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

TL;DR

This paper investigates the existence of steady-state solutions in fluid flow networks, establishing conditions for uniqueness and proposing alternative models for gases, which aids in computational diagnostics.

Contribution

It proves the existence and uniqueness of solutions for certain fluid equations and introduces an alternative system for gases with CNGA, improving computational reliability.

Findings

Unique solutions exist for scaled monomial equations of state.

An alternative system always has a unique solution for CNGA gases.

Results help distinguish algorithm failure from non-existence of solutions.

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.