithms [START_REF] Diffie | New directions in cryptography[END_REF]. Let F be the file to store, and N be a public RSA modulus, the product of two secret prime numbers p and q, which may be known by the data owner. a) Storage phase: The owner chooses a random integer a between 2 and N -2, and computes a checking value M = a F mod N where F denotes the value of the file F to be stored and is considered as a big integer.

b) Verification phase:

To check if the storer holds the file F, the data owner chooses a random value r (the domain is the same as a) and computes his challenge A = a r mod N , which is sent to the storer. The storer computes the proof B = A F modN using the challenge and the whole file. Then, the data owner verifies that M r = B, i.e. (a F ) r mod N = (a r ) F mod N . Note that in this scheme a does not need to be secret, and r is kept secret from the storer thanks to the discrete logarithm problem. We also note that the totient φ(N ) must also be kept secret from the storer.

IV. ANONYMOUS DATA ACCESS

We extend the previous protocol to be used on the Owner side. In our case, the value of a is secret and will be stored on the Owner side. It is denoted by s. The Storer now becomes the verifier. The aim of this protocol is to allow the owner of the stored data to protect access to the data without disclosing his identity. Only the owner is thus able to prove his eligibility to retrieve his data, change it or even delete it anonymously.

A. Exclusive Anonymous access

After the anonymous authentication described above, a peer can either store its (PUT operations) data or retrieve it (GET operations). These two processes should be made anonymously. All communications between the concerned peer may be done using an onion routing protocol like APFS [START_REF] Scarlata | Responder anonymity and anonymous peer-to-peer file sharing[END_REF]. To facilitate the description of the protocol, we do not present the tail nodes and we assume that the path established between the storer and the owner is anonymous.

a) Storage Phase: The owner chooses a random secret s and sends the file F with the associated credential S = s F mod N . This credential is a zero-knowledge value and will enable the owner to access or modify the file later.

b) Verification phase: The owner sends a GET request to recover his file F. The verifier (the Storer) chooses a random challenge r which is sent to the owner. The owner then computes the proof P = s r mod N using the challenge sent and the secret s. Using the stored file F and the value of S, the verifier checks the proof received, S r mod N = P F mod N , i.e.

(s F ) r mod N = (s r ) F mod N , before sending the file.

Note that in this scheme, the value s needs to be secret, whereas the exponent r is public. The security of the protocol is not strictly based on the discrete logarithm problem, but on the reverse discrete logarithm problem, which is known to be as hard as the prime factorization problem [START_REF] Bender | A Short Course in Discrete Mathematics[END_REF].

V. AUTHENTICITY OF DATA MODIFICATIONS

Here, we present an extension of the previous protocol to validate file modification by the owner. When the Owner demands to modify his file F , it sends F , the new value S , and the hash of the old file. The verifier chooses a random challenge r. The owner then computes the proof P using the challenge sent and the secret s. The hash allows the verifier to find the concerned file. Using the stored file F , the value of S and the new versions F and S , the verifier checks the received proof. The file will be replaced if the proof is correct. The new protocol is summarized in Figure 1.

Note that this protocol checks not only if the sender is the owner, but also checks the integrity of the new file sent. This protects us from a man in the middle attack, where the file is modified during the transfer.

VI. DATA SHARING AND MANAGEMENT OF ACCESS RIGHTS

In the previous schemes, only the peer who has stored the file (the Owner) is authorized to access and modify the file. In this section we expand our mechanism to allow data sharing between different nodes, where the kind of access (read, write, modify, delete .....) can be set for each peer.

For each stored file F , the Owner stores a second file ACL F describing the access control to F . This ACL file stores information about access rights and will be described below. The owner can therefore assign rights to other users. It can choose many secrets rather than one and calculates a zeroknowledge value for each secret. Each zero-knowledge value corresponds to one access right. For example, it can choose three secrets: one for reading, the second for writing, and the third for deleting. The file is stored with the pre-computed zero-knowledge values.

These chosen secrets corresponding, to each right can be sent than to every authorized peer selected by the owner. Transferring this secret can be done using securing channels.

A. Access Right Allocation

The owner can create files corresponding to each stored data, defining the access rights policy for each user. Only the owner and the data Storer are able to access or modify this file. This can be verified using the same method previously described in Section IV-A: The Owner associates a specific secret number to the access list file.

Users privacy should be preserved. Thus this file should not reveal any information about the identity of the listed users. Each user is represented by a random pseudo. The file contains the list of pseudos associated with the corresponding precomputed zero-knowledge values. These pseudos are public. Eavesdroppers cannot use these pseudos to access the stored data because they don't know the secrets.

B. Revocation

Thanks to the protocol described in Section V, the owner can safely modify or delete the list of authorized users if necessary. In this case, it asks the Storer to update the access rights policy. The Storer makes the necessary changes by assigning a new right to a previous user, removing rights, or adding new users.

To illustrate our solution, consider the following example, shown in Figure 2. The owner shares the data file F with nodes A and B according the following rights :

• node A can read, delete and modify the file F ;

• node B can read the file F . The Storer stores not only the data file but also the access rights file, which contains the list of users and the associated rights. This file is associated with a zero-knowledge value P AR computed by the owner. Only the owner is able to modify the file. Users store the secrets allowing them to prove their access rights. This is done using the protocol described in Section IV-A. Each right is represented by a zero-knowledge value: r for reading, w for modifying and d for deleting. Figure 3 illustrates the control access mechanism. The owner wants to deprive nodes B the ability to read the file. It should first modify the access control file and send it to the Storer. The Storer should verify the identity of the owner for each modification of the access rights file as described previously in Section V. Thus the owner should correctly response to the Storer challenge using the stored secret S AR . If this succeeds, the Storer stores the new version of the file and the new value P AR . Node B can no longer read the file.

VII. EVALUATION

We consider the encryption time to be an important metric for security costs. The computing time is related to the size of the stored file. The file should be fully considered to ensure that it was not modified by the Storer. For small files, like ACL files, this cost may be insignificant, but stored files may be several Megabytes or Gigabytes. Different methods may be used for this purpose, depending on the security and performance requirements of the system.

To estimate the computing time of the zero-knowledge values (i.e. s F , F being the big number represented by its bitfield), we use square and multiply algorithms and the GMP big number library. However, the time for checking access rights may take several minutes on a current computer for a ten Megabyte file, and this time is in the size of the file. This execution time is too long to be of practical use. Using hash files or elliptic curve encryption instead of the file may reduce significantly the time overhead: The time is indeed negligible using the hash of the file and is about a few hundred milliseconds for a file up to 80 Megabytes. Using elliptic curve encryption is 100 times faster than using the full file.

VIII. DISCUSSION

Our work offers the data owner the possibility of controlling access rights. In order to access the stored data, an attacker has know the secrets chosen by the data owner. The difficulty is defined by the choice of the RSA module. The size of this module should be large enough to ensure a high level of security. We have also shown that zero-knowledge value calculation can be reduced using the elliptic curve algorithm. The use of Diffie-Hellman is still possible and the time overhead can be reduced by splitting data by using erasure code, for example, like Reed Solomon. Small fragments are used instead of large data. The ACL file overhead will not noticeably affect the traffic speed or the storage space. Moreover, communication requires only a small number of exchanged messages. Our approach is therefore an acceptable alternative regarding the data overhead. IX. CONCLUSION A storage system based on distributed platforms faces several challenges, among which are access control requirements, which have been addressed in this paper. Users should own and control their data without compromising security or using trusted third party services. We have presented an access control protocol for distributed storage systems, which are environments mainly characterized by the lack of any centralized authority.

This protocol allows anonymous control of read/write access to data stored in a cloud/P2P storage system. It permits the system to delegate access rights to other users with revocation. Our system is also able to check the integrity of data transferred for modification.

We evaluated the cost of this protocol and showed that cryptographic primitives can be too costly for an efficient implementation. We proposed some improvements and have show that they reduce drastically this cost.

Our future investigation will be two-fold. On the cryptographic side, we would like to see how we can use cryptographic pairing techniques to solve these same problems, and what is the performance of such techniques. On the software side, the next step of our research will be to evaluate how such a system can be implemented in the usual cloud storage API, like Amazon S3 or OpenStack SWIFT.

FFig. 1 .

 1 Fig. 1. Anonymous modification with integrity checks.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Assigning access rights to different users


domain: math spi

text:
Yri Amandine Kambiri

Thierry Druot

Pascal Roches

Nicolas Peteilh

Nicolas Monrolin

Xavier Carbonneau

Energy consumption of aircraft with new propulsion systems and storage media

The transition from fossil fuels to the use of sustainable energy sources is an important challenge for aviation. A particularly critical question is to identify the best mix of energies for the future air transport system. In this quest, the capability to simulate various complex scenarios, varying assumptions, models, and objectives, will play an important part. This paper provides a simple and very rapid tool to evaluate the energy consumption of parametric airplane configurations with thermal propulsion powered by kerosene, methane, hydrogen, or electric propulsion powered by hydrogen fuel cells or batteries. The maturity of the different technologies is driven by a small number of global indices that are easy to connect with state of the art or forecasts. The user can freely select the capacity, range, speed, and propulsion system of the airplane, and simulate its operations over any missions within its payload-range envelope. The tool has been validated on its capability to reproduce the characteristic weights of a range of very different aircraft, from A380 to general aviation airplanes.

I. Introduction

A s any domain of activity, passenger air transport will have to drastically decrease its impact on the environment. Among all the negative effects, the contribution of air transport to radiative forcing is probably the most damaging [START_REF] Terrenoire | Impact of present and future aircraft NO 𝑥 and aerosol emissions on atmospheric composition and associated direct radiative forcing of climate[END_REF][START_REF] Snijders | Effect of cruise altitude and alternative aviation fuels on radiative forcing[END_REF]. The 𝐶𝑂 2 emissions contribute to radiative forcing, but other non-𝐶𝑂 2 effects of aviation are estimated to represent about 2/3 of the radiative forcing [START_REF] Lee | The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018[END_REF]. This contribution develops mainly while the aircraft is in service [START_REF] Boucher | On the contribution of global aviation to the CO2 radiative forcing of climate[END_REF], and more specifically during cruise [START_REF] Gao | Aircraft cruise phase altitude optimization considering contrail avoidance[END_REF][START_REF] Dallara | Aircraft design: Trading cost and climate mpact[END_REF]. Various studies have estimated that the carbon intensity of fossil kerosene ranges from 85 to 95 grams [START_REF] Bauen | CORSIA lower carbon aviation fuels: An assessment of the greenhouse gas emission reduction potential[END_REF][START_REF] Seber | Uncertainty in life cycle greenhouse gas emissions of sustainable aviation fuels from vegetable oils[END_REF] of carbon dioxide equivalent per megajoule (g CO 2 e/MJ) of fuel, of which approximately 73 g CO 2 e/MJ [START_REF] Bauen | CORSIA lower carbon aviation fuels: An assessment of the greenhouse gas emission reduction potential[END_REF][START_REF] Planès | Du dimensionnement de systèmes et d'architectures en conception avion à la simulation de scénarios prospectifs durables pour le transport aérien[END_REF] is due to the combustion of the fuel and the remainder to the extraction of the fuel, its processing in refineries and its transportation [START_REF] Pavlenko | Assessing the sustainability implications of alternative aviation fuels[END_REF]. The five main approaches, potentially combined together, to achieve Paris Climate Agreement proposed in the literature are 1) Change the energy carrier (hydrogen, bio-sourced or synthetic methane, ammonia [START_REF] Cabrera | Use of Sustainable Fuels in Aviation-A Review[END_REF])

2) Change the way to produce fuel (bio-sourced kerosene, synthetic kerosene) [START_REF] Marszałek | The future of sustainable aviation fuels[END_REF][START_REF] Chiaramonti | Sustainable Aviation Fuels: the challenge of decarbonization[END_REF][START_REF] Yusaf | Sustainable aviation-yydrogen is the future[END_REF] 3) Modify operations (especially during cruise to avoid contrails) [START_REF] Icao | Introduction to the ICAO basket of measures to mitigate climate change[END_REF][START_REF] Planès | Simulation and evaluation of sustainable climate trajectories for aviation[END_REF] 4) Adapt the airplane design to new practices (flight altitudes, speeds, new market segments) 5) Decrease the overall activity.

The last approach would be considered either if no alternative is found or as a consequence of the application of the other approaches. For example, a consequence of excessive demand for sustainable energy sources compared to the supply would lead to competition among the different economic sectors and competition inside the transportation sector itself. Among the other four approaches, the first and second are anticipated to bring up to 71% for the total expected gains (in terms of CO 2 emission) although the third and fourth ones would contribute to the rest [START_REF] Atag | Waypoint 2050[END_REF]. Changing the energy carrier may not only significantly change the technical and operational features of the aircraft itself but also deeply transform the full air transport system (ATS), and even the energy production and supply system. Therefore, it is of utmost necessity to carry out environmental impact analysis even beyond the ATS level, including the energy supply e.g., nuclear, solar, wind, or biomass sources. The various possible combinations of energy supply and energy vector (battery, hydrogen, synthetic kerosene, etc.) make the evaluation of the energy consumption and climate impact of the future air transport system a challenging task. One major difficulty in reaching this objective is the complexity of the aircraft design process itself with a broad design space. Classical design processes present two kinds of limits. The first limitation is that they are defined to address kerosene powered aircraft [START_REF] Raymer | Aircraft Design: A Conceptual Approach[END_REF][START_REF] Sadraey | Aircraft design : A system engineering approach, Aerospace series[END_REF][START_REF] Gudmundsson | General Aviation Aircraft Design: Applied Methods and Procedures[END_REF]. The second limitation is that, even the simplest tools proposed require geometrical and technological assessment as well as complex multidisciplinary analysis and optimization to evaluate one single aircraft configuration (MARILib, FAST-OAD, TASOPT [START_REF] Drela | TASOPT 2.00: Transport aircraft system optimization-Technical description[END_REF]), [START_REF] Yuhara | Conceptual design study on liquid hydrogen-fueled supersonic transport considering environmental impacts[END_REF][START_REF] Oak | Spectral project -Application of FAST-OAD code to the conceptual design of a hydrogen fuelled commercial aircraft[END_REF][START_REF] Maciel Monjon | Conceptual design and operating costs evaluation of a 19-seat all-electric aircraft for regional aviation[END_REF][START_REF] Yang | Conceptual design of a 10-passenger thin-haul electric aircraft[END_REF][START_REF] Nam | Conceptual design of an aviation propulsion system using hydrogen fuel cell and superconducting motor[END_REF][START_REF] Nicolay | Conceptual design and optimization of a general aviation aircraft with fuel cells and hydrogen[END_REF]. The use of higher fidelity tools require the use of high computational capabilities and long design time.

In the following, a simple aircraft design procedure that covers various energy vectors is presented. From as small set of requirements in terms of capacity, range, storage medium, and propulsion system, the model allows one to retrieve the fuel (or energy) consumption for any mission.

This paper is organized as follows. In Section II, the minimal classical approach to designing aircraft and determining its main mass and operational features is discussed with its limitations in the context of the ecological transition of the ATS. Then, the models developed to address these limitations are proposed in Section III. Section IV proposed the validation of our model. Then, in Section V, the results of our model are proposed. Finally, Section VI proposed the conclusion of our work.

II. Minimal classical aircraft design approach

According to Breguet equation shown in [START_REF] Terrenoire | Impact of present and future aircraft NO 𝑥 and aerosol emissions on atmospheric composition and associated direct radiative forcing of climate[END_REF], the performance of an aircraft in terms of fuel consumption (or energy), depends mainly on its characteristic weights (MTOW, OWE), its L/D in cruise, and its Specific Fuel Consumption (SFC).

𝑅𝑎𝑛𝑔𝑒 = 𝐿/𝐷 × 𝜂 × 𝑓 ℎ𝑣 𝑔 × log 𝑀𝑇𝑂𝑊 𝑀𝑇𝑂𝑊 -𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 (1)
=⇒ 𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 = 𝑓 1 (𝑀𝑇𝑂𝑊, 𝑅𝑎𝑛𝑔𝑒, 𝐿/𝐷, 𝑆𝐹𝐶, 𝑆 𝑝𝑒𝑒𝑑) (2)
The minimal classical approach for the computation of characteristic weights such as MTOW and OWE with a petrol propulsion system is based on the so-called mass-mission coupling process [START_REF] Raymer | Aircraft Design: A Conceptual Approach[END_REF]. In the context of very early stages of the design process such as the conceptual design phase, the objective is to estimate the optimized MTOW of the aircraft that ensures both the necessary amount of structure and energy on board. This can be presented by the following optimization problem: min

𝑋∈R 𝑘

MTOW

subject to structure and design mission fuel constraints [START_REF] Lee | The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018[END_REF] which is equivalent to solve

         𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 = 𝑓 1 (𝑀𝑇𝑂𝑊, 𝑅𝑎𝑛𝑔𝑒, 𝐿/𝐷, 𝑆𝐹𝐶, 𝑆 𝑝𝑒𝑒𝑑), 𝑀𝑇𝑂𝑊 = 𝑂𝑊 𝐸 + 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑇 𝑜𝑡𝑎𝑙𝐹𝑢𝑒𝑙, 𝑂𝑊 𝐸 = 𝑓 2 (𝐺𝑒𝑜𝑚𝑒𝑡𝑟 𝑦, 𝑀𝑇𝑂𝑊, 𝑁 𝑝𝑎𝑥 , 𝑃𝑟𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 𝑠𝑦𝑠𝑡𝑒𝑚),
where 𝑓 1 is defined by (1). This approach does not need any geometry definition. However this model is based on empirical statistical regressions with classical tube and wing architectures:

𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 = 𝑓 1 (𝑀𝑇𝑂𝑊, 𝑅𝑎𝑛𝑔𝑒, 𝐿/𝐷, 𝑆𝐹𝐶, 𝑆 𝑝𝑒𝑒𝑑), 𝑀𝑇𝑂𝑊 = 𝑂𝑊 𝐸 + 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 + Total Fuel,
where Total Fuel takes in account the mission fuel, reserve fuel, diversion and holding fuel.

Total Fuel = 𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 (1 + 𝐾 𝑟 ) + Diversion Fuel + Holding Fuel (4)

As shown in [START_REF] Gao | Aircraft cruise phase altitude optimization considering contrail avoidance[END_REF], the payload is computed with the number of passengers (𝑁 𝑝𝑎𝑥 ) and a mass allowance (𝑀 𝑝𝑎𝑥 ) presented in Table 1.

𝑃𝑎𝑦𝑙𝑜𝑎𝑑 = 𝑁 𝑝𝑎𝑥 × 𝑀 𝑝𝑎𝑥 (𝑅𝑎𝑛𝑔𝑒) (5)
In this classical minimum approach, the OWE can be divided into three parts. 1 Passenger mass allowance and furnishing mass.

𝑂𝑊 𝐸

• 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 is the part of the 𝑂𝑊 𝐸 related to the furnishing and operator items. This quantity can be approximated as a function of 𝑁 𝑝𝑎𝑥 and Range. • 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 is the part of the 𝑂𝑊 𝐸 related to the propulsion system. It depends on the type of the propulsion system and its maximum power. This last quantity will be approximated by the Generalized Power Index (GPI) defined in [START_REF] Pavlenko | Assessing the sustainability implications of alternative aviation fuels[END_REF]. • 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 is the part of the 𝑂𝑊 𝐸 that is independent of 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 and 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 in a conventional aircraft (structure and non-propulsive systems).

A. New propulsion systems

If the current kerosene-oriented propulsion system moves to electric propulsion with batteries or fuel cells and hydrogen tank, for instance, OWE computation in [START_REF] Dallara | Aircraft design: Trading cost and climate mpact[END_REF] should incorporate the weight of the battery or the weight of the fuel cell system and the hydrogen tank. The fuel cell system weight will be added to the propulsion weight as it depends on the installed power. The hydrogen tank or battery weights will be added to the equation as a new term:

𝑂𝑊 𝐸 = 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 + 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 + 𝑂𝑊 𝐸 𝑇 𝑎𝑛𝑘𝑆𝑦𝑠𝑡𝑒𝑚 (7)
𝑂𝑊 𝐸 𝑇 𝑎𝑛𝑘𝑆 𝑦𝑠𝑡𝑒𝑚 is the OWE related to the tank system and depends on the type of fuel and the storage mode (e.g., compressed or liquefied gas). It is approximated as a function of the amount of fuel (or energy) to be stored and the type of storage system.

III. Models

This section deals with the regression models that allow us to determine the aircraft characteristics (such as MTOW and OWE) for different propulsion systems: thermal engines powered by the combustion of kerosene, hydrogen, or liquid methane or electric engines powered by batteries or a fuel cell system. First, the total mass of the structure and standard systems (𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 ) that is required to build an aircraft is estimated for a given MTOW. Secondly, based on the (𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 ), we rebuild the OWE of an aircraft that is powered by classical or new propulsion systems. Lastly, we determine MTOW.

A. Empty weight breakdown based on statistical regressions

In this part, a database of existing aircraft with classical design is used to derive a correlation between the MTOW and the manufacturer weight empty (or the total mass of structure and standard systems), noted 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 . 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 is the part of 𝑂𝑊 𝐸 which is independent of 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 and 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 in the conventional aircraft, as expressed in [START_REF] Seber | Uncertainty in life cycle greenhouse gas emissions of sustainable aviation fuels from vegetable oils[END_REF]. The aircraft database of about 227 passenger aircraft of all categories, from general aviation to long haul, is used to estimate 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 .

𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 = 𝑂𝑊 𝐸 -𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 -𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 (8)

OWE related to payload

Operating weight empty related to payload, 𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 , is computed as follows: The weight of the furnishing, 𝑊 𝑓 𝑢𝑟 𝑛𝑖𝑠ℎ𝑖𝑛𝑔 , is estimated by a linear regression model, which is shown in Fig. 1. The model is built using the 38 types of aircraft data collected from [START_REF] Wild | Aircraft powerplants[END_REF][START_REF] Roux | Avions civils à réaction Plan 3 vues et donneées caractéristiques[END_REF]. In Fig. 1, we select a linear regression such that

𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 = 𝑊 𝑓 𝑢𝑟 𝑛𝑖𝑠ℎ𝑖𝑛𝑔 + 𝑊 𝑜 𝑝 𝑖𝑡𝑒𝑚𝑠 (9)
𝑊 𝑓 𝑢𝑟 = 𝑀 𝑓 𝑢𝑟 (𝑅𝑎𝑛𝑔𝑒) × 𝑁 𝑝𝑎𝑥,
where 𝑀 𝑓 𝑢𝑟 the furnishing mass budget per passenger. It is assumed to depend on the flight range as described in Table 1. For the estimation of operational items weight, we used the following function :

𝑊 𝑜 𝑝 𝑖𝑡𝑒𝑚𝑠 = 5.𝑒 -6 × 𝑁 𝑝𝑎𝑥 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
The operational items represent the set of crew provisions, passenger cabin supplies, potable water and toilet chemicals, safety equipment, oil residual fuel, and cargo handling equipment [START_REF] Torenbeek | Synthesis of subsonic airplane design : an introduction to the preliminary design of subsonic general aviation and transport aircraft, with emphasis on layout, aerodynamic design, propulsion and performance[END_REF].

OWE related to the propulsion system

The mass of the propulsion system components is assumed to depend on their "power". But the overall characteristics of turbofans (e.g., weight, dimensions, cost, etc.) are generally linked to their maximum thrust, which is most often quantified at low speed and low altitude. The Sea Level Static Thrust (𝑆𝐿𝑆𝑇) is most often used to drive the engine as a whole (e.g., thrust, weight, size, etc.). Meanwhile, the characteristics of turboprops or reciprocating engines are generally linked to their maximum shaft power: 𝑀𝑎𝑥𝑃𝑜𝑤𝑒𝑟. To gain in generality, the Generalized Power Index (GPI) is defined. This quantity, expressed in the unit of power (Watts), is a figure of interest to estimate the weight of an engine, whatever its type. It is defined as follow:

𝐺𝑃𝐼 = 𝑀𝑎𝑥𝑃𝑜𝑤𝑒𝑟 engine with propeller, 𝑃𝑊 turbofan, ( 10
)
where 𝑀𝑎𝑥𝑃𝑜𝑤𝑒𝑟 is the maximum shaft power of the turboprop or piston engine. To be consistent with all engine types, 𝑃𝑊 is created. It represents the maximum power of turbofan engine. This quantity is defined by equation [START_REF] Cabrera | Use of Sustainable Fuels in Aviation-A Review[END_REF].

𝑃𝑊 = 𝑛 𝑒 𝑆𝐿𝑆𝑇 × 𝑉 2𝑚𝑖𝑛 𝜂 𝑓 , where 𝑉 2𝑚𝑖𝑛 = 𝑉 𝑎 𝑝 𝑝 × 1.13 1.23 . ( 11
)
𝑉 2𝑚𝑖𝑛 is the minimum takeoff speed and 𝑉 𝑎 𝑝 𝑝 is the approach speed. Both are defined as a function of the 1g stall speed, 𝑉 𝑠1𝑔 , as follows.

𝑉 2𝑚𝑖𝑛 = 1.13 × 𝑉 𝑠1𝑔 and 𝑉 𝑎 𝑝 𝑝 = 1.23 × 𝑉 𝑠1𝑔

Fig. 2 Correlation between MTOW and GPI

The minimum takeoff condition, 𝑉 2𝑚𝑖𝑛 is the speed near sea-level at which the turbofan delivers its maximum thrust (SLST). It is computed from 𝑉 𝑎 𝑝 𝑝 , which is easily found in most aircraft databases. 𝑃𝑊 is representative of the shaft power delivered by the engine in takeoff conditions.

The total 𝐺𝑃𝐼 (including all engines) is correlated to the 𝑀𝑇𝑂𝑊 of all airplanes in the database. The graph Fig. 2 illustrates this correlation together with a second-order polynomial least square fit.

𝐺𝑃𝐼 = 8.31693845𝑒 -5 × 𝑀𝑇𝑂𝑊 2 + 2.03027049𝑒 2 × 𝑀𝑇𝑂𝑊 -1.05𝑒 5
The mass of the propulsion system, 𝑂𝑊 𝐸 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 is then estimated using the gravimetric indices (GI), which are represented with a unit in kW/kg. 𝑖 represents the type of propulsion system, and 𝑗 represents the type of thruster. The different gravimetric indices (GI) are computed as follows.

𝑂𝑊 𝐸

• 𝐺 𝐼 𝑡𝑢𝑟 𝑏𝑜 𝑓 𝑎𝑛 = 4.3𝑘𝑊/𝑘𝑔 (shown in Fig. 3) and 𝐺 𝐼 𝑡 𝑓 𝑎𝑛 = 15𝑘𝑊/𝑘𝑔 for a turbofan engine • 𝐺 𝐼 𝑡𝑢𝑟 𝑏𝑜𝑠ℎ𝑎 𝑓 𝑡 = 4.3𝑘𝑊/𝑘𝑔 (shown inFig. 4) and 𝐺 𝐼 𝑡 𝑝𝑟𝑜 𝑝𝑒𝑙𝑙𝑒𝑟 = 10𝑘𝑊/𝑘𝑔 for a turboprop engine • 𝐺 𝐼 𝑝𝑖𝑠𝑡𝑜𝑛 = 1.1𝑘𝑊/𝑘𝑔 (shown in Fig. 5) and 𝐺 𝐼 𝑡 𝑝𝑟𝑜 𝑝𝑒𝑙𝑙𝑒𝑟 = 10𝑘𝑊/𝑘𝑔 for a piston engine The GIs are computed by the regressions between dry weight and GPI, as shown in Figs. 3 to 5. The data used in these figures are from [START_REF] Roux | Avions civils à réaction Plan 3 vues et donneées caractéristiques[END_REF] for the current petrol-based propulsion systems and from manufacturers databases for new technologies.

Estimation of Basic 𝑀𝑊 𝐸

As shown in Fig. 6, the correlation between the OWE and the MTOW is very tight, regardless of the aircraft category. One possible explanation is that all these airplanes are designed to transport passengers, equipped with similar propulsion systems burning petrol derivatives (gasoline or kerosene). It can also be because the airplanes are designed using the same structural principles, and their designs are compliant with consistent regulation and economic pressure. Nevertheless, this regression that we observe includes two implicit links. The first one is between MTOW and the mass of the propulsion system and the second one is between OWE and payload. Different articles [START_REF] Hu | Preliminary sizing of electric-propulsion powertrains for concept aircraft designs[END_REF][START_REF] Riboldi | An integrated approach to the preliminary weight sizing of small electric aircraft[END_REF] derive mass relationship only for hybrid and all-electric as well as fixed-wing and VTOL configurations. For each airplane of the database, we have estimated the 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 as follows :

𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 = 𝑂𝑊 𝐸 -𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 (𝑁 𝑝𝑎𝑥, 𝑅𝑎𝑛𝑔𝑒) -𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 (𝐺𝑃𝐼 (𝑀𝑇𝑂𝑊)),
where 𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 , 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 and 𝐺𝑃𝐼 are computed as presented above in this chapter.

We found the following regression ( Fig. 7):

𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 = -3.18952359𝑒 -7 × 𝑀𝑇𝑂𝑊 2 + 4.22840552𝑒 -1 × 𝑀𝑇𝑂𝑊 -30 (12)

Correlation between GPI and piston engine mass

Basic 𝑀𝑊 𝐸 , which is computed as a function of MTOW, is independent of the number of passengers and of the propulsion system. We can use this quantity to estimate the OWE of airplanes with a non-petrol-based propulsion system and any passenger capacity. The freedom in the seating capacity is important because new propulsion technologies generally require putting additional on board systems compared to petrol-based architectures (e.g., liquid hydrogen tanks, fuel cells, etc.).

B. OWE related to electric and H2 propulsion

In Section III.A we already estimate 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 , 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 , 𝑂𝑊 𝐸 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 . However, to compute OWE related to electric and H2 propulsion, 𝑂𝑊 𝐸 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 needs to be reevaluated for electric and hydrogen technologies.

All electric with batteries

The weight of the electric propulsion system can be defined as the set of electric motor weight and electronics system weight.

𝑂𝑊 𝐸 = 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 + 𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑂𝑊 𝐸 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛𝑠𝑦𝑠𝑡𝑒𝑚 (13) For electric motors, we use the gravimetric index of Magnix 500 engine :

𝐺 𝐼 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 = 4.1𝑘𝑊/𝑘𝑔 *
This value might be quite pessimistic since the Siemens SP260D engine have gravimetric index of 5.2𝑘𝑊/𝑘𝑔 [START_REF] Siemens | Rekord-Motor SP260D und Extra 330LE[END_REF] and Kadyk et al. [START_REF] Kadyk | Analysis and Design of Fuel Cell Systems for Aviation[END_REF] forecasted that the gravimetric index of an electric motor airplane can reach up to 10𝑘𝑊/𝑘𝑔 in the future. Additionally, we use the following values for the gravimetric index of power electronics.

𝐺 𝐼 𝑝𝑜𝑤𝑒𝑟 𝑒𝑙𝑒𝑐 = 10𝑘𝑊/𝑘𝑔
In the estimation of 𝑀𝑇𝑂𝑊, 𝑇 𝑜𝑡𝑎𝑙 𝑓 𝑢𝑒𝑙 includes the battery weight estimated thanks to the energy density of battery.

hydrogen fuel cell aircraft

The weight of an electric propulsion system (motor + power electronics), 𝑂𝑊 𝐸 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 𝑠𝑦𝑠𝑡𝑒𝑚 , computed using [START_REF] Chiaramonti | Sustainable Aviation Fuels: the challenge of decarbonization[END_REF].

𝑂𝑊 𝐸 = 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 + 𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑂𝑊 𝐸 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 𝑠𝑦𝑠𝑡𝑒𝑚 + 𝑂𝑊 𝐸 𝑓 𝑢𝑒𝑙 𝑐𝑒𝑙𝑙 + 𝑂𝑊 𝐸 𝑇 𝑎𝑛𝑘𝑠𝑦𝑠𝑡𝑒𝑚
The weight of the fuel cell system must include not only the fuel cell stack but also the balance of plant (e.g. compressor, valves, manifolds, air inlet, humidity regulator etc.) and cooling devices (heat exchanger). The gravimetric index for a fuel cell is as follows:

𝐺 𝐼 𝑓 𝑢𝑒𝑙 𝑐𝑒𝑙𝑙 = 1𝑘𝑊/𝑘𝑔 This value is twice higher than existing packaging tailored for trucks. For exemple, Horizon Fuel Cell Technologies † is selling liquid-cooled fuel cell systems of 120 kW with a power density of around 0.5 kW/kg. Meanwhile, the literature seems to be rather optimistic: Dudfield [START_REF] Dudfield | Beyond Batteries: Hydrogen Fuel Cells for UAVs[END_REF] cited a report which forecasts 1.6 kW/kg for drones applications (air-cooling) while Kadyk et al. [START_REF] Kadyk | Analysis and Design of Fuel Cell Systems for Aviation[END_REF] estimated that metallic bipolar plates (with adequate corrosion coating) would strongly reduce the stack weight, enabling 8𝑘𝑊/𝑘𝑔 fuel cell systems in the future.

Tank system mass is related to the type of fuel and how it is packaged. At the granularity level of our models, gasoline or kerosene requires no special structures as they are stored in the existing structural parts. Hydrogen and methane require special devices and can be stored in compressed or liquefied form. The mass of the different tank types is defined using a gravimetric index [START_REF] Druot | Hydrogen Powered Airplanes, an exploration of possible architectures leveraging boundary layer ingestion and hybridization[END_REF] defined as follows:

𝐺 𝑖𝑛𝑑𝑒𝑥 =
𝑀 𝑓 𝑢𝑒𝑙 𝑀 𝑓 𝑢𝑒𝑙 + 𝑀 𝑡 𝑎𝑛𝑘 * https://cleantechnica.com/2019/10/24/750-horsepower-electric-aviation-engine-tested-by-magnix/ † www.horizonfuelcell.com, accessed on November 2023. VL II series liquid-cooled system.

For liquid hydrogen (∼ 20𝐾), The company Hypoint clamed in 2022 to be able to build LH2 tanks with a gravimetric index of 0.5, which is quite challenging ‡ . A study on hydrogen storage explains that the gravimetric index of liquid hydrogen could reach 0.7 [START_REF] Joaquim | Hydrogen-Powered Aircraft: Concepts, Technologies, and Environmental Impact[END_REF]. However, we take 0.4 to consider the possible difficulties in reaching the targeted value of 0.5. Considering that liquid methane is also cryogenic (∼ 111𝐾) [START_REF] Dorrington | Conceptual design of sustainable liquid methane fuelled passenger aircraft[END_REF] and is a bit less challenging to store compared to liquid hydrogen, we take 0.8 as the gravimetric index.

Liquid methane can be obtained at ambient temperature with a pressure of about 10 bars. So, its cryogenic storage is not considered, and only compressed storage has been evaluated. The gravimetric index of the cryogenic storages is as follows:

Liquid hydrogen : 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻 2 = 0.4, according to [START_REF] Druot | Hydrogen Powered Airplanes, an exploration of possible architectures leveraging boundary layer ingestion and hybridization[END_REF] Liquid methane : 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 = 0.8, presented in Section .B

The mass of pressurized vessels is computed according to the performance index expressed as follows:

𝑃 𝑖𝑛𝑑𝑒𝑥 = 𝑃 𝑖 × 𝑉 𝑖 𝑀 𝑡 𝑎𝑛𝑘
The performance index depends on the level of tank's technology. According to the Sheet 4.2 of France-Hydrogene § , the most efficient technology makes use of carbon fiber-reinforced plastic (CFRP) and have a performance index equal to 661 bar.L/kg. We take this value as a reference. Knowing the initial pressure of the gas into the tank, we can get the gravimetric index of the tank from the performance index:

𝐺 𝑖𝑛𝑑𝑒𝑥 𝑔𝑎𝑠 = 1 1 + 𝑃 𝑖 𝑃 𝑖𝑛𝑑𝑒𝑥 × 𝜌 𝑔𝑎𝑠 (𝑃 𝑖 )
In the estimation of 𝑀𝑇𝑂𝑊, we consider 𝑇 𝑜𝑡𝑎𝑙 𝑓 𝑢𝑒𝑙 as the weight of liquid hydrogen or liquid methane.

Empty weight for hydrogen (methane) fuelled aircraft

𝑂𝑊 𝐸 = 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 + 𝑂𝑊 𝐸 𝑝𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑂𝑊 𝐸 𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛𝑠𝑦𝑠𝑡𝑒𝑚 + 𝑂𝑊 𝐸 𝑇 𝑎𝑛𝑘𝑠𝑦𝑠𝑡𝑒𝑚 Liquid hydrogen or liquid methane engine are supposed to have the same mass as the conventional engine. But their Specific Fuel Consumption (SFC) [START_REF] Oak | Spectral project -Application of FAST-OAD code to the conceptual design of a hydrogen fuelled commercial aircraft[END_REF] is computed according to the fuel lower heating value (FHV) given in Table 2. The computing method of 𝑂𝑊 𝐸 𝑇 𝑎𝑛𝑘𝑠𝑦𝑠𝑡𝑒𝑚 is the same as those of hydrogen fuel cell airplanes.

Fuel

C. Determination of MTOW

In the context of our models, optmizing MTOW is equivalent to solving the following system of two equations:

𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 = 𝑓 1 (𝑀𝑇𝑂𝑊, 𝑅𝑎𝑛𝑔𝑒, 𝐿/𝐷, 𝑆𝐹𝐶, 𝑆 𝑝𝑒𝑒𝑑) 𝑀𝑇𝑂𝑊 = 𝑂𝑊 𝐸 + 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑇 𝑜𝑡𝑎𝑙𝐹𝑢𝑒𝑙 (14)
where, OWE, Payload, and Total fuel are already defined. In order to increase the precision of the model, two additional relations are introduced. ‡ https://hydrogen-central.com/hypoint-zero-emission-hydrogen-flight-range-ultralight-liquid-hydrogen-fuel-tanks/ accessed on November 2023 § https://www.france-hydrogene.org/fiches-techniques/ accessed November 2023.

𝜂 E-motors powered by fuel cells 𝜂 𝑝 × 𝜂 𝑒𝑙𝑒𝑐 × 𝜂 𝑓 𝑐𝑠 For reciprocating engines or turboprops 𝜂 ℎ × 𝜂 𝑝𝑤 Turbofans 𝜂 𝑡 ℎ × 𝜂 𝑝𝑟 Battery-based energy storage 𝜂 𝑝 × 𝜂 𝑒𝑙𝑒𝑐 Table 3 Propulsion system efficiencies • The first one relates 𝑆𝐹𝐶 with 𝐺𝑃𝐼 and the type of the propulsion systems :

𝑆𝐹𝐶 = 𝑓 9 (𝐺𝑃𝐼, 𝑃𝑟𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛𝑇 𝑦 𝑝𝑒), (15)
where 𝑓 9 is defined in Section .B. • The second one relates L/D and the MTOW:

𝐿/𝐷 = 𝑓 8 (𝑀𝑇𝑂𝑊), (16)
where 𝑓 8 is a correlation based on Fig. 9.

Generalized Breguet equation

When fuel consumption is involved, we use the following Breguet equation, which can be easily reversed to compute Fuel:

𝑅𝑎𝑛𝑔𝑒 = 𝐿/𝐷 × 𝜂 × 𝑓 ℎ𝑣 𝑔 × log 𝑀𝑇𝑂𝑊 𝑀𝑇𝑂𝑊 -𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙
For battery-based energy storage, we use the following equation:

𝑅𝑎𝑛𝑔𝑒 = 𝐿/𝐷 × 𝜂 × 𝑏𝑒𝑑 𝑔 × 𝑀 𝑏𝑎𝑡𝑡𝑒𝑟 𝑦 𝑀𝑇𝑂𝑊
As described in Table 3, the efficiency 𝜂 refers to the whole propulsion chain, from tank (or battery) to propulsive power. For a conceptual airplane design approach, the following efficiencies are assumed to be constant :

• 𝜂 𝑝 is the thruster efficiency. it can be a propeller ( 𝜂 ℎ = 0.80) or a fan (𝜂 𝑓 = 0.82) eventually driven by an electric motor. • The electric motor efficiency with its power electronics: 𝜂 𝑒𝑙𝑒𝑐 = 0.95 • The fuel cell system efficiency: 𝜂 𝑓 𝑐𝑠 = 0.5 PSFC of typical light aircraft piston engines is in the range between 0.24kg/h/kW and 0.5kg/h/kW [START_REF] Senzig | Rotorcraft performance data for AEDT[END_REF][START_REF] Kumar | Response surface methodology application in optimization of performance and exhaust emissions of RON 98, aviation gasoline 100LL and the blends in Lycoming O-320 engine[END_REF]. So we take the following constant as the power related 𝑆𝐹𝐶 of petrol piston engines : 𝑃𝑆𝐹𝐶 𝑝𝑖𝑠𝑡𝑜𝑛 = 0.25 𝑘𝑔/ℎ/𝑘𝑊, which leads to the following power efficiency:

𝜂 𝑝𝑤 = 0.335.
Power-related 𝑆𝐹𝐶 of petrol turboshafts has been taken from a regression on a set of 83 different engines , which is presented in Fig. 8. The following formula is derived from the regression:

𝑃𝑆𝐹𝐶 𝑡𝑢𝑟 𝑏𝑜 𝑝𝑟 𝑜 𝑝 = 5.54𝑒 -8 + 2.77𝑒 -6
𝐺 𝑃𝐼 1000 0.65 , which leads to the following power efficiency :

𝜂 𝑝𝑤 = 0.084 0.2 + 10 ( 𝐺 𝑃𝐼 1000 ) 0.65
.

Fig. 8 Correlation between PSFC and GPI Fig. 9 lift-to-drag ratio estimated from geometry in the database (dots) and fit curve of Table 4 (black line).

The thrust-related 𝑆𝐹𝐶 of petrol turbofans is computed using a constant 𝜂 𝑡 ℎ and a formula for 𝜂 𝑝𝑟 . The formula is explained in Appendix Section VI.

𝜂 𝑡 ℎ = 0.474, 𝜂 𝑝𝑟 = 1 1 2 + √ 1 4
+ 𝑟 × 𝜂 𝑡 ℎ × 𝑓 ℎ𝑣 2(1 + 𝐵𝑃𝑅)𝑉 2 0
with 𝑟 = 0.02.

Approximation of the Lift to Drag Ratio

Lift to drag ratio 𝐿/𝐷 is linked to the geometry of the airplane : aspect ratio, airfoil, roughness ... etc. Fig. 9 shows 𝐿/𝐷 estimated from a few geometric parameters described in the database as a function of the MTOW. The 𝐿/𝐷 estimation method is described in [START_REF] Raymer | Aircraft Design: A Conceptual Approach[END_REF][START_REF] Rebuffet | Aérodynamique expérimentale : Cours professé à l'Ecole Nationale Supérieure de l'Aéronautique[END_REF] and the code is available on the git repository ¶ .

For large civil transport aircraft (short, medium to long range) the correlation between 𝐿/𝐷 and MTOW is rather satisfying : the 𝐿/𝐷 remains between 15 and 21, probably because all this airplanes have a very similar overall geometry. But for lighter airplanes, large discrepancies are observed. Despite this, an arbitrary 𝐿/𝐷 is chosen with respect to MTOW as specified in Table 4. The resulting trend is drawn (black-line) on Fig. 9 MTOW (kg) 0 40000 200000 500000 1000000 L/D 13 16 19 20 20

Table 4 Interpolation of lift-to-drag

At this stage, all the parameters of the system [START_REF] Yusaf | Sustainable aviation-yydrogen is the future[END_REF] are known, and it can be solved numerically.

IV. Validation

The validity of the model is assessed with kerosene airplanes. For other architectures, very few data are currently available. The model requires a design range and passenger capacity to return the characteristic weights (e.g., MTOW, OWE, MWE, etc.) and other parameters such as fuel burn for the design mission. Fig. 10 (resp. 12) shows the estimated MTOW (resp. OWE) versus the real MTOW (resp. OWE). Fig. 11 (resp. 13) shows the relative error distribution 𝑀𝑇𝑂𝑊 𝑝𝑟 𝑒𝑑𝑖𝑐𝑡𝑒𝑑 -𝑀𝑇𝑂𝑊/𝑀𝑇𝑂𝑊 (resp. 𝑂𝑊 𝐸 𝑝𝑟 𝑒𝑑𝑖𝑐𝑡𝑒𝑑 -𝑂𝑊 𝐸/𝑂𝑊 𝐸.

The error distributions show that the model is able to predict MTOW and OWE with an accuracy equal or lower than 10% for 87% and 85% respectively of the aircraft in the database. For most airplanes the MTOW is predicted with less than 14% error. Of course this model is not intended to make accurate predictions, but it provides orders of magnitude based on previously observed trends.

V. Results

A. Use case 1 : Assessment of the airplane Alice from EVIATION

Alice is probably the full electrical, battery-powered transport airplane with the highest performances among the recent projects. Its design has been evolving a lot since the beginning of the project, and currently, very few data are published by EVIATION. On its website ‖ , we can find:

Making a full electric 9 passengers airplane flying over 460 km is very challenging, regarding the energy density of today's available batteries. To reach the required level of performance L/D and structural mass are probably MWE have probably been improved compared to conventional airplanes. But in which extend ?

The Generic Aircraft Model (GAM) can bring some elements to answer this question. In the model, both L/D and basic MWE (mainly structure + standard airplane systems) are coming from regressions over a full fleet of various 5 The specifications of Alice from EVIATION classical aircraft. They represent some sort of medium technological level. But Alice is clearly out of this. EVIATION engineers have most probably adopted several strategies to compensate for the huge battery weight. Increasing L/D can be obtained by increasing wing aspect ratio and smoothing all wetted surfaces. Reducing MWE can be made thanks to a wide use of CFRP together with clever structural optimization and system selection.

In the GAM model, the cruise L/D can be set directly. An improvement factor of the MWE can be added. Consequently, we have two unknowns to figure out. In the following we add a design constraint : the claimed MTOW of 8350 kg is most probably ceiled by the maximum weight allowed for CS23 certification which is 8600 kg, with a margin of 250 kg. If we consider a MTOW of 8350 kg as a target for the design, this new constraint gives a relationship between the required L/D and the improvement factor on MWE. The resultinf compromise is shown on Fig. 14.

To perform this calculation, the cruise speed has been set to 380 km/h and we fixed the passenger mass allowance to reach the given payload with 9 passengers on board. Let's note that one can change any of the parameters to match his own context. The resulting graph is presented in Fig. 14 for three values of the battery energy density. We can see that a 15% gain on the basic MWE requires an L/D of about 27, which is challenging for a commuter aircraft but achievable. A rapid analysis with the aerodynamic polar estimation method presented in [START_REF] Raymer | Aircraft Design: A Conceptual Approach[END_REF][START_REF] Rebuffet | Aérodynamique expérimentale : Cours professé à l'Ecole Nationale Supérieure de l'Aéronautique[END_REF] leads to an L/D of about 27.9. In the following analysis, we consider a 15% gain on the basic MWE and an L/D of 27 for the Alice airplane.

In the future, the required energy and the amount of materials is likely to become a major driver of sustainability. The required amount of materials and energy to offer an airborne transport service can be put in perspective to the PK product of a specific airplane design (the nominal number of passengers times the design travel distance). We can form two indicators: PK/E and PK/OWE, which can be respectively interpreted as the energetic efficiency and the structural efficiency of a given airplane flying a given route.

According to these criteria, we can compare the efficiencies of Alice with other aircraft. Fig. 15 presents the position of Alice regarding these efficiencies, for three different values of the battery energy density (250, 300, and 350 Wh/kg). The analysis is based on the nominal mission of each airplane to compute PK/E. The total energy stored in the kerosene for petrol airplane is set to 43.1 MJ/kg in our analysis . Fig. 15 is interesting because it clearly shows the tendencies of these efficiencies regarding the different categories of airplanes. We can see that Alice is competing with petrol commuters regarding energetic efficiency but is far less efficient regarding material efficiency. The physical characteristics of the battery-based electric power chain are responsible for that. We can also see that an energy density of 350 Wh/kg would push the airplane to the top level of energetic efficiency without entering the cloud for material efficiency. Of course, the use of PK/OWE as an indicator of material efficiency does not consider the type of material, which is critical when it comes to the batteries. However, our analysis, at least, gives an overview concerning the total amount of materials regardless of their types.

We also investigate how the PK/E evolves if we move the Capacity vs Range for Alice design. Fig. 16 displays this evolution of PK/E. The yellow frontier corresponds to MTOW limit of 8600 kg related to CS23 certification.

Fig. 16 Transport energy efficiency at Design Fig. 17 Transport energy efficiency in Operation

An interesting element of Fig. 16 is the existence of an optimum range related to the energetic efficiency. The optimum range lies on about 250 km for a capacity of 9 passengers and slightly increases with the capacity. This implies that EVIATION company put priority on the maximum possible range. The existence of this optimum can be explained as follows. For ranges lower than the optimal one, the PK product increases more rapidly than the required mission energy, while the required mission energy increases more rapidly than the PK product for ranges higher than the optimal one.

Let's keep in mind this optimum range corresponds to the design mission (each point in Figure xxx corresponds to a different aircraft, designed with the related capacity and range). If we consider now Alice in an of design context, the evolution of mission PK/E is driven by on-board passengers and range as shown in Fig. 17 Interestingly, the evolution of design mission PK/OWE also reveals an optimum range about 320 km, which is longer than that of PK/E. This optimum has the same trend as the optimum range from the PK/E. For ranges lower than the optimal one, the PK product increases more rapidly than the required empty weight, while the required empty weight increases more rapidly than the PK product for ranges higher than the optimal one. Fig. 19 displays this frontier for a capacity of 9 passengers. To be consistent with the classic representation of the Pareto front where PK/E and PK/OWE are supposed to be independently minimized, we take the inverse of previous indicators, which are represented as E/Pk and OWE/PK in the figure .  According to the Pareto front, a good compromise between energetic and material efficiency could be at about 300 km for the design range. The anchor point for material efficiency at 320 km, however, would not be a bad choice as it makes the corresponding range a relevant indicator for the global efficiency of the design.

If now, we consider the range that optimizes the material efficiency PK/OWE, we can compute the evolution of this maximum when the battery energy density varies. Fig. 20 shows, in green color, the variation of the best range while varying the energy density of the battery. Blue lines refer to the optimum range if we accept a variation of ±10% around the optimal PK/OWE. We can see that , ranges approximately between 750 and 1000 km would be achievable with an energy density of about 400 Wh/kg, which is claimed to be reachable within a decade [citation].

As shown in the presentation of GAM model, an Alice airplane model can be set up as follows:

Power system Design mission energy type : battery category : commuter engine count :2 payload : 9 engine type : emotor speed : 380 km/h thruster type : propeller range : 460 km bpr : None altitude : 3048 m Table 6 Step 1: Characteristic of Alice airplane This study illustrates a possible application of the GAM model to identify some main characteristics of an aircraft from limited initial information. Here, we focuse on a full electric commuter from EVIATION, and we evaluate the required level of lift to drag ration, 𝐿/𝐷, and structural alleviation that allow to fly with 9 passengers over the maximum Specific data maximum fuel factor (max fuel factor) : 1 battery energy density (battery enrg density) : 250Wh/kg stdm factor: 0.84 lift-to-drag (lod) : 27 7 possible range (460 km) according to the maximum take off weight allowed in CS23.

B. Use case 2

Potential propulsion technologies for a 19-passenger commuter are listed as follows:

1) Electric motor with battery 2) Electric motor with hydrogen fuel cell 3) Turboshaft burning hydrogen 4) Turboshaft burning methane 5) Turboshaft burning kerosene Except for turboshaft burning kerosene, which is considered mature, all other propulsion technologies present at least one critical characteristic that would require some improvements listed as follows:

• Battery energy density (currently ≈ 250 Wh/kg) • 1000 V barrier for electric motor (currently ≈ 700 to 800 V limiting the power to ≈ 1 to 2 MW)

• Fuel cell efficiency (currently ≈ 50% to 60% max resulting in huge heat power to dissipate)

• Fuel cell overall power density (currently ≈ 1kW/kg due to a huge balance of plant)

• Gravimetric index of liquid hydrogen & methane tanks (currently respectively ≈ 0.1 and 0.2)

• Cryogenic fuel management (20K for liquid hydrogen and 111K for liquid methane)

The most important evolution in the near future will be on the battery energy density, fuel cell power density, and cryogenic tanks' gravimetric index. Some companies (like HyPoint) are claiming that they can produce tanks with the gravimetric index of 0.5 (without a balance of plant), and a lot of research have been undertaken to reach a battery energy density of about 500 Wh/kg [START_REF] Hepperle | Electric Flight -Potential and Limitations[END_REF]. Concerning fuel cell integration, a more compact balance of plant should bring significant improvement of fuel cell energy density at system level.

The GAM model allows a quick assessment of the possible benefit in terms of airplane performances from the evolutions of technological efficiencies. Our focus is on CS23 commuters (capacity lower or equal to 19 passengers). We first look for the maximum capacity that can be obtained at the range that maximizes the criterion PK/OWE. For this study, we apply the same technological improvements that we found from the Alice airplane (L/D = 27, Basic 𝑀𝑊 𝐸 factor = 0.85) to the battery airplanes. For all other airplanes, we let the standard estimations performed by the model, e.g. L/D from regression as presented in Section III.C.2 , and mass aleviation factor = 1. These assumptions implies that airplane shapes and aerodynamic cleanliness are more driven by the manufacturing and production costs of classical metallic structures, which can be seen as a bit conservative.

As mentionned in the use case I presented in Section V.A, we considered the optimum range according to PK/OWE as an indicator of the efficiency of a given technology. Fig. 21 shows these optimal ranges together with the maximum possible passenger capacity (MTOW limited at 8600kg), for different values of battery energy densities and cryogenic tank gravimetric index. It is noted that a gravimetric index of 0.2 for hydrogen corresponds to 0.6 for methane and 0.4 for hydrogen corresponds to 0.8 for methane. We can get at least four interesting insights from the figure, which are listed as follows:

• Good performance of battery-powered airplanes regarding the maximum passenger capacity, provided the energy density of batteries can be pushed to 500 Wh/kg. This is due to the improvement in L/D and MWE versus other technologies.

• Impossibility for state of the art fuel cell airplanes to fly 19 passengers, even if the tank gravimetric index reaches its highest value. • Except for battery-powered airplanes, all other technologies reveal optimal ranges higher or equal to 1500 km. Let's keep in mind that the presented ranges are the ones that optimize the PK/OWE of each propulsive technology. All types of airplanes can be designed for shorter ranges, but with lower PK/OWE. Table 9 gives the main figures for the different concepts, and Fig. 22 Let's note that the efficiency (Eff) in the table corresponds to the ratio of propulsive power (Thrust . Speed) over primary power (Fuel flow . Fuel Heating Value or power sucked from the battery). • When fuel cell energy density reaches about 2 kW/kg, hydrogen fuel cell-powered airplanes are showing similar ranges as turboprop burning hydrogen. • We observe the very good performance of hydrogen-fueled thermal technology, which shows a potential even higher than petrol, provided the tank gravimetric could reach about 0.4 kg of H2 per kg of H2+tank. • We also observe the very good performance of methane-powered airplanes, even with a tank gravimetric of about 0.6 kg of CH4 per kg of CH4+tank (which corresponds to 0.2 for hydrogen). If we plot all these concepts in a graph PK/OWE -PK/E, we obtain Fig. 23. We observe that:

• Reaching a cryogenic tank gravimetric index of about 0.4 for hydrogen and 0.8 for methane would put these technologies at the same level of efficiency as petrol. • Both TGI of 0.4 or power density of 2 kW/kg are not able, individually, to unlock the fuel cell technology. If combined, these evolutions could push the fuel cell technology to a high level of energetic efficiency and a certain level of material efficiency. • According to PK/E, the high efficiency of battery airplanes is partially due to the L/D and MWE improvements. If those improvements were applied to other propulsive technologies than full electric, they would bring improvements as well. Fig. 24 shows the transport efficiencies of the studied configurations in comparison with those coming from the airplanes in the data base.

Fig. 24 Transport efficiencies vs database

We observe that:

• The energetic efficiency of battery airplanes (1 and 2), enhanced by structural and aerodynamic improvements, are far above the cloud of point, whatever the aircraft category. • A global gravimetric energy density of fuel cell systems of about 2 kW/kg pushes the fuel cell-powered airplanes on top of the cloud of points (5 and 6). • Only thermal configurations with highly efficient tanks (8 and 9) are able to compete with the material efficiency of petrol-powered airplanes. Main characteristics of airplanes designed for a nominal range of 1000 km (when possible) are presented in Table 10.

• Battery airplanes as well as state of the art fuels cell concepts are not able to reach the required distance. An energy density of 500 Wh/kg is able to provide a range. • Material efficiency (PK/OWE) is reduced due to the range decrease in comparison to the values in Table 9 The presented study gives some inside view into the potential, in terms of range and capacity, of different propulsion technologies in the commuter segment. Studied propulsion systems were:

1) Electric motor with battery 2) Electric motor with hydrogen fuel cell 3) Turboshaft burning hydrogen 4) Turboshaft burning methane 5) Turboshaft burning kerosene The study gives a global overview of the different technologies according to energy efficiency (PK/E), material efficiency (PK/OWE), maximum capacity, and maximum range. It enlightened the high potential of fuel cell-powered airplanes • At TGI of 0.7 would put the hydrogen airplane in a very good position in the efficiency domain, better than most of the existing long ranger. Even very simple model presented in this study can reveal an already-known result about the domain of applicability of hydrogen-based thermal propulsion. This result gives additional evidence of the applicability of our model.

VI. Conclusion

This paper introduced a simple and very fast model of airplane design and operation. This model allows one to capture some important characteristics of an airplane, such as design weights, range, capacity, speed, fuel, or energy consumption, without the need to define the geometry of the aircraft. The model can simulate various propulsive technologies based on electric motors, batteries, fuel cells, or thermal engines burning kerosene, methane, or hydrogen. The global technological indices proposed in this work enable modeling such propulsion technologies with easy adjustment towards the technical evolution or forecasts. Three possible applications have been presented. The first one allows the identification of the technical challenges in the Alice airplane from the company called EVIATION. The second one has presented an overview of the transport potential, in terms of range and passenger capacity, of different propulsive technologies applied to the 19-passenger, CS23, commuter segments. The last application allows one to identify the technological limit of thermal hydrogen-powered airplanes in relation to the maturity of the components of the propulsive system. All the models are available open source according to the CeCILL-C Free Software License Agreement on a git repository * * * * https://gitlab.com/m6029/genericairplanemodel.git commit number cc5683feff706f483e3632171feda4ad0df8edfc

Appendix C

In the following we assume that • the technologies for the cryogenic tank of liquid hydrogen and liquid methane are the same.

• the stored volume of fuel (liquid methane or hydrogen) is the same.

1 𝑔

 1 Finally, three use cases are presented. The first one allowed us to identify the level of technological challenge of the Alice aircraft project from EVIATION. The second one is a short mapping of the transportation potential of five different propulsion systems for the segment of 19-passenger commuter airplanes. The third use case puts the thermal hydrogen aircraft in the perspective of existing kerosene airplanes in terms of energy and structural efficiency, and reveals an optimum of the hydrogen technology. Nomenclature 𝑀𝑇𝑂𝑊 = Maximum takeoff weight 𝑂𝑊 𝐸 = Operating weight empty 𝑀𝑊 𝐸 = Manufacturer's weight empty 𝑆𝐹𝐶 = Specific fuel consumption (kg/s/N) 𝐿/𝐷 = Lift to drag ratio 𝐹 = Thrust 𝑉 = Velocity, true airspeed 𝑊 = Power 𝐸 = Energy 𝑃𝑆𝐹𝐶 = Power related specific fuel consumption 𝑇 𝑆𝐹𝐶 = Thrust related specific fuel consumption 𝑓 𝑓 = Mass fuel flow (kg/s) 𝑓 ℎ𝑣 = Fuel heating value 𝐾 𝑟 = Fuel reserve factor, proportion of the mission fuel for reserve 𝑉 0 = Airplane speed, true airspeed 𝑉 𝑎 𝑝 𝑝 = Approach speed 𝑉 2𝑚𝑖𝑛 = Minimum speed at 35ft for takeoff 𝑉 𝑠1𝑔 = Stall speed 𝑛 𝑒 = Number of engines of the airplane 𝑟 = Fuel ratio 𝜂 𝑓 = 0.82 : Fan efficiency 𝑊 𝑠ℎ𝑎 𝑓 𝑡 𝜂 𝑒𝑙𝑒𝑐 = Power efficiency of the electric motor, including its controller 𝑊 𝑠ℎ𝑎 𝑓 𝑡 𝑊 𝑒𝑙𝑒𝑐 𝜂 𝑓 𝑐𝑠 = Global power efficiency of the fuel cell system, including balance of plant 𝑊 𝑒𝑙𝑒𝑐 𝑓 𝑓 × 𝑓 ℎ𝑣 𝑀 𝑏𝑎𝑡𝑡𝑒𝑟 𝑦 = Battery mass 𝑏𝑒𝑑 = Battery energy density 𝑚 = Airflow 𝑔𝑖 = Gravimetric index 𝑃 𝑖𝑛𝑑𝑒𝑥 = Performance index 𝑇𝐺 𝐼 = Gravimetric index of liquid hydrogen tanks 𝑃 𝑖 = Initial internal pressure of the tank 𝑉 𝑖 = Internal volume of the tank 𝑀 𝑡 𝑎𝑛𝑘 = Mass of the empty tank 𝐺 𝑖𝑛𝑑𝑒𝑥 = Tank gravimetric index 𝜌 𝑔𝑎𝑠 = Gas density at 𝑃 𝑖 𝜂 𝑝𝑤 = Power efficiency of the engine 1 𝑝𝑠 𝑓 𝑐 × 𝑓 ℎ𝑣 = 𝑊 𝑠ℎ𝑎 𝑓 𝑡 𝑓 𝑓 × 𝑓 ℎ𝑣 𝜂 𝑡 ℎ = Thermal efficiency 𝜂 𝑝𝑟 = Propulsive efficiency 𝑀𝑖𝑠𝑠𝑖𝑜𝑛𝐹𝑢𝑒𝑙 = The fuel or energy required at nominal range

Fig. 1

 1 Fig. 1 Correlation between number of passenger and furnishing weight. Data from ref. [28, 29].

  𝑝𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛 = 𝑊 𝑒𝑛𝑔𝑖𝑛𝑒 + 𝑊 𝑡 ℎ𝑟𝑢𝑠𝑡𝑒𝑟 , where 𝑊 𝑒𝑛𝑔𝑖𝑛𝑒 = 𝐺𝑃𝐼 𝐺 𝐼 𝑖 and 𝑊 𝑡 ℎ𝑟𝑢𝑠𝑡𝑒𝑟 = 𝐺𝑃𝐼 𝐺 𝐼 𝑡 𝑗 .

Fig. 3

 3 Fig. 3 Correlation between GPI and turbofan mass Fig. 4 Correlation between GPI and turboshaft mass

Fig. 6 Fig. 7

 67 Fig. 6 Correlation between MTOW and OWE

Fig. 10

 10 Fig. 10 Comparison between estimated MTOW and published MTOW Fig. 11 Relative error distribution

Fig. 14

 14 Fig. 14 Trade : Aero-Structure Fig. 15 Transport efficiencies

  .

Fig. 18

 18 Fig. 18 displays this evolution of PK/OWE by varying capacity and range.Interestingly, the evolution of design mission PK/OWE also reveals an optimum range about 320 km, which is longer than that of PK/E. This optimum has the same trend as the optimum range from the PK/E. For ranges lower than the optimal one, the PK product increases more rapidly than the required empty weight, while the required empty weight increases more rapidly than the PK product for ranges higher than the optimal one. Fig.19displays this frontier for a capacity of 9 passengers. To be consistent with the classic representation of the Pareto front where PK/E and PK/OWE are supposed to be independently minimized, we take the inverse of previous indicators, which are represented as E/Pk and OWE/PK in the figure.According to the Pareto front, a good compromise between energetic and material efficiency could be at about 300 km for the design range. The anchor point for material efficiency at 320 km, however, would not be a bad choice as it makes the corresponding range a relevant indicator for the global efficiency of the design.

Fig. 18

 18 Fig. 18 Transport energy efficiency at Design

Fig. 22

 22 Fig. 22 Maximum range for CS23 Fig. 23 Transport efficiencies

Fig. 25

 25 Fig. 25 Transport efficiencies map. Small dots stand for for kerosene aircraft in the database. Large blue dots stand for an aircraft with a liquid-hydrogen tank, with different gravimetric index and design range.

( 1 +

 1 𝑡 𝜂 𝑝𝑟 𝑓 ℎ𝑣 (B.9) (B.7)&(B.9) =⇒ 𝑆𝐹𝐶 = 2𝑉 0 + 𝛿𝑉 2𝜂 𝑡 𝑓 ℎ𝑣 (B.10) (B.5)&(B.6) =⇒ 𝐹 = 𝑚 𝛼 𝑏 𝑝𝑟)𝛿𝑉 (B.11) (B.8)&(B.11) =⇒ 𝛿𝑉 = 𝛼 𝑆𝐹𝐶 (1 + 𝑏 𝑝𝑟) (B.12) (B.10)&(B.11) =⇒ 2𝜂 𝑡 𝐹𝐻𝑉 × 𝑆𝐹𝐶 = 2𝑉 0 + 𝛼 𝑆𝐹𝐶 (1 + 𝑏 𝑝𝑟) (B.13)Finally, we have(B.13) ⇐⇒ 2𝜂 𝑡 (1 + 𝑏 𝑝𝑟) 𝑓 ℎ𝑣 × 𝑆𝐹𝐶 2 -2𝑉 0 𝑆𝐹𝐶 (1 + 𝑏 𝑝𝑟) -𝛼 = 0 (B.14) 𝑆𝐹𝐶 2 -𝑉 0 𝜂 𝑡 𝑓 ℎ𝑣 𝑆𝐹𝐶 -𝛼 2𝜂 𝑡 𝑓 ℎ𝑣(1 + 𝑏 𝑝𝑟) = 0 (B.15)By solving the second-degree polynomial in (B.15), we get :

  = 𝐵𝑎𝑠𝑖𝑐 𝑀𝑊 𝐸 + 𝑂𝑊 𝐸 𝑃𝑎𝑦𝑙𝑜𝑎𝑑 + 𝑂𝑊 𝐸 𝑃𝑟 𝑜 𝑝𝑢𝑙𝑠𝑖𝑜𝑛

	Category	𝑀 𝑝𝑎𝑥 (kg/pax) 𝑀 𝑓 𝑢𝑟 (kg/pax)
	General	95	18
	Commuter	105	18
	Regional	110	22
	Short-medium	115	22
	Long-range	120	30
	Table
			(6)
	where

Table 2 Fuel lower heating value used in this study.

 2

		FHV (MJ/kg)
	Kerosene	43.1
	Hydrogene	121
	Methane	50.3

Table 7 Step 2 : Add specific data to the Alice airplane

 7

	Distance	460 km
	Take Off Weight 8350 kg
	Payload	1125 kg
	Total energy	862 kWh
	Mission energy 638 kWh
	Reserve energy 224 kWh

Table 8 Results of Alice model considering the characteristics in Table 6 and Table

 8

Table 9 Aircraft characteristics for fixed MTOW

 9 illustrates the maximum ranges.

	Name	Range	L/D	Eff OWE MTOW PK/OWE PK/E
	1 Battery: 250 Wh/kg	185	27.5 0.72 6605	8600	0.53	5.62
	2 Battery: 500 Wh/kg	694	27.5 0.72 6605	8600	2 10.74
	4 FC H2: gi=0.4 pwd=1kW/kg	122 13.63 0.36 6548	8600	0.35	1.36
	5 FC H2: gi=0.2 pwd=2kW/kg	1093 13.63 0.36 6412	8600	3.24	3.31
	6 FC H2: gi=0.4 pwd=2kW/kg	2192 13.63 0.36 6260	8600	6.65	3.66
	7 Thermal H2: gi=0.2	1388 13.63	0.2 6205	8600	4.25	1.99
	8 Thermal H2: gi=0.4	2859 13.63	0.2 5861	8600	9.27	2.19
	9 Thermal CH4: gi=0.60	1839 13.63	0.2 5433	8600	6.43	2.16
	10 Thermal CH4: gi=0.80	2509 13.63	0.2 5091	8600	9.36	2.28
	11 Thermal Petrol	2745 13.63	0.2 4733	8600	11.02	2.35

  𝛼𝜂 𝑡 𝑓 ℎ𝑣 2𝑉 2 0 (1 + 𝑏 𝑝𝑟) , consider P 𝑐𝑖𝑛 = 1 2 𝑚 𝑎 (𝑉 2 𝑗 -𝑉 2 0 ) with 𝑉 𝑗 defined as in (A.B.1). 𝑚 𝑎 = 𝑚 𝑐 + 𝑚 𝑓 (B.1) =⇒ 𝑚 𝑎 = 𝑚 𝑐 (1 + 𝑏 𝑝𝑟) (B.4) (B.3)&(B.4) =⇒ 𝑚 𝑎 = 𝐹 = 𝑚 𝑎 𝛿𝑉 with 𝛿𝑉 = 𝑉 𝑗 -𝑉 0 (B.6)

	𝑏 𝑝𝑟 =	𝑚 𝑓 𝑚 𝑐				(B.2)
	𝑚 = 𝛼 𝑚 𝑐				(B.3)
	(B.1)&(B.2) 𝑚 𝛼	(1 + 𝑏 𝑝𝑟)		(B.5)
	𝜂 𝑝𝑟 =	𝐹𝑉 0 P 𝑐𝑖𝑛	=	𝑚 𝑎 𝛿𝑉 × 𝑉 0 2 𝑚 𝑎 (𝑉 2 1 𝑗 -𝑉 2 0 )	=	1 1 + 𝛿𝑉

  The gravimetric indices of liquid hydrogen and liquid methane, 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2 and 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 are expressed as follows.𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2 = 𝑀 𝐿𝐻2 𝑀 𝐿𝐻2 + 𝑀 𝑡 𝑎𝑛𝑘 𝐿𝐻2 and 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 = 𝑀 𝐿𝐶 𝐻 4 𝑀 𝐿𝐶 𝐻 4 + 𝑀 𝑡 𝑎𝑛𝑘 𝐿𝐶 𝐻 4For a given volume of fuel, and for a given cryogenic tank technology, the above equations can be rearranged as follows.=𝜌 𝐿𝐻2 𝜌 𝐿𝐶 𝐻 4 and 𝑀 𝑡 𝑎𝑛𝑘 𝐿𝐶 𝐻 4 = 𝑀 𝑡 𝑎𝑛𝑘 𝐿𝐻2 = 𝑀 𝑡 If we divide (C.2) by (C.1), we obtain the following expression.

	𝑀 𝐿𝐻2	=	𝑀 𝐿𝐶 𝐻 4	=⇒	𝑀 𝐿𝐻2
	𝜌 𝐿𝐻2		𝜌 𝐿𝐶 𝐻 4			𝑀 𝐿𝐶 𝐻 4
				𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2 =	1 1 + 𝑀 𝑡 𝑀 𝐿𝐻2	=⇒	𝑀 𝑡 𝑀 𝐿𝐻2	=	1 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2	-1	(C.1)
				𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 =	1 1 + 𝑀 𝑡 𝑀 𝐿𝐶 𝐻 4	=⇒	𝑀 𝑡 𝑀 𝐿𝐶 𝐻 4	=	1 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4	-1	(C.2)
	𝑀 𝐿𝐻2 𝑀 𝐿𝐶 𝐻 4	=	1 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 1 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2	-1 -1	=	𝜌 𝐿𝐻2 𝜌 𝐿𝐶 𝐻 4	=⇒ 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐶 𝐻 4 =	1 +	1 𝐺 𝑖𝑛𝑑𝑒𝑥 𝐿𝐻2 1	-1 𝜌 𝐿𝐻2

𝜌 𝐿𝐶 𝐻 4

¶ https://gitlab.com/m6029/genericairplanemodel.git

‖ https://www.eviation.com/aircraft/ accessed on November 2023

provided the required level of maturity for fuel cell power density and cryogenic tanks gravimetric index. The study has shown the interest of methane-powered airplanes, provided this methane is coming from renewable sources.

C. Hydrogen powered airplane efficiency

A crucial question for hydrogen as an aviation fuel is the range capability of hydrogen-powered airplanes in comparison to petrol-based one. Two aspects of the hydrogen technology are particularly critical:

• The weight of the tank ( not the weight of fuel)

• The required volume to store the fuel In this study, the proposed model can represent the tank weight, but not its volume. Still, it can give a trend of the hydrogen airplane energetic efficiency. This parameter is measured by 𝑃𝐾/𝐸, passenger-kilometer per fuel energy reacquired for the design mission.

First a reference kerosene aircraft is defined:

Power system Design mission energy type : kerosene category : long range engine count :2 payload : 350 engine type : turbofan speed : 875 km/h thruster type : fan range : 14000 km bpr : 12.5 altitude : 35000 ft Table 11 Requirements for the reference long-range keresosene aircraft.

Then, with the same set of requirements, the energy type is set to liquid hydrogen. The design range is varied from 4000 km to 14000 km. In all cases, the gravimetric index of liquid hydrogen tanks (TGI) has also been varied from 0.4 to 0.7, but the actual gravimetric index is more likely equal to 0.2 [START_REF] Xisto | Design and performance of liquid hydrogen fuelled aircraft for year 2050 EIS[END_REF]. According to the company HyPoint, 0.4 (including the balance of plant) is already a challenging value. 0.7 appears in some theoretical publications as a feasible proposition for an entry into service in 12-15 years [START_REF] Huete | Impact of tank gravimetric efficiency on propulsion system integration for a first-generation hydrogen civil airliner[END_REF]. The results are shown on Fig. 25.

• The PK/OWE criterion seems to split the global fleet into three domains:

-General, commuter and some regional airplanes, below 10 pax.km/kg (green, yellow and orange dots) -Short and medium range airplanes, between 10 and 20 pax.km/kg (purple dots) -Long range airplanes, beyond 20 pax.km/kg (red dots) • For a Tank Gravimetric Index equals to 0.4 (TGI =0.4), liquid hydrogen-powered airplanes can not escape from the short-medium range domain. • The range of optimal PK/OWE evolves from about 7000km (for TGI=0.4) to about 14000km (for TGI=0.7).

Additionally, we found that a TGI higher than 0.5 is necessary to penetrate into the efficiency domain of long range kerosene aircraft.

B. Demonstration

The following derivation is used in Section .B for turbofan engines. First, the thrust 𝐹 is expressed as a function of the exhaust speeds in the engine core and engine fan ? Kinetic power P 𝑐𝑖𝑛 .

The relative error between 𝑉 𝑗 and 𝑉 𝑗 ′ is evaluated using the difference between 𝑉 𝑗 and 𝑉 𝑗 ′ , as presented as follows.

𝑎

Hence, the relative error between 𝑉 𝑗 and 𝑉 𝑗 ′ is expressed as follows.

𝑉

Appendix B

To demonstrate that 𝜂 𝑝𝑟 =


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