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On the Existence of Steady States for Blended Gas Flow with Non-Constant Compressibility Factor on Networks

Simone Göttlich, Michael Schuster, Alena Ulke

TL;DR

The paper addresses steady-state transport of hydrogen-natural gas mixtures in pipeline networks where the compressibility factor $Z(\eta,p)$ depends on composition. It develops an implicit pressure representation along pipes and a cycle-cutting continuity approach to handle the lack of explicit pressure-variation formulas for non-constant $Z$, proving existence of steady states on tree networks and on networks with a single cycle (with compressors) under subsonic flow. The key contributions include a rigorous existence proof for a broad class of $Z$, continuity lemmas for flow, mass fraction, and pressure on cut graphs, and a numerical study showing significant model-dependent differences in steady states. The results provide a mathematical foundation for future optimization and control of hydrogen-enriched gas networks under realistic real-gas effects, and highlight the sensitivity of steady states to the chosen compressibility model. $\,$

Abstract

In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states.

On the Existence of Steady States for Blended Gas Flow with Non-Constant Compressibility Factor on Networks

TL;DR

The paper addresses steady-state transport of hydrogen-natural gas mixtures in pipeline networks where the compressibility factor depends on composition. It develops an implicit pressure representation along pipes and a cycle-cutting continuity approach to handle the lack of explicit pressure-variation formulas for non-constant , proving existence of steady states on tree networks and on networks with a single cycle (with compressors) under subsonic flow. The key contributions include a rigorous existence proof for a broad class of , continuity lemmas for flow, mass fraction, and pressure on cut graphs, and a numerical study showing significant model-dependent differences in steady states. The results provide a mathematical foundation for future optimization and control of hydrogen-enriched gas networks under realistic real-gas effects, and highlight the sensitivity of steady states to the chosen compressibility model.

Abstract

In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states.
Paper Structure (13 sections, 13 theorems, 63 equations, 8 figures, 2 tables)

This paper contains 13 sections, 13 theorems, 63 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction and Motivation
  2. Modeling Gas Flow of Mixtures on Networks
  3. Gas Flow through Pipes
  4. Gas Flow across Junctions
  5. Operation of Compressors
  6. Full Gas Flow Model
  7. Existence of Steady-State Solutions
  8. Tree-Shaped Networks
  9. Networks With One Cycle
  10. Continuity of Flow, Mass Fraction, and Pressure
  11. Existence of a Suitable Cut Edge
  12. A Numerical Example on GasLib-11
  13. Conclusion

Key Result

Lemma 2.2

Let $D_F \subseteq \{ (\eta, q, p) \in [0,1] \times \mathbb{R} \times [0,\infty) \mid Z(\eta, p) > 0\} \cap D_{\mathrm{subsonic}}$ and define Then, the function $F$ is differentiable and strictly monotonically increasing on $D_F$.

Figures (8)

  • Figure 1: Simulation results for the compressibility factor models \ref{['eq:compressibilityConstant']} - \ref{['eq:compressibilityQuadratic']} on a single pipe with $L = 50$ km, $\lambda_{\mathrm{fr}} = 0.05$ and $D = 0.5$ m. The gas parameters are given by $R = 8.3145$ J/(mol K)$^{-1}$, $T = 283.15$ K, $p_{c,\text{H}_2} = 13.15$ bar, $p_{c,\text{NG}} = 46.01$ bar, $T_{c,\text{H}_2} = 33.19$ K and $T_{c,\text{NG}} = 204.62$ K.
  • Figure 2: Key idea to prove existence of steady-states, and step at which the argumentation of Ulke2025 for the ideal gas case fails for an arbitrary compressibility factor $Z = Z(\eta, p)$ (highlighted in red).
  • Figure 3: A cut network with the path connecting the node $v_1 = v^\ast$ to the node $v_6 = v$. The path and the numbering are highlighted in red.
  • Figure 4: A gas network with cut edge $e^c$ and $n_{\mathrm{cycle}}=6$ (left), and the corresponding cut graph with nodes and edges of the former cycle being numbered (right).
  • Figure 5: The computation of $\beta_e$ traverses the nodes in ascending order starting at $v_1$ (left), while $\tilde{\beta}_e$ is computed in descending order starting at $v_6$ (right).
  • ...and 3 more figures

Theorems & Definitions (31)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • Theorem 3.1: Existence of Solution for Trees
  • proof
  • Theorem 3.2: Existence of Solution for Networks with one Cycle
  • ...and 21 more