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Steady State Blended Gas Flow on Networks: Existence and Uniqueness of Solutions

Alena Ulke, Michael Schuster, Simone Göttlich

Abstract

We prove an existence result for the steady state flow of gas mixtures on networks. The basis of the model are the physical principles of the isothermal Euler equation, coupling conditions for the flow and pressure, and the mixing of incoming flow at nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical properties of the model allowing us to prove the existence of solutions in particular for tree-shaped networks and networks with exactly one cycle. Numerical examples illustrate the results and explore the applicability of our approach to different network topologies.

Steady State Blended Gas Flow on Networks: Existence and Uniqueness of Solutions

Abstract

We prove an existence result for the steady state flow of gas mixtures on networks. The basis of the model are the physical principles of the isothermal Euler equation, coupling conditions for the flow and pressure, and the mixing of incoming flow at nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical properties of the model allowing us to prove the existence of solutions in particular for tree-shaped networks and networks with exactly one cycle. Numerical examples illustrate the results and explore the applicability of our approach to different network topologies.

Paper Structure

This paper contains 16 sections, 16 theorems, 63 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction and Motivation
  2. The Model Equations on Networks
  3. The Isothermal Euler Equations for Mixtures
  4. Boundary and Coupling Conditions for the Nodes
  5. The Full Model on Graphs
  6. Properties of the Gas Flow
  7. The Existence and Uniqueness of Steady States on Networks
  8. Steady States on Tree-Shaped Networks
  9. Steady States on Networks with One Cycle
  10. Properties of the Flow and the Nodal Composition
  11. Properties of the Functions $H_p$ and $H_{\eta}$
  12. Numerical Examples
  13. A Network with One Cycle
  14. A Network with Multiple cycles
  15. Conclusion, Discussion and Future Work
  16. ...and 1 more sections

Key Result

Lemma 2.4

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a connected graph and let $(\mathcal{P}, \mathcal{E}_\mathcal{P})$ be a path starting at node $v_0$ and ending at node $v_k$ where $\mathcal{P} = \{v_0, \ldots, v_k\}$ and $\mathcal{E}_\mathcal{P}$ contains the edges connecting two consecutive nodes where the node $v_e$ is the node of edge $e$ that is closer to the start node $v_0$ of the path, i.

Figures (13)

  • Figure 1: The mixing of gas for a supply node (left) and a demand node (right). The supply and demand of a node can be seen as an invisible pipe with fixed flow.
  • Figure 2: The four possible relations between flow and edge direction, cf. \ref{['eq: flow vs edge']}.
  • Figure 3: Illustration of a circular flow moving clockwise ,, starting“ from node $v_0$. The edge direction is marked in black and the flow direction in blue.
  • Figure 4: A directed acyclic graph (left) and a topological ordering of its nodes (right).
  • Figure 5: A directed graph $\mathcal{G}$ in black, given flow $q_e$ on the edges in blue (left) and the graph $\mathcal{G}_{\mathrm{flow}}$ with edges according to the flow direction (right).
  • ...and 8 more figures

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Pressure Loss Along a Path
  • proof
  • Corollary 2.5
  • proof
  • Definition 3.1: Topological Ordering Jensen2008
  • Remark 3.2
  • Lemma 3.3: Existence of the Composition
  • ...and 29 more