FLOW VISUALISATION OF POLYMER MELT CONTRACTION FLOWS FOR VALIDATION OF NUMERICAL SIMULATIONS

Transcription

1 FLOW VISUALISATION OF POLYMER MELT CONTRACTION FLOWS FOR VALIDATION OF NUMERICAL SIMULATIONS R Spares, T Gough, M T Martyn, P Olley and P D Coates IRC in Polymer Science & Technology, Mechanical & Medical Engineering, University of Bradford, Bradford UK Abstract Full field velocity measurements in planar contraction geometries are reported - these having been evaluated using the techniques of particle image and particle streak velocimetry for HPDE and LDPE melt flows. Measurements were made in a specially designed flow cell that enables laser sheet lighting to be used to illuminate planes of the polymer at precise locations across the die, mounted on a Betol BK38 extruder. Stress measurements obtained by flow birefringence complement the velocity field information. Steady state and start up flow regimes were investigated with results presented in a form suitable for direct comparison with ongoing numerical simulations. 1. Introduction Work previously conducted[1-5] has shown flow visualisation techniques to be useful in the understanding of the flow regimes within abrupt contraction die geometries. Such geometries have also become a standard test for researchers seeking numerical predictions of polymer melt flows[6-9]. Here we show the methods employed in the shift in emphasis in this work from rheological characterisation to providing data suitable for validation of numerical programs. Normally, experimental studies provide too few measurement locations compared with numerical simulation element sizes, particularly near to flow boundaries where the meshes may be highly refined to accurately describe the high velocity gradients. 2. Experimental The flow behaviour of several polyethylenes passing through an abrupt contraction geometry was studied using a flow visualisation cell. This paper will focus on results for Dow LD150 LDPE. The cell was fitted to a computer monitored 38mm extruder (Betol BK38). The cell has replaceable inserts allowing different convergent geometries to be studied. The research reported here involved a 180 entry slit with a contraction ratio of 5:1. Two different geometries were used; 0.6 and 3mm slit heights with 6 and 30mm slit lengths respectively. In each case the depth of the die was 25mm. A photograph of the die with 3mm slit is shown in Figure 1. Images taken on a Pulnix TM-86 camera are stored on a video recorder. This video is then transferred to a computer by the use of a BT848 based PC-TV card for subsequent analysis. These cards allow loss-less capture and storage of a large quantity of contiguous frames, typically 768x576 pixels, and it is the move to this method of capture from boards that store a limited number of frames in memory that has enabled increased data to be produced. The associated optics change depending on the inserts and how much of the die is being studied so scaling is determined based on the location of any two corners within the die. For the 0.6mm slit height positional accuracies of better than ±0.005 mm can be achieved. Melt flow velocities were quantitatively determined within the convergent region using particle image velocimetry. This was achieved by using laser sheet lighting to illuminate a ~ 0.8 mm thick plane at the centreline of the flow. Glass bead tracer particles (40µm and 2,500kg/m 3 ) were introduced into the melt. Velocity fields were obtained by analysing successive video frames of the seeded flowing melt. Additionally the paths of individual particles have been tracked in the vortex region. The high magnification required to study these geometries leads to a shallow depth of field. This may be further reduced by fully opening the aperture of the lens and setting an appropriate level of back illumination. With suitably low concentrations all other particles are sufficiently out of focus to enable single particles to be studied to gain the three components of velocity within the vortex.

2 3. Analysis Velocity Determination The velocity field is determined by the analysis of captured video. Previously our work has been based on particle streak velocimetry[10] (PSV), here we have also employed digital particle image velocimetry (DPIV) and particle tracking velocimetry (PTV) techniques. The DPIV method calculates the velocity of the fluid by matching an 'interrogation window', typically 64x64 pixels, on one frame with a similar window in a subsequent frame. The method is efficiently implemented using a 2-D crosscorrelation[11]. The offset of the second image when the maximum correlation is achieved, shown in Figure 2, gives the displacement, the velocity is trivially calculated based on the time between frames, 0.04s in this case. The images used are the same as for the PSV method and the seeding concentration, in the experiments, is not as high as would normally be used for DPIV. Higher concentrations of smaller particles exist on the centreline and the method has been used successfully for the calculation of centreline velocity during start-up flows. PTV has been implemented using a method whereby the particle location is determined by the overlap between the streaks of pixels illuminated during the exposure of each field, see Figure 3, which was constructed using 2 frames containing 4 fields. For the camera used, this results in a particle's location being recorded every 0.02s. PTV methods usually suffer from the difficulty in associating the many recorded positions in each frame with the correct particle in the next frame. In this method the streak is used to avoid this ambiguity. When sufficient locations have been recorded for a given particle, a curve is fitted to give position as a function of time, and differentiation then gives velocity as a function of time. Velocity can then be calculated at known positions. Curve Fitting The PTV and PSV methods produce data at random locations, and this along with any spurious measurements, renders the data unsuitable for direct comparisons with numerical simulations, where results are calculated at predetermined locations controlled by the chosen mesh. Even for DPIV, where the calculated velocity is assumed to be in the middle of the interrogation window and a systematic scan over the entire image will produce a regular array of data points, it is unlikely that the measurement locations will coincide with the nodal positions of a finite element mesh. To enable direct comparisons between experimental and numerical values, and hence give an error measurement for the numerical simulation, we fit the experimental data to a bi-linear triangular element mesh. If the numerical work is performed on quadrilateral and/or higher order elements, then these elements may be adequately approximated by three node elements. For each data point, p, located at x, y ), we search ( p p through the mesh and find the element it lies within, Figure 4. Using standard finite element shape functions, ψ x, y ), i ( p p expressed in a global cartesian co-ordinate system[12] suited to the planar flows modelled, it is possible to write down an equation of the form, where ϕ = φ ψ + φ ψ + φψ, p ϕ p is the measured value and j j k k l l φ i are the unknown nodal values. Repeating this for all the measured values, and expressing in matrix form we have, K φ = ϕ. Solution to this is achieved by re-writing in the form T T K Kφ = K ϕ Prior to the solution of the set of equations it is also possible to add velocity boundary conditions, typically u u = 0 at the wall and u = 0 on the symmetry plane, x = y where u x is the component of velocity in the x direction and u x that in the y direction. If no equations exist for any node then the matrix will be singular and a solution may not be found. In this case an equation for each such node is generated to make it a weighted average of the nodes connected to it, effectively coarsening the mesh in the region of the node. This approach leads to a scheme sufficiently robust for small amounts of data that minimises the effect of spurious measurements with increased data. 4. Results and Discussion Figure 5 shows the velocity profile along the centreline of the mid-plane during start-up, for Dow LD150 LDPE at 180 C in the 3mm slit. When the flow is stable the maximum velocity is 3.1mm/s at an apparent shear rate of 28s -1. The development y

3 of this profile is dependent upon the time for which the extruder is idle before the beginning of the test, which influences the time taken for the corner vortices, Figure 6, to form in the die. Sample results following the curve fitting are presented in Figure 7a. Measurements were taken after the above test had become stable and are based on 1 minute of video (1500 frames), producing in excess of 300,000 data points. As a further post-processing step, streamfunction can be calculated based on these velocities. Figure 7b shows contours of streamfunction, valid only on the mid-plane where the flow is two dimensional. This calculation also gives an experimental measure of vortex intensity which is a measure typically used in numerical simulations. For these conditions the vortex intensity is 2.1%. 5. Nakason C, Kamala M, Martyn M T and Coates P D. Annual Technical Conference-ANTEC, Vol 3,p , (1997). 6. Goublomme A, Crochet M J, J. Non-Newtonian Fluid Mech. 47 (1993) Goublomme A, Drailly B, Crochet M J, J. Non-Newtonian Fluid Mech. 44 (1992) Park H J, Kiriakidis D G, Mitsoulis, J. Rheol. 36(8) (1992) Kiriakidis D G, Park H J, Mitsoulis, Vergnes B, Agassant J-F, J. Non-Newtonian Fluid Mech. 47 (1993) Kamala M, PhD Thesis, University of Sheffield (1992) 11. Willert C and Gharib M, Exp. Fluids 10 (1991) Zienkiewicz O C and Taylor R L, The finite element method, Vol 1, (1988). 5. Conclusions An automated, systematic, method for resolving polymer melt velocity throughout a flow visualisation cell has been developed. Primarily aimed towards producing data for direct comparison with numerical simulations, the increased data produced by this approach yields greater confidence in the measurements. Data at specific locations may still be extracted for material characterisation. Acknowledgements The support of the IRC in Polymer Science and Technology, and Bradford University is gratefully acknowledged. References 1. Nakason C, Kamala M, Martyn M T and Coates P D, Flow visualisation for extensional viscosity assessment, SPE Tech Paps XLIV, (1998). 2. Coates P D, Nakason C, Kamala M and Martyn M T, Visualisation and on-line sensing for polymer processing, Proc. PPS-14, Keynote Paper S401-KN, , Yokohama, Japan (1998). 3. Nakason C, PhD Thesis, University of Bradford (1997). 4. Nakason C, Kamala M, Martyn M T and Coates P D, Process Engineering 97, Coates P.D., Ed., Inst. of Materials, London, (1997).

4 Figure 1 Photograph of flow visualisation cell a)first interrogation window b)second window shifted to match the first Figure2 Velocity measurement by DPIV Figure3 Particle location determination for PTV

5 Figure 4 Typical triangular element containing data point Figure 5 Centreline velocity during start up Figure 6 Laser sheet illuminated plane showing corner vortex for the LDPE a) Calculated velocity vectors b) Contours of streamfunction Figure 7 Processed measurements Keywords: velocimetry, contractions, flow visualisation, numerical simulation