Melt-Mixing by Novel Pitched-Tip Kneading Disks
in a Co-Rotating Twin-Screw Extruder

The no-slip condition on the velocity at the barrel and screw surfaces is assumed. The velocities at inlet and outlet boundaries were obtained according to the same cross-sectional velocity profile under the given volumetric flow rate of $Q=60$ cm³/s ($\approx$159 kg/h). The temperatures on the barrel surface and at the inlet boundary were set to 473 K and 453 K, respectively. The natural boundary conditions for the temperature equation in the exit boundary plane and the screw surface were assigned.

To incorporate the incompressibility condition (1) in momentum conservation (2), the penalty method is utilized: the pressure in Eq.(2) is eliminated by imposing the relation

with a constant $\lambda\gg 1$(Zienkiewicz, 1977). The penalty term in the momentum conservation effectively acts as a very large bulk viscous stress; therefore, the resulting velocity field satisfies the incompressibility condition (1) in the limit as $\lambda\to\infty$.

The channel in a TSE is decomposed by the hexahedral element. The velocity and the temperature are approximated by tri-quadratic interpolation with the 27-node in each hexahedral element. The momentum equations (2) and (8) were discretized by the Galerkin finite element method. The temperature equation (3) was discretized by streamline-upwind/Petrov–Galerkin method (Brooks and Hughes, 1982; Marchal and Crochet, 1987) to stabilize the advection term.

The time evolution of the velocity and temperature fields was constructed with the converged fields for every three degrees of screw rotation. This computation was the most costly. For reference, one simulation takes about 68 hours on a computer with an Intel${}^{\text{\textregistered}}$ Core${}^{\text{TM}}$ 2 Extreme processor.

The trajectories of the passive tracers were determined based on the solved velocity field. The time evolution of the tracer density was utilized to assess the flow pattern in the melt-mixing zone with a ptKD. The set of tracer trajectories was utilized to compute the residence time distribution (Levenspiel, 1998) and to investigate the flow history and mixing process. The Lagrangian-history average of a quantity $f$ over the trajectory of $\alpha$th tracer is defined as

where $T_{\alpha}$ and $\mbox{\boldmath$X$}_{\alpha}(.)$ are the residence time and position of the $\alpha$th tracer, respectively, and $\delta(.)$ is the Dirac delta distribution. The statistical distribution of $\overline{f_{\alpha}}^{T_{\alpha}}$ characterizes the process in the melt-mixing zone.

Initially, 2000 points were uniformly distributed in a certain section relative to the axial position, which is arbitrarily set in the second disk. They were advected until they reach the outlet section. When computing the tracer advection, some tracers went out of bounds because the time resolution of the velocity field was limited. To circumvent the effect of the lost tracers on the statistics, we set the number of initial points of 2000, which ensured that a sufficient number of points reached the outlet. In our advection computation, at least 93% of the tracers reached the outlet.

To assess the effect of pitched tips on the pressurization capability, the pressure drop from the inlet to the outlet was calculated under a fixed screw rotation speeds and various volumetric flow rates (Fig.3). For a fixed volumetric flow rate, the pressure drop was largest in Bs-Bt, followed by Bs-Ft, Fs-Bt, and Fs-Ft. The results in Fig.3 show that the disk stagger mainly controls the drag ability of ptKD and that the pitched tips only experience slight modification. It is noted that the pressure drop in Fs-Ft ptKD was negative for a certain range of the volumetric flow rate, which indicates that the pressure-driven flow on average occurred in the direction from the outlet to the inlet.

Figure 4 shows the average temperature profile along the extrusion direction. For all types of ptKDs, the temperature rose from the inlet temperature to the barrel temperature over one block. Next, the average melt temperature increased two or three degrees beyond the barrel temperature due to viscous heating effects. Because the difference in the results with different types of ptKD was very small, we assume that the temperature did not have a significant effect on the mixing ability.

Figures 5 and 6 show the joint probability density function (PDF) of the residence time and the Lagrangian-history average of stress magnitude under $Q=60$ cm³/s and 200 rpm. To measure the stress magnitude, a second-order invariant of $\tau$ is considered ³³3 Note that although one of the authors (T.K.)  defined “characteristics shear stress” as $\displaystyle\sqrt{4\left(\tau_{xy}^{2}+\tau_{yz}^{2}+\tau_{zx}^{2}\right)+\left(\tau_{xx}-\tau_{yy}\right)^{2}+\left(\tau_{yy}-\tau_{zz}\right)^{2}+\left(\tau_{zz}-\tau_{xx}\right)^{2}},$  (Kajiwara et al., 1996) which was used in Yoshinaga et al. (2000) and Ishikawa et al. (2001, 2002), it is not an invariant of a second-rank tensor; therefore, it is not adequate to quantify the magnitude of the stress tensor.. Although $\mbox{\boldmath$\tau$}=2\eta\mbox{\boldmath$D$}$ is traceless by the incompressibility condition (1), a small but finite $\mathrm{tr}\mbox{\boldmath$\tau$}$ is unavoidable in numerical simulations with finite arithmetic. Therefore, we use the traceless part of $\tau$ to compute the stress magnitude, $\sigma$, as

where $I$ is the unit tensor.

Spatial density of the tracer as a function of time, $p(\mbox{\boldmath$x$},t)$, is computed from the trajectories of the tracers, $\mbox{\boldmath$X$}_{i}(t)~(i=1,\ldots,N)$. To investigate the axial tracer kinetics, the tracer density is projected in the axial ($z$-) direction as

Figure 4.2.1 shows the evolution of the axial distribution for the four ptKDs, provides detailed information on the transport mechanism, and explains the stress-residence-time joint PDF in the previous section.

RTD has been discussed often in connection with the axial mixing (Ishikawa et al., 2002; Zhang et al., 2009). However, a detailed investigation of the axial transport for ptKDs revealed that the source of the residence time fluctuation differs among the four types of ptKDs, and that broadness of RTD does not necessarily means good mixing capability.
Figure 7: Time evolution of the tracer density on the axial direction for (a) Fs-Bt, (b) Fs-Ft, (c) Bs-Ft, and (d) Bs-Bt. The initial axial position of the tracer is depicted by the vertical line on the second disk. The different times correspond to the different lines indicated in the legend.

For effective mixing, two points that are close initially must separate rapidly over time to promote points distribution in space, stretch a line element, and stretch an area element. To quantify the mixing capability, we considered two aspects of this kinetic process: the potential line-stretching capability and the type of strain-rate field in which the line-stretching occurs.

To quantify the potential line-stretching capability, we used the maximum line-stretching rate

The details of the quantity are described in A. The statistical distribution of the Lagrangian-history average of the maximum line-stretching rate, $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$, is depicted in Fig.8 For co-pump ptKDs, namely Fs-Ft and Bs-Bt, PDFs of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ are broad. In particular, for Fs-Ft, a shoulder at large $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ is observed. The large fluctuation of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ is a result of the pipeline flow. In the trajectory by the pipeline flow, $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ is small, and the larger $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ comes from the surroundings. The large fluctuation of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ indicates that gross mixing might be inhomogeneous in co-pump ptKDs. It is noted that the fingerprint of the pipeline flow is clearer in the PDF of the maximum line-stretching rate than in that of the residence time.

For counter-pump ptKDs, namely Fs-Bt and Bs-Ft, PDFs of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ are unimodal, which indicates that no special structure exists in the flow pattern in counter-pump ptKDs, and mixing over whole channel is expected.

It is noted that the mean values of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ indicated in Fig.8 are determined by the disk-stagger direction and are useless for characterizing the mixing capability. To characterize the mixing capability, the fluctuation of $\overline{\left(\lambda_{M}\right)_{\alpha}}^{T_{\alpha}}$ should be taken into account.

To identify a type of strain-rate field, the strain-rate-type identifier as a function of the invariants of the strain rate, $\mbox{\boldmath$D$}(\mbox{\boldmath$x$},t)$, is introduced as

where $n$ is a unit vector. The strain-rate-type identifier takes values specific to the strain-rate state in incompressible flow,

The details of the strain-rate-type identifier are summarized in A.

The statistical distribution of the Lagrangian-history average of the strain-rate-type identifier, $\overline{\left(f_{D}\right)_{\alpha}}^{T_{\alpha}}$, is depicted in Fig.15. The modes of the PDFs for the four ptKDs are located near $\overline{\left(f_{D}\right)_{\alpha}}^{T_{\alpha}}=\sqrt{1/2}\approx 0.707$, which is a general characteristic of screw extruders in which the contribution from circumferential shear flow is almost $f_{D}=\sqrt{1/2}$. A close look at the modes in the inset of Fig. 9 reveals that the ptKD characteristics were found. For co-pump ptKDs, namely Fs-Ft and Bs-Bt, the modes of their PDFs indicate that the history of the strain-rate type is planar shear flow. In contrast, for counter-pump ptKDs, namely Fs-Bt and Bs-Ft, the contribution of biaxial elongational flow at the pitched tips causes mode to shift to a lower value of $\overline{\left(f_{D}\right)_{\alpha}}^{T_{\alpha}}$. The result suggests that the counter-pump ptKDs to generate the three-dimensional deformation, which promotes efficient mixing, are more effectively than co-pump ptKDs.