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The SPE Extrusion Division

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Figure 2(a) shows a chronology of a typical erosion process. Similarly, figure 2(b) is a visualization of a rupture process. Both erosion and rupture examples shown in figure 2 are for the same level of shear stress but for two different initial sizes of the pellets. Based on a large set of experiments, where we modified initial sizes of pellets and shear stresses, we found out that the critical shear stress needed to get rupture of a pellet is inversely proportional to its initial size, figure 3:

For erosion mechanisms, different laws describing the evolution of the pellets size are available in the literature. Those laws have been determined with artificially built pellets suspended in a low viscosity Newtonian matrix. For example, we can cite the law of Kao and Mason [4]:

and the law of Rwei at al. [5]:

where R

Contrary to studies found in literature, we must note that, in our experiments, the pellets are commercial ones, introduced in the device untreated; moreover the matrix is highly viscoelastic, fig. 4. Erosion kinetics of N234 carbon black pellets in the SBR matrix seems to follow the model proposed by Kao and Mason. It allows gathering all the measurements on a single curve, as shown on fig. 5. Furthermore, it has been possible to express the coefficient c

where 𝜏 is the shear stress and α and 𝜏

while the Rwei at al. law is:

A special treatment is made for rupture since it should be considered as a discontinuous phenomenon. Let us assume that at time t, for a material point X, the local shear stress is T. Based on fig. 3, we get immediately the sizes of agglomerates that are breaking apart at this stress T. We assume that all of the agglomerates are broken into two equal fragments. Accordingly, the mass density function is modified.

To validate our approach we ran some experiments with rubber masterbatches containing around 20% in volume of carbon black. After having sheared the samples in a Mooney chamber for various times, we used the Dispergrader 1000NT

where (y with dot) is the shear rate, n

A wealth of information may be extracted from the results. For example in table 2, by combining fig. 3 with shear stress along a given trajectory, we evaluate the fraction of N234 agglomerates of a given size broken in each mixer during the ten rotations of the rotors. Almost all particles above 30 μm are broken for the three mixers. Yet, some big agglomerates still exist after ten rotations. When looking at smaller particles, we observe the better performance of intermeshing rotors. Only 1/3 of the 15 μm particles are broken in tangential mixers while we reach 60% of broken particles with PES3 case. Using (5), we evaluate the evolution of the mass density function of agglomerate sizes attached to each material point. Next we average all those functions at each time to get the mean mass density function representative of the dispersive mixing in the whole mixer. Fig. 10 shows the evolution of this mean function for the PES3 case, based on trajectories of 500 material points. The initial distribution size is between 25 and 50 μm. Rupture has a big effect. The initial peak almost disappears and is shifted to medium size agglomerates. Then erosion takes place where the peak maximum (at 15μm) decreases slowly while amounts of aggregates (fragments with size less than 2 μ m) increase dramatically. By applying the same procedure to the three cases, we compare their final mean mass density functions (fig. 11): in all cases, we observe a small residual fraction of large agglomerates (smallest obtained with the OS case). The ZZ2 and OS cases have the same mean final curves, while the PES3 case presents the best dispersive mixing with the smallest fraction of agglomerates of size above 2 μm.

Distributive mixing is also evaluated. In a first approach, we count the fraction of material points that left its initial chamber, table 3. The ZZ2 case has the slowest transfer rate between chambers, while the PES3 is the best, followed closely by the OS. Another parameter is the stretching capability of the mixer, based on the evaluation of the area stretch ratio along trajectories. In table 4, we present the average of the natural logarithm of the area stretch ratio as a function of time for the three mixers: PES3 has the lead, followed by ZZ2 and OS has the lowest stretching performance. Thus, clearly the intermeshing PES3 case appears to provide the best dispersive and distributive mixing [9].

Various numerical techniques have been combined to simulate mixing in batch mixers. Mesh superposition technique to determine the flow field, particle tracking, dispersion model and statistical analysis to simulate and quantify dispersive and distributive mixing. These techniques not only help in understanding the complex phenomena occurring during the dispersive and distributive mixing, but also enable testing of new ideas and eventually improve the mixing process. Clearly, the intermeshing rotors show better performances to disperse and distribute carbon black agglomerates. The simulations also show that a small fraction of large agglomerates remain undispersed, as is the case in reality.

2. V. Collin, Etude rhéo-optique des mécanismes de dispersion du noir de carbone dans des élastomères, Thèse de Doctorat, Ecole des Mines de Paris, Sophia Antipolis, France, (2004).

3. V. Collin, E. Peuvrel-Disdier, Elastomery, 9, p. 9 (2005).

4. S. V. Kao, S. G. Mason, Nature, 253, p. 619 (1975).

5. S. P. Rwei, D. L. Feke, I. Manas-Zloczower, Polym. Eng. Sci., 31, p. 558 (1991).

6. B. Alsteens, Mathematical Modelling and Simulation of Dispersive Mixing, PhD Thesis, Université Catholique de Louvain, Belgium (2005).

7. POLYFLOW v3.10.0 User’s Guide, chapter 20, Fluent Benelux, Wavre, Belgium (2003).

8. J.M. Ottino, The kinematics of mixing: stretching, chaos and transport, Cambridge University Press, (1989).

9. T. Avalosse, B. Alsteens, V. Legat, Elastomery, 9, (2005).

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