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- Design of calibrators for extruded profiles. Part I: modelling the thermal interchangesPublication . Nóbrega, J. M.; Carneiro, Olga; Covas, José A; Pinho, Fernando; Oliveira, Paulo J.The parameters influencing the calibration/cooling stage of profile extrusion are discussed, and a numerical finite-volume method code to model the heat transfer is described and validated. For this purpose, the numerical predictions are compared with the analytical solution of a simple problem, with results available in the literature and with those produced by a commercial software. The routines developed are then used to identify the main process parameters and boundary conditions and to estimate their relative importance. The investigation clearly shows the advantages of using several calibrators separated by annealing zones, relative to a single calibrator of the same total length, and the large impact of the contact resistance between extrudate and cooling unit while showing negligible impact of other boundary conditions. A decrease of the extrudate velocity is seen to be also an effective control parameter, but it decreases the production rates.
- Analytical solution for fully developed channel and Pipe Flow of Phan-Thien/Tanner FluidsPublication . Oliveira, Paulo J.; Pinho, FernandoAnalytical expressions are derived for the velocity vector, the stress components and the viscosity function in fully developed channel and pipe flow of Phan-Thien/ Tanner (PTT) fluids; both the linearized and the exponential forms of the PTT equation are considered. The solution shows that the wall shear stress of a PTT fluid is substantially smaller than the corresponding value for a Newtonian or upperconvected Maxwell fluid, with implications for comparing predicted and measured values in a non-dimensional form.
- Accounting for temperature-dependent properties in viscoelastic duct flowsPublication . Nóbrega, J. M.; Pinho, Fernando; Oliveira, Paulo J.; Carneiro, OlgaA numerical and theoretical study is conducted to evaluate the effect of temperature-dependent properties on the hydrodynamic and thermal characteristics of viscoelastic fluid flow. The rheological constitutive equation of the fluid under consideration follows a common form of the PTT model, which embodies both influences of elasticity and shearthinning in viscosity. A large number of simulations were carried out for a developing channel flow with an imposed constant wall temperature by varying the parameters controlling elasticity (Weissenberg number) and viscous dissipation (Brinkman number). The resulting Nusselt number and friction factor were determined from the numerical results, for both conditions of constant and temperature-dependent properties. The properties that were allowed to vary with temperature were the viscosity, thermal conductivity, specific heat and relaxation time of the PTT model. From the results it was possible to determine how the usual correlations for Nu and Cf have to be modified, following the property-correction method, in order to represent variable-property flow of this viscoelastic fluid. An alternative method to account for variable properties, based on the definition of an equivalent temperature to be used with the constant property Nu and Cf expressions, is also proposed and shown to be less sensitive to the influence of viscous dissipation. The corrections are highly non-linear and strongly depend on eWe2 and Br especially when viscous dissipation is weak.
- Plane contraction flows of upper convected Maxwell and phan-ThienTanner fluids as predicted by a finite-volume methodPublication . Oliveira, Paulo J.; Pinho, FernandoA finite-volume (FV) procedure is applied to the prediction of two-dimensional (2-D) laminar flow through a 4 : 1 planar contraction of upper convected Maxwell (UCM) and simplified Phan-Thien±Tanner (SPTT) fluids. The method incorporates general coordinates, indirect addressing for easy mapping of complex domains, and is based on the collocated mesh arrangement. Calculations with the UCM model at a Reynolds number of 0.01 were carried out with three consecutively refined meshes which enabled the estimation of the accuracy of the predictions of the main vortex characteristics through Richardson's extrapolation. Converged solutions with the first-order upwind differencing scheme for the convective terms were obtained up to at least De à 8 in the finest mesh, but were limited to De 1, De 3 and De 5 for the fine, medium and coarse meshes, respectively, when using the second-order linear upwind scheme. The predicted flow patterns for increasing Deborah numbers with the UCM model resemble the well known lip vortex enhancement mechanism reported in the literature for constant-viscosity fluids in axisymmetric contractions and shear-thinning fluids in planar contraction, but very fine meshes were required in order to capture the described vortex activity. Predictions with the SPTT model also compared well with the behaviour reported in the literature.
- A general correlation for the local loss coefficient in newtonian axisymmetric sudden expansionsPublication . Oliveira, Paulo J.; Pinho, Fernando; Schulte, A.Results from numerical simulations and guidance from an approximated corrected-theory, developed by Oliveira and Pinho (1997), have been used to arrive at a correlation expressing the irreversible loss coefficient for laminar Newtonian flow in axisymmetric sudden expansions. The correlation is valid for the ranges 1.5 < D2/D1 < 4 and 0.5 < Re < 200 with errors of less than 5%, except for 25 < Re < 100 where the error could be as much as 7%. The recirculation bubble length is also presented for the same range of conditions and the pressure recovery coefficient was calculated for Reynolds numbers above 15.
- Viscoelastic flow in a 3D square/square contraction: Visualizations and simulations.Publication . Alves, M. A.; Pinho, Fernando; Oliveira, Paulo J.The inertialess three-dimensional (3D) flow of viscoelastic shear-thinning fluids in a 4:1 sudden square-square contraction was investigated experimentally and numerically and compared with the flow of inelastic fluids. Whereas for a Newtonian fluid the vortex length remains unchanged at low Reynolds numbers, with the non-Newtonian fluid there is a large increase in vortex length with fluid elasticity leading to unstable periodic flow at higher flow rates. In the steady flow regime the vortices are 3D and fluid particles enter the vortex at the middle plane, rotate towards its eye, drift sideways to the corner-plane vortex, rotate to its periphery, and exit to the downstream duct. Such dynamic process is reverse of that observed and predicted with Newtonian fluids. Numerical predictions using a multimode Phan-Thien–Tanner viscoelastic model are found to match the visualizations accurately and in particular are able to replicate the observed flow reversal. The effect of fluid rheology on flow reversal, vortex enhancement, and entry pressure drop is investigated in detail.
- Flow balancing in extrusion dies for thermoplastic profiles. Part III: Experimental AssessmentPublication . Nóbrega, J. M.; Carneiro, Olga; Pinho, Fernando; Oliveira, Paulo J.A computer code, previously developed by the authors for the automatic die design, is used to optimise the flow distribution in a profile extrusion die using two alternative strategies: one based on length optimisation and the other on thickness optimisation. The numerical predictions are then compared with experimental data gathered during extrusion experiments. The numerical predictions and the experimental results agree within the experimental uncertainty thus showing the effectiveness of the computer code, the optimisation algorithm and the design strategies implemented. Generally speaking, measured and predicted values of pressure drop and flow rate are in good agreement (within 8% and 6%, respectively). It also is confirmed that optimisation based on thickness control leads to final profiles that are more prone to distortion.
- The log-conformation tensor approach in the finite-volume method frameworkPublication . Afonso, A. M.; Oliveira, Paulo J.; Pinho, Fernando; Alves, M. A.The log-conformation formulation, proposed by Fattal and Kupferman [J. Non-Newt. Fluid Mech. 123 (2004) 281], has helped to provide further insights into the High-Weissenberg Number Problem. In this work, we investigate the performance of the log-conformation formulation in the Finite Volume Method (FVM) framework for creeping flows of viscoelastic fluids in steady and unsteady flows around a confined cylinder. The Oldroyd-B and Phan-Thien–Tanner (PTT) constitutive equations were used to assess the effect of different rheological behaviour on the flow patterns and solution stability. The calculation of the polymer stress contribution is carried out with both the standard technique and with the logconformation methodology. For all test cases, up to the critical conditions when both methods converge to a steady solution, the use of the log-conformation technique provides solutions with similar accuracy as the standard approach. In terms of stability the log-conformation formulation is found to be significantly more robust, and solutions could be obtained at higher Deborah number flows.
- Numerical Simulation of Non-linear Elastic Flows with a General Collocated Finite-Volume MethodPublication . Oliveira, Paulo J.; Pinho, Fernando; Pinto. G. A.This paper reports the development and application of a finite-volume based methodology for the calculation of the flow of fluids which follow differential viscoelastic constitutive models. The novelty of the method lies on the use of the non-staggered grid arrangement, in which all dependent variables are located at the center of the control volumes, thus greatly simplifying the adoption of general curvilinear coordinates. The pressure–velocity–stress decoupling was removed by the development of a new interpolation technique inspired on that of Rhie and Chow, AIAA J 82 (1982) 998. The differencing schemes are second order accurate and the resulting algebraic equations for each variable are solved in a segregated way (decoupled scheme). The numerical formulation especially designed for the interpolation of the stress field was found to work well and is shown to be indispensable for accurate results. Calculations have been carried out for two problems: the entry flow problem of Eggleton et al., J. Non-Newtonian Fluid Mech. 64 (1996) 269, with orthogonal and non-orthogonal meshes; and the bounded and unbounded flows around a circular cylinder. The results of the simulations compare favourably with those in the literature and iterative convergence has been attained for Deborah and Reynolds numbers similar to, or higher than, those reported for identical flow problems using other numerical methods. The application of the method with non-orthogonal coordinates is demonstrated. The entry flow problem is studied in more detail and for this case differences between Newtonian and viscoelastic fluids are identified and discussed. Viscoelasticity is shown to be responsible for the development of very intense normal stresses, which are tensile in the wall region. As a consequence, the viscoelastic fluid is more intensely decelerated in the wall region than the Newtonian fluid, thus reducing locally the shear rates and the role of viscosity in redeveloping the flow. A layer of high stress-gradients is formed at the wall leading edge and is convected below and away from the wall; its effect is to intensify the aforementioned deviation of elastic fluid from the wall.
- Numerical procedure for the computation of fluid flow with arbitrary stress-strain relationshipsPublication . Oliveira, Paulo J.; Pinho, FernandoA finite-volume method is presented that allows for general stress-strain constitutive equations to be incorporated into a standard momentum± pressure-correction procedure. The method is sequential and segregated in nature, the various equations for mass and momentum conservation and for the evolution of the stress tensor are solved following a predefined order, and one of its features is the use of nonstaggered, and generally nonorthogonal, computational meshes. Two types of constitutive equations are used to test the method: the standard explicit and algebraic Newtonian model, and one of the simplest implicit differential equations, the upper-convected Maxwell model. In spite of its apparent simplicity, this latter model is known to pose the most severe numerical difficulties. However, the results in this article show the method to be effective in solving the equations for the flow of Newtonian and viscoelastic fluids through abrupt planar contractions with an area reduction of 4 to 1, one typical benchmark problem. The results are compared with available data and with solutions from a standard and validated code, and good agreement and consistency is found. A new formulation to evaluate stresses at cell faces is presented and shown to lead to improved results.
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