## Appendix C Liquid Hydrogen Thermal Expansion Coefficient Computation  
The LH\({}_{2}\) volumetric thermal expansion coefficient knowledge is necessary to estimate the natural convection inside the liquid phase of the saturated H\({}_{2}\). Unfortunately, this characteristic is not present in the H\({}_{2}\) properties database [36]; however, it is possible to derive it from other properties. The volumetric thermal expansion coefficient \(\beta\) is defined as follows:  
\[V_{2}=V_{1}+\Delta V=V_{1}(1+\beta\,\Delta T)\] (A11)  
where \(V_{1}\) and \(V_{2}\) are the substance volumes before and after a temperature change and \(\Delta T=T_{2}-T_{1}\) is the difference between the final temperature and the initial one. Equation (A11) could be rearranged as in Equation (A12), considering that temperature increase is acting on the same unit mass of fluid ( \(M=\rho_{1}\,V_{1}=\rho_{2}V_{2}\) ). \[\beta=\frac{1}{\Delta T}\bigg{(}\frac{V_{2}}{V_{1}}-1\bigg{)}=\frac{1}{ \Delta T}\bigg{(}\frac{\rho_{1}}{\rho_{2}}-1\bigg{)}\] (A12)  
By applying Equation (A12) between two arbitrary temperatures, it is possible to determine the volumetric thermal expansion coefficient of the saturated LH\({}_{2}\), since the density data are known from [36]. * _[PERSON] et al. (2021)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Innovative box-wing aircraft: Emissions and climate change. _Sustainability_**2021**, _13_, 3282. [CrossRef]
* _[PERSON] (1924)_ [PERSON] _NACA-TN-182. Induced Drag of Multiplanes_; NACA: Cleveland, OH, USA, 1924. * _[PERSON] et al. (2020)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Aerodynamic analysis and optimization of a boxwing architecture for commercial airplanes. In Proceedings of the AIAA SciTech 2020 Forum, Orlando, FL, USA, 6-10 January 2020. [CrossRef]
* _NASA Project Bee (B57-B). 1965. Available online: [[https://www.grc.nasa.gov/WWW/portal/apps/naca/](https://www.grc.nasa.gov/WWW/portal/apps/naca/)]([https://www.grc.nasa.gov/WWW/portal/apps/naca/](https://www.grc.nasa.gov/WWW/portal/apps/naca/)) (accessed on 1 July 2022). * [PERSON] (1957) [PERSON] _NACA-RM-E56K08a. Experimental Study of Foam-Insulated Liquified-Gas Tanks_; NACA: Cleveland, OH, USA, 1957. * [PERSON] (1956) [PERSON] _NACA-RM-E56E14. Turbojet Performance and Operation at High Altitudes with Hydrogen and JP4_; NACA: Cleveland, OH, USA, 1956. * [PERSON] (1955) [PERSON] _NACA-RM-E55F22. Aircraft-Fuel-Tank Design for Liquid Hydrogen_; NACA: Cleveland, OH, USA, 1955. * [PERSON] (1955) [PERSON] _NACA-RM-E55C28a. Liquid Hydrogen as a Jet Fuel for High-Altitude Aircraft_; NACA: Cleveland, OH, USA, 1955. * [PERSON] (1972) [PERSON] _NASA-CR-112204. A Fuel Conservation Study for Transport Aircraft Utilizing Advanced Technology and Hydrogen Fuel_; NASA: Hampton, VA, USA, 1972. * [PERSON] (1975) [PERSON] _NASA-CR-132559. Study of the Application of Hydrogen Fuel to Long Range Subsonic Transport Aircraft_; NASA: Burbank, CA, USA, 1975; Volume 2. * [PERSON] (1974) [PERSON] _NASA-TM-X-71994. A Study of Subsonic Transport Aircraft Configurations Using \(H_{2}\) and \(CH_{4}\)_; NASA: Hampton, VA, USA, 1974. * [PERSON] (1976) [PERSON] _NASA CR-2700. LH2 Airport Requirements Study_; NASA: Burbank, CA, USA, 1976. * _CRYOLANE-Liquid Hydrogen Fuelled Aircraft--System Analysis. Final Technical Report (Published Version), Contract G4 RD-CT-2000-00192. Project GRD1-1999-10014_; Airbus Deutschland GmbH: Hamburg, Germany, 2003. * _[PERSON] et al. (2015)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Final Results of Advanced Cryo-Tanks Research Project CHATT. In Proceedings of the 6 th EUCASS Conference, Krakau, Poland, 29 June-3 July 2015. * _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Modelling and Designing Cryogenic Hydrogen Tasks for Future Aircraft Applications. _Energies_**2018**, _11_, 105. [CrossRef]
* [PERSON] (2009) [PERSON] The Potential of Liquid Hydrogen for Long Range Aircraft Propulsion. Ph.D. Thesis, Cranfield University, Cranfield, UK, 2009.