Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

Shriram Srinivasan, Nishant Panda, Kaarthik Sundar

TL;DR

By giving a complete answer to the question of existence of solutions, this result enables correct diagnosis of algorithmic failure, problem stiffness and non-convergence in computational algorithms.

Abstract

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision-support tools for system design and operation. While the nonlinear system is known to have a unique solution (when one exists), the absence of a definite result on the existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter, we show that for any fluid whose equation of state is a scaled monomial, a unique solution exists for such nonlinear systems if the term solution is interpreted in terms of potentials and flows rather than pressures and flows. However, for gases following the CNGA equation of state, while the question of existence remains open, we construct an alternative system that always has a unique solution and show that the solution to this system is a good approximant of the true solution. The existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, our result enables correct diagnosis of algorithmic failure, problem stiffness, and non-convergence in computational algorithms.

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

TL;DR

By giving a complete answer to the question of existence of solutions, this result enables correct diagnosis of algorithmic failure, problem stiffness and non-convergence in computational algorithms.

Abstract

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision-support tools for system design and operation. While the nonlinear system is known to have a unique solution (when one exists), the absence of a definite result on the existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter, we show that for any fluid whose equation of state is a scaled monomial, a unique solution exists for such nonlinear systems if the term solution is interpreted in terms of potentials and flows rather than pressures and flows. However, for gases following the CNGA equation of state, while the question of existence remains open, we construct an alternative system that always has a unique solution and show that the solution to this system is a good approximant of the true solution. The existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, our result enables correct diagnosis of algorithmic failure, problem stiffness, and non-convergence in computational algorithms.
Paper Structure (9 sections, 9 theorems, 15 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Model Description
  4. Results and Implications
  5. Ideal Gas Equation of State
  6. Non-ideal Gas with CNGA Equation of State
  7. Generalizations
  8. Use of Existence Result in Optimization
  9. Conclusion

Key Result

Proposition 1

If $\mathrm{deg}(\boldsymbol{f}, \mathcal{D}, \boldsymbol{0}) \neq 0$, then $\boldsymbol{f}$ has a zero/root in $\mathcal{D}$.

Figures (2)

  • Figure 1: The plots above show the magnitude and distribution of the relative error $\mathrm{e}_{\mathrm{rel}} (\alpha, p)$ in \ref{['eq:rel_error']} for two different choices of $g(\alpha)$ for the CNGA equation of state. The colour bars demonstrate how the error is reduced drastically by choosing the optimal $g(\alpha)$ in \ref{['eq:gamma-opt']}. With this choice, we may write $\pi(\alpha p) \approx \gamma_{\mathrm{opt}}\pi(p)$ so that the system \ref{['eq:GF-p']} for the CNGA equation of state is approximated by \ref{['eq:GF-pi']}.
  • Figure 2: Cumulative distribution function for GasLib-40 showing the residual $O(10^{-2})$ (red) as well as the relative error $O(10^{-3})$ (blue) defined by \ref{['eq:approx-measure']} for the non-ideal gas.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Proposition 2: Homotopy invariance
  • Definition 3
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • ...and 11 more