In this work a discussion on the particularities of the pressure drop equations being used in the design of natural gas pipelines will be carried out. Several versions are presented according to the different flow regimes under consideration and through the presentation of these equations the basic physical support for each one is discussed as well as their feasibility.

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Two problems of laminar-forced convection in pipes and channels, under fully developed conditions, are solved for an imposed constant temperature at the wall, with fluids obeying the simplified Phan-Thien-Tanner (SPTT) model. The fluid properties are taken as constants and axial conduction is negligible. The first case represents the asymptotic behaviour of the Graetz problem for the SPTT fluid, i.e., equilibrium between axial convection and radial conduction of thermal energy with negligible viscous dissipation. The solution is given by an analytical expression but it is only approximate (within 0.3%) as it was obtained with an algebraic method based on successive approximations. The second problem has an exact analytical solution representing the equilibrium between viscous dissipation and radial heat conduction, with negligible axial convection and a constant wall temperature. Ó

a b s t r a c tThe flow of a Newtonian fluid and a Boger fluid through sudden square-square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streakline photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re 2 ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De 2 ) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De 2 ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.

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Measurements of pressure on the cylinder surface for the flow of Newtonian and non-Newtonian fluids around a circular cylinder were carried out, from which several parameters were calculated: the form drag coefficient (C D ), the pressure rise coefficient (C pb − C pm ) and the wake angle (θ w ). The non-Newtonian fluids were aqueous solutions of CMC and tylose with varying degrees of shear-thinning and elasticity, at weight concentrations of 0.1-0.6% and the experiments encompassed the transition and shear-layer transition regimes. For low Reynolds numbers flows elasticity on the shear layers was responsible for an increase in drag reduction with polymer concentration. Within the shear-layer transition regime the increase of the wake angle and pressure rise coefficient for the more concentrated solutions reduced C D by narrowing the near wake. For the tylose solutions a good correlation was found between the elasticity number and the mean pressure rise coefficient.