Entropy Time Evolution in a Twin Flight Single Screw Extruder and its Relationship to Chaos

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