P.M. Coelho

Considerations about equations for steady state flow in natural gas pipelines

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. Keywords: natural gas flow , pressure drop, gas pipelines, Renouard

Vortex shedding in cylinder flow of shear-thinning fluids II. Flow characteristics

A detailed experimental study on the flow characteristics of various vortex shedding regimes was carried out for the flow of non-Newtonian fluids around a cylinder. The fluids were aqueous solutions of carboxymethyl cellulose (CMC) and tylose at weight concentrations ranging from 0.1 to 0.6%, which had varying degrees of shear-thinning and elasticity. Two cylinders of 10 and 20 mm diameter were used in the experiments, defining an aspect ratio of 12 and 6 and producing blockages of 5 and 10%, respectively. The Reynolds number (Re) ranged from 50 to 9 × 10 3 . Shear-thinning gave rise to a decrease of the cylinder boundary-layer thickness and to a reduction of the diffusion length (ld), which raised the Strouhal number, St. In the laminar shedding regime, a modified Strouhal number was successful at overlapping the shedding frequency variation with the Reynolds number for the various solutions. In contrast, fluid elasticity was found to increase the formation length ( lf ), and this contributed to a decrease of the Strouhal number. The overall effect of shear-thinning and elasticity was an increase in the Strouhal number. The increase in polymer concentration and the corresponding increase in fluid elasticity were responsible for the reduction of the critical Reynolds number marking the sudden decrease of the formation length, Relf . In the shear layer transition regime, the formation length and Strouhal number data collapsed onto single curves as function of a Reynolds number difference, which confirmed Coelho and Pinho (J. Non-Newtonian Fluid Mech. (2003), accepted for publication) finding that an important effect of fluid rheology was in changing the demarcations of the various flow regimes.

Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model

A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm–Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge–Kutta method. The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter � ) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.

Fully developed forced convection of the Phan-Thien–Tanner fluid in ducts with a constant wall temperature

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. 2002 Elsevier Science Ltd. All rights reserved.

On the Generalized Brinkman Number Definition and Its Importance for Bingham Fluids

In this technical note we discuss the importance of using a generalized Brinkman number definition for laminar pipe flow of a Bingham fluid, when viscous dissipation effects are relevant. We show that adapting the Brinkman number definition commonly used for Newtonian fluids directly to the more general class of non-Newtonian fluids does not calculate correctly the ratio between heat generated by viscous dissipation and heat transfer at the wall and leads to a distortion of the graphical representation of the Nusselt number, Nu, rendering difficult, if not impossible, the comparisons of the Nu behavior between different Brinkman numbers. The use of the proposed generalized Brinkman number removes these problems and simultaneously it has the merit of being independent of any reference apparent viscosities.

Viscoelastic fluid flow through 3D square-square expansions

In this work we present an experimental study of the 3D laminar flow of Newtonian and Boger fluids through square‐square expansions with expansion ratios of 1:2.4, 1:4, 1:8 and 1:12. Visualizations of the flow patterns were performed using streak line photography and the velocity field of the flow was measured in detail using Particle Image Velocimetry (PIV). The experimental results obtained with the Newtonian fluid are compared with numerical predictions. The numerical code used is based on a Finite‐Volume method and an excellent agreement is found between experimental and numerical results. For all expansion ratios studied, a Moffatt corner vortex is observed downstream of the expansion and an increase in the flow inertia leads to an enhancement of the vortex size. On the other hand, the viscoelastic fluid flow also reveals the existence of a corner vortex downstream of the expansion, which decreases in size and strength when the Deborah number is increased. The vortices in square‐square expansion flow...

Estimating Heat Transfer Coefficients and Friction Factors in Non-Newtonian Flows between Parallel Plates

ABSTRACT A simplified method, successfully tested previously for flow in circular pipes, is used in this work to estimate the friction factor and Nusselt number in fully-developed laminar flow between parallel plates of non-Newtonian fluids. Both constant wall temperature and constant wall heat flux cases are considered. The methodology was tested using several constitutive equations, including generalized Newtonian fluids, such as the Herschel–Bulkley, Bingham, Casson, and Carreau–Yasuda models, and also the simplified Phan-Thien–Tanner viscoelastic model. The error of the approximate methodology was found to be small, below 3.4%, except for the fluids with yield stress for which the maximum error increased to 8.4% for the cases analyzed, which cover a wide range of shear viscosity curves.

Electrical discharge machining plasma developed in a solid or gaseous medium

The article aims to show that the electrical discharge machining plasma can be developed in solid or gaseous medium, through the numerical and experimental evaluation of process performance. The plasma channel developed in gaseous medium is based on an electrical discharge developed in a gas bubble and the plasma channel developed in solid medium is based on underwater explosions. The main electrical difference between both mediums is on its electrical resistivity. However, if the radius of plasma channel increases, its electrical resistivity should decrease because its electrical resistance and applied current intensity are constant, or in other words, the applied electrical power is constant during discharge duration. Thus, the plasma channel is developed in gaseous and solid mediums, with same electrical resistivity and Joule factor, because the radius of plasma channel is considered constant during discharge duration. The comparison of numerical results of electrical discharge machining performance obtained through an electrical discharge machining plasma developed in gaseous and solid mediums shows high agreement with the experimental results. Therefore, the electrical discharge machining plasma developed in solid and gaseous mediums is reliable when hydrocarbon oil is used as a dielectric fluid due to the high degree of agreement of numerical and experimental results of electrical discharge machining performance.

Heat Transfer of Bingham Fluids in an Annular Duct with Viscous Dissipation

ABSTRACT Analytical expressions for the velocity and temperature profiles in a fully-developed laminar Poiseuille flow through a concentric annular duct of a Bingham fluid with constant wall heat flux at the inner and outer wall, in the presence of viscous dissipation are deduced and presented. It is found that the proportion of the heat generated by viscous dissipation near the outer wall increases with an increase of the dimensionless flow parameter, and a decrease of the duct radius ratio. The Nusselt numbers are first calculated based on a single bulk temperature for the entire duct cross section. The possibility of performing calculations of the relevant parameters discussed in this work is available via the Supplementary Material as an Excel file. Also in this work a new approach is employed, where two different bulk temperatures are used, one for each side of the radial location in the temperature profile whose derivative is zero. With this new approach the Nusselt number behavior is free of either unphysical discontinuities or negative values. As a consequence, the Nusselt number values better reflect the actual heat transfer coefficient at the walls and are more comparable with the heat transfer inside ducts when the temperature profile is symmetric.