The number of ice crystals formed in nascent contrails strongly influences contrail-cirrus life cycle and radiative forcing. Previous studies on contrails from hydrogen combustion focused on microphysical processes that affect the ice crystal number. These studies, however, paid less attention to engine-related aspects. To fill this gap, we investigate how the exhaust plume evolution is thermodynamically influenced by (i) the engine's overall propulsion efficiency, (ii) the engine exit conditions due to varying ambient conditions, (iii) the engine size and exit jet speed, and (iv) the explicit treatment of kinetic energy dissipation and entrainment of enthalpy initially contained in the bypass flow of a turbofan engine. Based on simulations with the box model version of the Lagrangian Cloud Module, we investigate how these aspects influence the contrail formation process and derive suitable (scaling) relations for the number of ice crystals formed <em>N</em>_ice,fon entrained ambient aerosols for hydrogen combustion. These relations help to derive an expression of <em>N</em>_ice,fthrough a functional relationship that relies on a reduced set of input parameters, while ensuring a generalized parametrization of <em>N</em>_ice,fin contrails from hydrogen combustion.