On a port-Hamiltonian formulation and structure-preserving numerical approximations for thermodynamic compressible fluid flow

The high volatility of renewable energies calls for more energy efficiency. Thus, different physical systems need to be coupled efficiently although they run on various time scales. Here, the port-Hamiltonian (pH) modeling framework comes into play as it has several advantages, e.g., physical properties are encoded in the system structure and systems running on different time scales can be coupled easily. Additionally, pH systems coupled by energy-preserving conditions are still pH. Furthermore, in the energy transition hydrogen becomes an important player and unlike in natural gas, its temperature-dependence is of importance. Thus, we introduce an infinite dimensional pH formulation of the compressible non-isothermal Euler equations to model flow with temperature-dependence. We set up the underlying Stokes-Dirac structure and deduce the boundary port variables. We introduce coupling conditions into our pH formulation, such that the whole network system is pH itself. This is achieved by using energy-preserving coupling conditions, i.e., mass conservation and equality of total enthalpy, at the coupling nodes. Furthermore, to close the system a third coupling condition is needed. Here, equality of the outgoing entropy at coupling nodes is used and included into our systems in a structure-preserving way. Following that, we adapt the structure-preserving aproximation methods from the isothermal to the non-isothermal case. Academic numerical examples will support our analytical findings.