Characterization of Distributive Mixing in Polymer Processing Equipment using Renyi Entropies

A new method for characterization of distributive mixing in processing equipment, based on Renyi entropies, was developed. This method was applied to a twin-flight single screw extruder, in which tracer positions were determined through computer simulations of the flow field. The various entropies were calculated using particle concentrations in equal area domains of the mixer. Renyi entropies, which are function of a parameter β, were calculated for extruders of different lengths. We discuss the merit of using Renyi entropies for different values of β by pointing to the different mixing characteristics they probe. The relative Renyi entropy varies between 0 and 1 and represents a measure of distributive mixing quality, with 1 corresponding to perfect mixing and 0 corresponding to poorest mixing. We compare this new method of distributive mixing characterization to traditional ones based on the concepts of Scale and Intensity of Segregation, and the calculations based on Pairwise Correlations and Correlation Sums. The results show good agreement between the relative Renyi entropy and the traditional methods. Other advantages of the Renyi entropy such as reduced calculation time and geometric independence are discussed. For the case of a twin-flight single screw extruder, it is shown that a longer extruder is not necessarily more beneficial to distributive mixing.

Study of distributive mixing in a journal bearing flow geometry

We implicitly assess the distributive mixing of generalized Newtonian fluids with shear-thinning behavior in a journal bearing flow geometry. For this purpose, we firstly develop a finite element code to calculate the flow field parameters. Our numerical algorithm splits the viscous stress tensor into arbitrary Newtonian stress and a source term, which grows gradually during the iterative solution. Therefore, we get a better converging solution than the Picard method, especially for highly shear-thinning fluids. Secondly, considering two inert fluids in the mixing domain, we employ a Lagrangian-Eulerian approach to predict the shape of the interface between two fluids. The results of our numerical analysis provide us the required information to evaluate three implicit mixing criteria: the concentration variance, the striation thickness, and the mean strain function. Then we conduct a parametric study to investigate the effects of different parameters (geometry and rheology) on the distributive mixing state. In addition, we discuss which mixing criteria provide a better evaluation for distributive mixing.

Study of Dispersive and Distributive Mixing in a Converging Pipe

The creeping flow of a polymeric melt in a converging pipe is analyzed using an in-house finite element code. An introduced power-law model considers the shear thinning behavior of the viscous liquid. Furthermore, distributive and dispersive mixing processes are evaluated with the mean strain function and the Manas-Zloczower mixing index, respectively. Using a parametric study, the dependency of the mixing criteria on the rheological and geometrical factors is investigated. Finally, a new analytical expression for the evaluation of the distributive mixing is suggested, based on simplifying assumptions and a curve fitting method.

Quantifying evenly distributed states in exclusion and nonexclusion processes

Spatial-point data sets, generated from a wide range of physical systems and mathematical models, can be analyzed by counting the number of objects in equally sized bins. We find that the bin counts are related to the Pólya distribution. New measures are developed which indicate whether or not a spatial data set, generated from an exclusion process, is at its most evenly distributed state, the complete spatial randomness (CSR) state. To this end, we define an index in terms of the variance between the bin counts. Limiting values of the index are determined when objects have access to the entire domain and when there are subregions of the domain that are inaccessible to objects. Using three case studies (Lagrangian fluid particles in chaotic laminar flows, cellular automata agents in discrete models, and biological cells within colonies), we calculate the indexes and verify that our theoretical CSR limit accurately predicts the state of the system. These measures should prove useful in many biological applications.