Hydrodynamics of finite-length pipes at intermediate Reynolds numbers

Abstract

Extensive studies of the hydraulics of pipes have focused on limiting cases, such as fully-developed laminar or turbulent flow through long conduits and the accelerating flow through an orifice, for which there exist laws relating pressure drop and flow rate. We carry out experiments on smooth, circular pipes for dimensions and flow rates that interrogate intermediate conditions between the well-studied limits. Organizing this information in terms of dimensionless friction factor, Reynolds number and pipe aspect ratio yields a surface$f_D(Re,\alpha )$that is shown to match the three laws associated with developed laminar, developed turbulent, and orifice flows. While each law fails outside its applicable range of$(Re,\alpha )$, we present a hybrid theoretical–empirical model that includes inlet, development and transition effects, and that proves accurate to approximately 10 % over wide ranges of$Re$and$\alpha$. We also present simple formulas for the boundaries between the three hydraulic regimes, which intersect at a triple point. Measurements show that sipping through a straw is an everyday example of such intermediate conditions not accounted for by existing laws but described accurately by our model. More generally, our findings provide formulas for predicting frictional resistance for intermediate-$Re$flows through finite-length pipes.