Entropy Time Evolution In A Twin-flight Single-screw Extruder And Its Relationship To Chaos

Abstract

The flow fields in polymer processing exhibit complex behavior with chaotic characteristics, due in part to the non-linearity of the field equations describing them. In chaotic flows fluid elements are highly sensitive to their initial positions and velocities.A fundamental understanding of such characteristics is essential for optimization and design of equipment used for distributive mixing.In this work we analyze the flow in a twin-flight single screw extruder, obtained through 3-D FEM numerical simulations. We study particle motion and, implicitly, mixing in the extruder. Here, particles are massless points whose presence does not affect the flow field or other particle motion.We visualize chaos through Poincaré sections and calculate Lyapunov exponents as a measure of divergence of initial conditions, signaling chaotic features of flow. We use entropic measures to probe disorder or system homogeneity. The time evolution of the Renyi entropy of β = 1 for the 3-D spatial distribution of particles using different initial conditions are followed. The Kolmogorov-Sinai entropy rate, calculated by the sum of positive Lyapunov exponents, is correlated with the rate of evolution of entropy.In the same context we also examine the eccentric Couette flow.We find that the Renyi entropy dependence on time is logarithmic. To gain further understanding of this numerical observation, we analyze analytically the diffusion with drift entropy and find that it also depends logarithmically on time. Using the logarithmic coefficient of the Shannon entropy (ß = 1), as a measure of the overall rate of 2 mixing, we find that the eccentric Couette device has the highest rate of mixing, followed by the twin-flight single screw extruder, and by the 1-D diffusion with drift.

Transport Phenomena in Chaotic Laminar Flows

“…For example, the use of impellers on an orbiting rotor (familiar from classic Kitchen Aid R mixers) (27) produces a time-dependent, quasi-2D flow that mixes efficiently. Polymer extruders (28) and static mixers such as the Kenics R design (29) can achieve a close approximation of the extensional baker's transformation (Figure 2a). In these cases, varying the shape of the cross section of a conduit generates secondary flows that stretch, cut, and reassemble the fluid as it moves in the mean flow.…”

Mixing Through Stirring of Steady Flow in Small Amplitude Helical Tubes

“…Information entropy was first defined by Shannon (1948), and can be interpreted as a measure of disorder, naturally leading to its use in 8 such diverse fields as mixing in polymer processing (Wang et al, 2003), (Wang et al, 2005b), (Wang et al, 2005a), chaotic micromixers (Kang and Kwon, 2004) and aerosol mixing in the lung (Butler and Tsuda, 1997). Here the formulation used is that introduced by Kang and Kwon (2004).…”

Numerical Characterization of Mixing in a Kneading Element Channel of Dual‐Speed Corotating Non‐Twin Screws Using Lagrangian Statistics Method

“…[26] A series of indicators were previously proposed to characterize the mixing performance. [27][28][29][30][31][32] Elongation flow field is regarded as an efficient way to improve dispersive mixing even for the immiscible system difficult to disperse where the viscosity ratio of dispersive phase to matrix is beyond 4.0 and so that pure shear action is nearly unable to achieve the dispersive mixing. Cheng and Manas-Zloczower proposed so-called mixing index 𝜆 to identify how far the flow field away from the pure elongation flow.…”

“…As far as the distributive mixing is concerned, there are a variety of ways, including tracer pattern, bin counting, nearestneighbor distances, and correlation coefficients. [29,30] In order to explore the spatial uniformity in chaotic flows, Phelps and Tucker proposed a measure based on the bins number of flow domain along the line of previous works initially by Danckwerts, called mixing variance index. [31,32] This indicator is naturally related to the intensity of segregation.…”