Simultaneous Extrema in the Entropy Production for Steady-State Fluid Flow in Parallel Pipes

TL;DR

This paper explores how steady-state fluid flow in parallel pipes can simultaneously satisfy two different entropy production extremum principles, revealing complex interactions depending on flow conditions and boundary constraints.

Contribution

It demonstrates the coexistence of two contradictory entropy production principles in steady-state pipe flow and clarifies their applicability under different conditions.

Findings

MinEP principle applies to flow stability in constant flow rate conditions.

MaxEP principle relates to turbulence formation at higher Reynolds numbers.

Modified MaxEP and inverted MaxEP principles emerge under constant head conditions.

Abstract

Steady-state flow of an incompressible fluid in parallel pipes can simultaneously satisfy two contradictory extremum principles in the entropy production, depending on the flow conditions. For a constant total flow rate, the flow can satisfy (i) a pipe network minimum entropy production (MinEP) principle with respect to the flow rates, and (ii) the maximum entropy production (MaxEP) principle of Ziegler and Paltridge with respect to the choice of flow regime. The first principle - different to but allied to that of Prigogine - arises from the stability of the steady state compared to non-steady-state flows; it is proven for isothermal laminar and turbulent flows in parallel pipes with a constant power law exponent, but is otherwise invalid. The second principle appears to be more fundamental, driving the formation of turbulent flow in single and parallel pipes at higher Reynolds numbers. For constant head conditions, the flow can satisfy (i) a modified maximum entropy production (MaxEPMod) principle of \v{Z}upanovi\'c and co-workers with respect to the flow rates, and (ii) an inversion of the Ziegler-Paltridge MaxEP principle with respect to the flow regime. The interplay between these principles is demonstrated by examples.