Efficient Modeling and Numerical Approximation of Thermodynamic Compressible Fluid Flow in Energy Networks
Core Concepts
The authors introduce an infinitedimensional portHamiltonian (pH) formulation of the compressible nonisothermal Euler equations to model temperaturedependent fluid flow in energy networks. They establish the underlying StokesDirac structure, derive the boundary port variables, and incorporate energypreserving coupling conditions to enable structurepreserving coupling of the fluid flow system with other network components.
Abstract
The key highlights and insights of the content are:

The authors introduce an infinitedimensional portHamiltonian (pH) formulation of the compressible nonisothermal Euler equations to model temperaturedependent fluid flow in energy networks. This allows for efficient coupling of the fluid flow system with other network components, as the pH framework encodes physical properties in the system structure and enables easy coupling of systems running on different time scales.

The authors derive the underlying StokesDirac structure and the boundary port variables for the nonisothermal Euler equations. The boundary port variables represent the mass flow and total specific enthalpy at the pipe ends, enabling the incorporation of energypreserving coupling conditions.

The authors incorporate energypreserving coupling conditions, such as mass conservation and equality of total enthalpy, into the pH formulation in a structurepreserving way. This ensures that the coupled network system remains in pH form.

The authors adapt structurepreserving numerical approximation methods from the isothermal to the nonisothermal case, including space discretization, model order reduction, and complexity reduction. These methods preserve the pH structure and are tailored to the thermodynamics of the system.

Numerical examples demonstrate the influence of the interconnection operator on model order reduction and compare pipewise and networkwise reduction of the gas networks.
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On a portHamiltonian formulation and structurepreserving numerical approximations for thermodynamic compressible fluid flow
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The authors use the following key figures and metrics to support their work:
The compressible nonisothermal Euler equations, which depend on the mass density ρ, velocity v, and internal energy density e.
The ideal gas laws, relating pressure p, temperature T, density ρ, internal energy ϵ, and total energy E.
The pipe diameter d, friction factor λ, and thermal conductivity coefficient kω.
The ambient temperature T∞.
Quotes
"The portHamiltonian (pH) modeling framework has lately been widely used in the modeling of energy networks, as it has various advantages. As energy is used as a lingua franca, it brings the different scales on a single level, e.g., gas, power and district heating. This makes the coupling of these individual systems easier."
"Furthermore, when using energy preserving coupling the pH character is inherited during the coupling of the individual systems."
Deeper Inquiries
How can the proposed pH formulation and structurepreserving numerical methods be extended to model more complex fluid flow phenomena, such as turbulence or multiphase flow, in energy networks
The proposed pH formulation and structurepreserving numerical methods can be extended to model more complex fluid flow phenomena, such as turbulence or multiphase flow, in energy networks by incorporating additional terms and operators in the governing equations. For turbulence modeling, the NavierStokes equations can be modified to include turbulence models like Reynoldsaveraged NavierStokes (RANS) or Large Eddy Simulation (LES) terms. These terms account for the turbulent fluctuations in the flow field and can be integrated into the pH framework to capture the effects of turbulence on energy transport and conversion processes.
In the case of multiphase flow, the pH formulation can be extended to include additional phases such as liquid or solid particles along with the compressible fluid phase. This extension would involve introducing new state variables, equations of state, and coupling conditions to describe the interactions between different phases. By incorporating multiphase flow models like the EulerEuler or EulerLagrangian approaches, the pH framework can simulate the behavior of complex multiphase systems in energy networks.
The numerical methods used for solving the extended pH formulations would need to be adapted to handle the increased complexity of the models. This may involve developing specialized algorithms for solving the coupled equations, implementing efficient solvers for the additional terms, and validating the numerical schemes against experimental or computational data for turbulent or multiphase flows.
What are the potential challenges and limitations of the pH framework in handling nonlinear couplings or nonlocal interactions between different network components
The pH framework may face challenges and limitations when handling nonlinear couplings or nonlocal interactions between different network components in energy systems. One challenge is the complexity of modeling nonlinear interactions, especially in systems with feedback loops or highly nonlinear behavior. Ensuring the stability and convergence of the numerical methods in the presence of strong nonlinearities can be a significant challenge.
Another limitation is the scalability of the pH framework to largescale energy networks with numerous interconnected components. As the number of network nodes and edges increases, the computational complexity of solving the coupled pH equations grows, requiring efficient parallelization strategies and model reduction techniques to maintain computational tractability.
Handling nonlocal interactions, where the effects of one component propagate over long distances in the network, can also pose challenges. Incorporating delayed or distributed interactions into the pH formulation may require advanced modeling techniques and numerical algorithms to accurately capture the dynamics of the system.
Overall, while the pH framework offers advantages in preserving the structure and energy properties of network systems, addressing nonlinear couplings and nonlocal interactions effectively requires careful consideration of the model assumptions, numerical methods, and computational resources available.
Given the importance of hydrogen in the energy transition, how could the presented approach be adapted to model the flow of hydrogennatural gas mixtures, accounting for the different thermodynamic properties of hydrogen compared to natural gas
To adapt the presented approach to model the flow of hydrogennatural gas mixtures in energy networks, several modifications and considerations need to be made. Since hydrogen and natural gas have different thermodynamic properties, including different equations of state and energy conversion characteristics, the pH formulation would need to account for these differences.
One approach would be to extend the pH formulation to include separate state variables and equations for hydrogen and natural gas components in the network. This would involve defining specific Hamiltonians, effort variables, and coupling conditions for each gas type to accurately represent their thermodynamic behavior and interactions within the network.
Additionally, the boundary conditions and external inputs in the pH formulation would need to be tailored to reflect the properties of hydrogennatural gas mixtures, such as varying compositions, pressures, and temperatures. By incorporating the specific thermodynamic properties of hydrogen and natural gas into the pH framework, the model can simulate the flow dynamics and energy transport in networks with mixed gas components effectively.