Improved model of isothermal and incompressible fluid flow in pipelines versus the Darcy–Weisbach equation and the issue of  friction factor

• Published in: Journal of fluid mechanics Vol. 891
• Main Authors: Kowalczuk, Zdzisław, Tatara, Marek S.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 25.05.2020
• Subjects: Additives; Approximation; Average flow; Coefficient of friction; Computational fluid dynamics; Empirical equations; Flow velocity; Fluid flow; Fluids; Friction; Friction factor; Geometry; Incompressible flow; Incompressible fluids; Inlets (waterways); Mathematical analysis; Pipes; Pressure; Pressure distribution; Pressure drop; Reynolds number; Submarine pipelines
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.131

In this article, we consider the modelling of stationary incompressible and isothermal one-dimensional fluid flow through a long pipeline. The approximation of the average pressure in the developed model by the arithmetic mean of inlet and outlet pressures leads to the known empirical Darcy–Weisbach equation. Most importantly, we also present another improved approach that is more accurate because the average pressure is estimated by integrating the pressure along the pipeline. Through appropriate transformation, we show the difference between the Darcy–Weisbach equation and the improved model that should be treated as a Darcy–Weisbach model error, in multiplicative and additive form. This error increases when the overall pressure drop increases. This symptomatic phenomenon is discussed in detail. In addition, we also consider four methods of estimating the coefficient of friction, assess the impact of pressure difference on the estimated average flow velocity and, based on experimental data, we show the usefulness of new proposals in various applications.