Homogeneity of multilayers produced with a Static Mixer

Transcription

1 Homogeneity of multilayers produced with a Static Mixer J.C.vanderHoeven,R. Wimberger-Friedl,H.E.H.Meijer Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands DPI, P.O. Box 513; 5600 MB Eindhoven, The Netherlands Abstract: A multiflux static mixer can be used to produce multilayered structures. The flow is repeatedly cut, stretched and stacked by mixing elements in the channel of such a device. In the standard design, however, the obtained layer thicknesses are inhomogeneous. The causes for the multiflux static mixer s deviation from ideal behaviour are identified by 3D numerical simulations as unequal pressure drops in the separating flows. Changes in the arrangement of the elements are proposed and their effects are verified by simulations and experiments. A significant improvement of the layer homogeneity is achieved by introducing additional elements with separating walls at the inlets and at the outlets of the mixing elements. Keywords Static mixer, multilayer, layer homogeneity, extrusion, finite element analysis, Fidap, SEPRAN, layer thickness distribution Introduction To achieve the most efficient mixing process when mixing highly viscous fluids, the application of the bakers transformation is essential. In this transformation a fluid is successively stretched, cut and stacked, in the same way that a baker rolls and folds his dough. The bakers transformation can be realized in a continuous process, for instance by using a multiflux static mixer. In the channel of this device, several mixing elements are present, in each of which the flow undergoes a bakers transformation. The geometry of the multiflux static mixer was first described by Sluijters [1]. It is the most compact design one can think of for this purpose. It has a low pressure drop and is easily produced at low cost. The operating principle of a mixing element is demonstrated in figure, showing several cross sections through this element, using a white and a coloured fluid. In cross section a the two input flows are kept separate by a horizontal wall, in cross section b the flows have been stacked and cut in two by a vertical wall. In cross sections c, d, e and f the flows are stretched and finally stacked (acceleration and deceleration), resulting in a four layered flow. In a second element, the number of layers can be doubled to eight and so on. The final number of layers equals m 2 n,wheremis the number of layers fed into the mixer and n is the number 1

2 of mixing elements used. Van der Sanden et.al. [2], Mueller et.al. [3] and Baer et.al. [4] used the multiflux static mixer to make multilayers for research purposes. Schilo and Ostertag [5] experimentally compared the mixing performance of the multiflux static mixer and the Kenics static mixer. Numerical investigations into the Kenics mixer performance have been done by among others: Bryde and Sawley [6], Hobbs et.al. [7], Hobbs and Muzzio [8, 9, 10], Avalosse and Crochet [11], Ling and Zhang [12] and Arimond and Erwin [13]. The Ross mixer, see Schrenk [14] for instance, is yet another static mixer design. In stead of cutting the flow in two flows, the Ross mixer cuts the flow in four before stretching and stacking. Schrenk [15] used a Ross mixer to make multilayers with sub-micrometer thicknesses which had interesting optical properties. Ebeling et.al. [16] used co-extruded multilayer sheets, supplied by The Dow Chemical Company, for their research into the effect of the peel rate and temperature on the delamination toughness of polycarbonate styrene-acrylonitrile micro-layers. It is important to have control over the layer thicknesses and to have straight interfaces for such applications. This is, however, not generally achieved. Both, the thicknesses of the individual layers, as well as the thicknesses of the layers relative to each other, are inhomogeneous. This investigation studies relevant processes and parameters which determine the layer distribution in the extrudate of the multiflux static mixer. It is not the mixing quality that is considered, but the ability of the mixer to produce multilayers with straight interfaces and layers with equal thicknesses. Minagawa and White [17] performed an experimental study on the influence of the viscosity ratio and the die cross-section on the interface shape in co-extrusion. Schrenk et.al. [18] used a critical interfacial shear stress criterion for the prediction of the onset of interfacial instabilities. Mavridis [19] added to this a contribution of the elasticity difference between the layers. Wilson and Khomami [20, 21] studied interfacial instabilities in co-extrusion of polypropylene and high-density polyethylene. The problems described above are less relevant for the present investigation, since the materials used here have equal rheological properties. This approach, in which interfacial tension effects will also be ignored, facilitates a study of the basic functioning of the multiflux mixer. The flow in the static mixer was simulated with 2D and 3D numerical models, indicating several causes of the layer inhomogeneities. The layers in the static mixer were visualised and compared with experimental results for validation of the models. Ways of improving the homogeneity of the layer thickness distribution, henceforth referred to as the layer distribution, by changing the geometry of the mixer channel, will be investigated, both, numerically and experimentally. Experimental The experimental setup is equipped with two velocity controlled pistons to feed the mixer. In this way, the fluxes can be controlled accurately, relative to feeding the mixer with extruders. The cylinders and the mixer are provided with heating elements and thermocouples for controlling the temperature of the polymer melt. The mixer is assembled from separate rectangular steel blocks that leave a quadrilateral channel of mm 3. This channel can be filled with separate mixing elements of 1 cm 3 volume 2

3 which are supported by the channel walls only. This allows a random variation of all kinds of elements within the length of the channel. Different extrusion dies can be used at the exit. The polymers used in this work were grey and white polycarbonate, PC, (Lexan 121R, General Electric Plastics) and blue and white ABS (Ronfalin FA-50, DSM). Figure 2 depicts the viscosities of the materials as determined using a plate-plate rheometer (Rheometric Scientific SR-5000) under the indicated conditions. Before insertion into the feeding cylinders of the static mixer, the polymers were compression molded into rods. PC was extruded at 280 C and ABS at 240 C, both with a total volume flux of 0.38 cm 3 s 1. ABS can be used for a frozen flow investigation by removing the material from the mixing channel in a non-destructive way. As ABS is rubbery at around 90 C, the multilayered flow can be isolated by disassembling the static mixer at the end of an experiment. The frozen flow can be sliced in order to study the thickness of the layers and the interface shapes. PC is used for its Newtonian behaviour in the low shear regions. The experiments in which PC is used will closely resemble the 3D numerical simulations, in which a constant viscosity is used. As the frozen PC flow could not be isolated from the mixer, only the layer distribution in the PC extrudate of the mixer is studied in the experiments. Numerical simulations For numerical models with two or more fluids with different rheological properties, techniques described by Galaktionov et.al. [22], Karagiannis et.al. [23], Mavridis et.al. [24] and Gifford [25], amongst others, may be useful. Calculation of moving boundaries using the pseudo concentration method is reported by, for instance, Haagh [26] and Thompson [27]. In the experiments performed in this work, however, the fluids are two polymer melts, which only have different colours. Having confirmed that the rheological properties of the melts are not influenced by the pigments, the flow in the static mixer is modelled with a single fluid only. The following assumptions are used: Both fluids have the same rheological properties The fluids do not react or diffuse There is no interfacial tension The flow is isothermal There is no slip along solid walls Gravity can be neglected Assuming a parabolical velocity profile, the velocity of the fluid in the smallest cross section of the mixer has a maximum value U max =21 mm s 1. The minimum height in the mixer channel is H min = 4.5 mm, the density of the melts is ρ 1200 kg m 3 and the viscosities of the polymer melts are η PC 2000 Pa s at 240 Candη ABS Pa s at 180 C (see also figure 2). This results in maximum values of the Reynolds numbers Re and for the PC and the ABS melt, respectively, and inertia is consequently negligible. The governing equations are given by: η u = p 3 (1)

4 u=0 (2) These equations were solved with the finite element analysis packages Fidap 1 and SEPRAN 2. In the Fidap models, 9 node quadrilateral elements are used for the 2D models and 27 node bricks for the 3D models of the static mixer. The 3D SEPRAN model uses spectral elements. The Powerlaw, Carreau and Bingham viscosity models used in the 2D Fidap simulations are given by equation 3, 4 and 5, respectively. η( γ)= η 0 γ n 1 (3) η( γ)= η 0 (1+λ 2 γ 2 ) 1 n 2 (4) τ = η γ + τ y τ τ y (5) Viscous heating is estimated using equation 6. When the pressure drop over the mixer with three mixing elements is estimated by using the model given by Sluiters [1] and the specific heat C p = J kg 1 K 1, the temperature increase of the melt for PC is 0.1 K. Viscous heating is therefore neglected. P v = ρ C p T v T = P ρ C p (6) The layers in the Fidap model are visualised with the help of particle tracking. First the velocity field is calculated. In a separate calculation, the trajectories of 640 mass and size-less particles, released at the inlets, are determined. The velocity field in the mixer is thus not influenced by the particles. The layer distribution in any cross section through the mixer channel can be visualised by calculating the intersections of the particle trajectories with the cross section plane. The particles released in the upper inlet can be given a different colour from the particles released in the lower inlet, in order to visualise the layers. Related methods were used by Bryde and Sawley [6], Hobbs and Muzzio [8, 9, 10], Avalosse and Crochet [11], Ling and Zhang [12], and many others. In the SEPRAN model an adaptive front tracking technique was implemented. Details of this technique are described by Galaktionov et.al. [22]. Results and discussion Introduction In the use of the multiflux device for the production of regular multilayered stacks, the layer distribution after leaving the extrusion die is important. The transition from a pressure driven flow to a free flow affects the layer distribution. Similarly, the transitions from separated to recombined flows at the ends of separating walls parallel to the layer-stack are also of importance. Figure 3 shows the simulation of a 2D multilayered flow. Obviously, the homogeneous layer distributions in the two inlets are disturbed by the recombination (see layer distribution at cross section 1) and again by 1 Supported by FLUENT, 2 Ingenieursburo SEPRA, Leidschendam, The Netherlands, 4

5 the extrusion process 3 (see cross section 2). From this simple calculation it can be concluded that the layer distribution in the extrudate of the static mixer can not be made homogeneous when the layer distribution in the mixer channel is homogeneous. In an extruded homogeneous layer-stack all layers have equal cross section surfaces. Also, the velocity is constant over the cross section. In other words, the volume fluxes of the individual layers are equal. The layer distribution in the extrudate of the multiflux static mixer should thus be made homogeneous by making the volume fluxes of the individual layers in the mixer equal. Typical mixer configuration Figure 4 shows, both, the FEM mesh of the static mixer as used in the Fidap simulations and the typical mixer configuration. The two inlet flows are separated by a horizontal wall. This element of the static mixer is called an H-element. Subsequently, there are two types of mixing elements. In the first element, the flow at the righthand side from top to bottom is led downwards (and vice versa at the lefthand side), while in the second element the flow at the lefthand side is led downwards. The element types are abbreviated by E R and E L, respectively. Before leaving the static mixer, the flow is led through an open element, referred to as an O-element. The configuration of the static mixer is denoted by giving the abbreviations of the adjacent element types. For example, the static mixer in figure 4 has an H-E R E L E R -O configuration. In the cross section between the inlet H-element and the first mixing element (cross section 1, see figure 5), four openings of equal size can be distinguished. It is also clear, however, that when the cross sections perpendicular to the flow velocity are considered, there are two larger openings (perpendicular to the mixer axis) and two smaller openings (not perpendicular to the mixer axis). The consequences of this are shown in figure 6: in cross section 1 the volume fluxes through the smaller openings are smaller, resulting in unequal layer thicknesses in the extrudate. The four openings perpendicular to the flow velocity related to cross section 2 are of equal size (see figure 5). This is the result of the L and R-type alternation of adjacent elements. In the model, the calculated volume fluxes through these openings are equal. Figure 6 demonstrates that the differences between the flow velocities through the four openings in cross section 1 are much larger than in cross sections 2 and 3. If all elements in the mixer were of either L or R-type only, the homogeneity of the final layer thickness distribution would be much worse. Figure 6 also demonstrates that the shapes of the four velocity profiles in cross section 2 (and cross section 3) are not equal. This could also result in layer thickness differences. The velocity profiles of the flows entering the O-element (cross section 4) are asymmetrical as well. This causes a kind of twisting behaviour of the extrudate in experiments. Figure 7 shows a cross section through the extrudate of the static mixer in the H-E R E L E R -O configuration, obtained from extrusion of white and grey PC without a die. Apart from the deformed cross-section due to extrudate swell (the flow does not solidify immediately after extrusion from the square channel), the photo demonstrates that the layer distribution is not only strongly inhomogeneous, but is twisted as well. 3 The extrusion process is modelled as a simple transition from no slip at the wall to full wall slip. 5

6 Figures 8a and 9a show cross sections through the frozen ABS flows, isolated from the mixer, halfway through the first and the second mixing elements, respectively, showing the layer distributions. In figures 8b and 9b the layers in the same cross sections visualised with particle tracking in the 3D Fidap model of the mixer are given. The layer distributions calculated with the SEPRAN model are given in figures 8c and 9c. The differences between the experimental and the numerical layer distributions are remarkably small. Clearly, the numerical simulations using a constant viscosity, give very good predictions of the layer distribution in the mixer channel. Improved mixer configurations Considering the results shown so far, an obvious way of improving the layer distribution is by regulating the asymmetry in the flow-fields of the separate arms of the elements. The size of the bigger openings in cross section 1 of the static mixer (figures 4 to 6) can be adapted, for instance with a so-called extra resistance. Figure 10 shows a 2D FEM model of the static mixer with an extra resistance, representing a side view of the inlet H-element and a part of the first mixing element. Equal volume fluxes through the upper and lower inlet, Q 1 /Q 2 = 1, are desired at equal inlet pressures p 1 =p 2. In figure 11, the values of the calculated flux ratios Q 1 /Q 2 are plotted for several heights of the resistance h r /h mix for a constant viscosity and for viscosities modelled with the powerlaw (see equation 3). It is clear from this result that the height of the extra resistance h r necessary to result in equal fluxes depends on the rheological properties of the fluid. Thus, such a design is not a robust solution. Another way of balancing the fluxes is by shifting the inclining, compressing wall further downstream, in this way increasing the effective cross section of the smaller opening. Practically, this can be achieved by inserting an extra element with a vertical midplane (called an I-element) between the inlet H-element and the first mixing element. Figure 12 shows a 2D FEM model of this arrangement. In figure 13, the calculated flux ratios Q 1 /Q 2 at equal inlet pressures p 1 =p 2 are plotted for several lengths l I /h mix of the I-element with a constant viscosity, two powerlaw viscosities, a Carreau viscosity model (see equation 4) and a Bingham viscosity model (see equation 5). Regardless of the rheological properties of the fluids, the fluxes become equal when the length of the I-element is at least one times the height of a mixing element l I /h mix 1. The same modification can be used to avoid the twisting of the extrudate at the exit. When an H-element is used between the last mixing element and the O-element, the velocity profiles can develop into symmetrical profiles before the two flows are stacked. The length of this H-element should also be at least equal to the height of the mixing element l H /h mix 1. Figure 14 shows a cross section through the extrudate of the static mixer in the H-I-E L E R E L -H-O configuration, obtained from extrusion of white and grey PC without a die. The use of the I and the H-element clearly improves the layer quality. The layer distribution is still not completely homogeneous, however, but is much better than in the extrudate of the mixer in the H-E R E L E R -O configuration (compare figures 7 and 14). In addition, the layers are less twisted. The remaining deviations from an ideal distribution may be explained by the differences in the shapes of the velocity profiles in the cross sections between adjacent mixing elements (see also figure 6). The 6

7 influence of this on the layer homogeneity is demonstrated by placing an H-element at the outlet of each mixing element. The outflows of the mixing elements can develop, just like in the H-element between the last mixing element and the O-element of the mixer in the H-I-E L E R E L -H-O configuration. When this is done, I-elements should be used in front of all mixing elements, just like between the inlet H-element and the first mixing element of the mixer in the H-I-E L E R E L -H-O configuration. Note that in this H-IEH-IEH- IEH-O configuration there is no need to use alternating L and R-type mixing elements. Figure 15 shows the FEM mesh of the static mixer in the H-IEH-IE configuration. Figure 16 demonstrates that the shapes of the velocity profiles of the flow through the four openings on cross section 1 and 2, where the actual cutting takes place, are equal, indicating equal volume-flux divisions. Figure 17 shows a photograph of a cross section through the frozen outflow of the mixer in the H-IEH-IEH-IEH-O configuration, obtained from extrusion of white and grey PC without a die. Obviously, the symmetry of the velocity profiles are important for a perfect layer homogeneity, as the homogeneity of the layer distribution has improved with respect to the mixer in the H-I-E L E R E L -H-O configuration. A disadvantage of the H-IEH- IEH-IEH-O configuration is that it makes the mixer three times longer, compactness being one of the great advantages of the static mixer. The H-I-E L E R E L -H-O configuration can be used as a good compromise, as the layer quality is considerably better and the mixer length only increases by two element lengths. Conclusions The layer distribution in the mixer is different from the layer distribution in the extrudate, as a result of the difference between the velocity profiles in the static mixer and in the extrudate. It should be stressed that the layer distribution in the extrudate is determined by the volume fluxes of the individual layers in the static mixer. When these volume fluxes are unequal, the layer distribution in the extrudate will also be unequal. The flux division can be corrected by adding extra elements with a vertical wall in front of the mixing elements, and adding extra elements with a horizontal wall behind the mixing elements. The length of these extra elements should at least be equal to the total height of a mixing element. A disadvantage is that the improved mixer becomes three times longer than the standard mixer. A more compact modification is obtained when an element with a vertical wall is added before the first mixing element, an element with a horizontal wall is added behind the last mixing element and adjacent mixing elements are alternatingly of the L and R-type. However, the improvement of the layer distribution is somewhat less than that obtained with the former modification of the mixer geometry. Acknowledgements The authors would like to thank Hans de Bruin, who built the experimental setup and performed all experiments, and Peter Kruijt and Patrick Anderson who did the SEPRAN simulations. The help and advice of Wim Zoetelief and Ramin Badie are also gratefully acknowledged. 7

8 References [1] R. Sluijters, Chemische Techniek, 3, 33 (1965) [2] M.C.M. van der Sanden, L.G.C. Buijs, F.O. de Bie and H.E.H. Meijer, Polymer, 35, 2783 (1994) [3] C.D. Mueller, S. Nazarenko, T. Ebeling, T.L. Schuman, A. Hiltner and E. Baer, Polym. Eng. Sci., 37, 355 (1997) [4] E. Baer, D. Jarus, and A. Hiltner, Proc. SPE ANTEC Tech. Papers, 3947 (1999) [5] D. Schilo and K. Ostertag, Verfahrenstechnik, 6, 45 (1972) [6] O. Bryde and M.L. Sawley, Computers & Fluids, 28, 1 (1999) [7] D.M. Hobbs, P.D. Swanson and F.J. Muzzio, Chem. Eng. Sci., 53, 1565 (1998) [8] D.M. Hobbs and F.J. Muzzio, Fluid Mechanics and Transport Phenomena, 43, 3121 (1997) [9] D.M. Hobbs and F.J. Muzzio, Chem. Eng. Sci., 53, 3199 (1998) [10] D.M. Hobbs and F.J. Muzzio, AIChE Journal, 43, 3121 (1997) [11] Th. Avalosse and M.J. Crochet, AIChE Journal, 43, 588 (1997) [12] F.H. Ling and X. Zhang, Chem. Eng. Comm., 136, 119 (1995) [13] J. Arimond and L. Erwin, Chem. Eng. Commun., 37, 105 (1985) [14] W.J. Schrenk, R.K. Shastri, R.F. Ayres and D.J. Gosen, U.S. Patent (1992) [15] W.J. Schrenk, U.S. Patent (1993) [16] T. Ebeling, A. Hiltner and E. Baer, Polymer, 40, 1525 (1999) [17] N. Minagawa and J.L. White, Polym. Eng. Sci., 15, 825 (1975) [18] W.J. Schrenk, N.L. Bradley, Jr. T. Alfrey and H. Maack, Polym. Eng. Sci., 18, 620 (1978) [19] H. Mavridis and R.N. Shroff, Polym. Eng. Sci., 34, 559 (1994) [20] G.M. Wilson and B. Khomami, J. Non-Newtonian Fluid Mech., 45, 355 (1992) [21] G.M. Wilson and B. Khomami, J. Rheology, 37, 315 (1993) [22] A.S. Galaktionov, P.D. Anderson and G.W.M., Peters, Vol of Lecture Notes in Computer Science, (1997) [23] A. Karagiannis, A.N. Hrymak and J. Vlachopoulos, Rheol. Acta, 29, 71 (1990) [24] H. Mavridis, A.N. Hrymak and J. Vlachopoulos, AIChE Journal, 33, 410 (1987) [25] W.A. Gifford, Polym. Eng. Sci., 37, 315 (1997) [26] G.A.A.V. Haagh and F.N. van de Vosse, Int. J. Numer. Methods Fluids, 28, 1355 (1998) [27] E. Thompson, Int. J. Numer. Methods Fluids, 6, 749 (1986) 8

9 a. b. c. d. e. f. Figure 1: Several cross sections through the first element of the static mixer demonstrating its working principle by using a stratified white and coloured fluid. 1.0E E+05 η* [Pa s] PC (Grey) PC (White) ABS (Blue) ABS (White) 1.0E E E E E E E E+03 ω [rad/s] Figure 2: The melt viscosities measured with a Rheometric Scientific SR-5000 in parallel plate configuration, at 240 C for grey and white PC and at 180 C for blue and white ABS. 9

10 Inlet 2 Inlet 1 Y Free outflow X section 1 section 2 Figure 3: 2D simulation demonstrating the influence of stacking and extruding on the homogeneous layer distributions at the inlets. 10

11 Inlets (H-element) section 1 st 1 element section 2 nd 2 element section 3 rd 3 element section 4 Outlet (O-element) Figure 4: The FEM mesh demonstrating the geometry of a static mixer in the H-E R E L E R -O configuration. H-element section 1 st 1 element section 2 nd 2 element section 3 rd 3 element section 4 O-element Figure 5: Several cross sections perpendicular to the flow velocity through the static mixer channel demonstrating the different sizes of the openings. H-element section 1 1 st element section 2 2 nd element section 3 3 rd element section 4 O-element Figure 6: Several cross sections through the static mixer with three elements showing the contours of the dimensionless velocity. Black contours are related to the highest velocities, white contours to zero velocity. 11

12 Figure 7: A photograph of a cross section through the extrudate of the static mixer in the H- E R E L E R -O configuration, obtained from extrusion without a die at 280 C using white and grey PC. 12

13 z[mm] y[mm] a. b. c. Figure 8: The layer distributions in cross sections through the middle of the first element of the static mixer: a: Isolated frozen ABS flow, b: Fidap particle tracking, c: SEPRAN adaptive front tracking z[mm] y[mm] a. b. c. Figure 9: The layer distributions in cross sections through the middle of the second element of the static mixer: a: Isolated frozen ABS flow, b: Fidap particle tracking, c: SEPRAN adaptive front tracking. 13

14 p Q 1 1 p 2 Q 2 h h r h 1 2 h mix Figure 10: A 2D model of the static mixer with an extra resistance adjusting the openings in the channel to obtain equal volume fluxes Q 1 and Q 2 through the upper and lower inlet at equal inlet pressures p 1 =p 2. 1HZWRQLDQ 3RZHUODZÃQÃ Ã 3RZHUODZÃQÃ Ã 3RZHUODZÃQÃ Ã 4 ÃÃ4 K U ÃÃK PL[ Figure 11: The calculated ratios of the volume fluxes through the upper inlet and the lower inlet Q 1 /Q 2 of the 2D static mixer model with an extra resistance for several heights of the resistance h r /h mix using different viscosity models. 14

15 `I p Q 1 1 p 2 Q 2 h H-element I-element Mixing-element h 1 2 h mix Figure 12: A 2D model of the static mixer with a vertical wall (I-element) of length l I between the two inlets and the mixing element in order to obtain equal volume fluxes Q 1 and Q 2 through the upper and lower inlet at equal inlet pressures p 1 =p 2. 4 ÃÃ4 1HZWRQLDQ 3RZHUODZÃQÃ Ã 3RZHUODZÃQÃ Ã &DUUHDX %LQJKDP O, ÃÃK PL[ Figure 13: The calculated ratios of the volume fluxes through the upper and lower inlet Q 1 /Q 2 of the 2D static mixer model for several lengths l I /h mix of the vertical wall (I-element) using different viscosity models (Carreau model with λ =0.25 and n=0.25, Bingham model with τ y = 2.7 [kpa]. η 0 = 2000 [Pa s]) 15

16 Figure 14: A photograph of a cross section through the extrudate of the static mixer in the H- I-E R E L E R -H-O configuration, obtained from extrusion without a die at 280 C using white and grey PC. 16

17 Inlets I-element st 1 element H-element I-element nd 2 element Figure 15: The FEM mesh of the static mixer with an I-element in front of each element and an H-element behind each element, the H-IEH-IE configuration. Inlets section 1 I-element 1 st element H-element section 2 I-element 2 nd element Figure 16: The velocity profiles on several cross sections through the static mixer in the H-IEH-IE configuration, demonstrating equal volume fluxes through the four openings on the cross sections X 1 and X 2 where the cutting of the flow takes place. The shapes of these four velocity profiles are also equal, indicating that the flows of the individual layers being cut are equally divided. 17

18 Figure 17: A photograph of a cross section through the extrudate of the static mixer in the H- IEH-IEH-IEH-O configuration, obtained from extrusion without a die at 280 C using white and grey PC. 18