SIMULTANEOUS CHARACTERIZATION OF DISPERSIVE AND DISTRIBUTIVE MIXING IN A SINGLE SCREW EXTRUDER Kirill Alemaskin1, Ica Manas-Zloczower1, Miron Kaufman2 1 Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio, 44106, USA 2 Physics Department, Cleveland State University, Cleveland, Ohio, 44115, USA Abstract ratios: Qp/Qd = 0 and Qp/Qd = -0.5. FIDAP, a computational fluid dynamics software from Fluent, was used to obtain the Simulation of the dispersion of solid agglomerates velocity field for an incompressible Newtonian fluid with a along their trajectories in a single screw extruder with density of 1000 kg/m3 and a viscosity of 10 Pa⋅s. sequential tracking of parents and fragments allow us to A particle tracking technique was employed to observe assess conditions appropriate for dispersive mixing and the dynamics of the mixing process. We were able to extend simultaneously account for the spatial distribution of all our calculations to an extruder of infinite length based on particles in the system. A new mixing index based on the the single screw extruder periodicity, by tracking fluid calculation of Shannon entropy for different size fractions elements/particles between periodic parallel planes. of the minor component present in the system and giving Agglomerates were considered to be massless and non- preference to smaller sizes was developed. interactive with each other or with the flow field. Inertia and gravitational forces were neglected. 1. Introduction 2.2. Renyi Entropies Mixing is an important and essential component in Entropy is a rigorous measure of order/disorder in a most polymer processing operations. Material properties system. A well-distributed multi-component system will and the quality of the final products are highly dependent have high entropy. Renyi entropies are defined by: on equipment mixing performance. S( β ) = − ⋅ ln ∑ piβ , One can distinguish two mixing mechanisms, namely β − 1 i =1 1 M distributive and dispersive mixing. However, in most (1) polymer processing operations they occur at the same where β is the Renyi parameter (0≤β≤∞), M is the total time, and therefore for adequate evaluation of the mixing process we need to characterize both of them simultaneously. number of boxes (or bins) we divide our system to, and pi is Wang, Manas-Zloczower, and Kaufman a used Renyi the concentration of particles in bin i. All bins are equal in entropies for distributive mixing characterization. parameter β we focus on different aspects of system size with an aspect ratio close to 1. By changing the Dispersive mixing was characterized separately via particle size distributions b. homogeneity: voids for β = 0 to highest concentration The goal of this work is to develop a new mixing regions for β → ∞. When β=1, the Shannon (information) criterion, which will allow for simultaneous evaluation of entropy is obtained: S(1) = −∑ pi ln pi dispersive and distributive mixing efficiency. Such a criterion can be used for process control and mixing M equipment optimization. (2) i =1 2. Procedure The relative entropy S(β)/ln(M) defines an index of 2.1. Simulation and Particle Tracking homogeneity, which is 1 for total disorder or homogeneity and is small for poorly distributed systems. In our simulation we used a four-pitch single screw Calculating Renyi entropies at different cross sections extruder with three different designs. We analyzed the along the extruder will provide information about the dynamics of the mixing process at two different throttle dynamics of mixing in the system. a Wang, W., Manas-Zloczower, I., Kaufman, M.: Intern. Polym. Process. 16, p. 1 (2001) b Wang, W., Manas-Zloczower, I. , Polymer Engineering and Science, vol. 41(6), 1068-1077 ( 2001)  2.3. Erosion Kinetic Model extruder. Operating at negative throttle ratios enhances dispersive mixing. At the same time an increase in helix To account for agglomerate dispersion along their angle also leads to faster erosion. trajectories, an erosion model recently developed by Scurati c was adopted into the particle-tracking algorithm. 3.2. Distributive Mixing Characterization via In this model the rate of erosion is proportional to the Renyi Entropies excess of hydrodynamic force acting on the agglomerate relative to its cohesive force and to the rate at which Renyi entropy calculations were based on examining specific surface positions of the agglomerate experience the spatial distributions of all particles, regardless of their conditions favorable for erosion. In simple shear flow, the size, at different cross sections of the extruder. Figure 4 erosion rate can be calculated from shows the evolution of relative Shannon entropy along the extruder length for different screw designs. Increasing the − ∝ (Fh − Fc ) , . helix angle from 10 to 17 degrees is beneficial. However, dR γ (3) further increase to 24 degrees does not prove to be effective. dt 2 3.3. Evaluation of Renyi Entropies for where R is the agglomerate radius, Fh is the Agglomerates of Various Sizes hydrodynamic force acting on it, Fc is the cohesive force γ ⋅ While dispersion occurs in the system we can calculate resisting dispersion, and /2 is the agglomerate rotational not only the total entropy of the system but also evaluate the speed (with γ the shear rate). ⋅ entropy associated with the spatial distribution of particles of given size c. In this work we considered the kinetic model Sc (β ) =− ⋅ ln ∑ p βj / c β − 1 j =1 parameters for a system of silica agglomerates dispersing 1 M (4) in a PDMS fluid of viscosity 10 Pa.s. During erosion we assumed that parent agglomerates and eroded fragments maintain a spherical shape. The initial agglomerate radius was set at 0.5 mm, while the radius of the smallest where pj/c is the probability of finding a particle in bin #j, eroding fragment was set at 0.11 mm, thus rendering a considering only particles of size c. We are going to call maximum of 93 fragments to erode from a single this entropy fractional entropy. Sc(β)/ln(M), the relative agglomerate. The erosion kinetic model does not account fractional entropy, is positive and less than one. for breakage of initial agglomerates into large fragments. Figure 5a shows the evolution of relative fractional entropy for different size fractions along the extruder length at zero throttle ratio. We observe that as the concentration of 3. Results and Discussion a particular fraction increases, its entropy also increases. However as a particular fraction disappears/erodes in time, 3.1 Dispersive Mixing Characterization via its entropy also decreases. Agglomerate Size Distributions Figure 5b plots the evolution of entropies along the extruder line for a negative throttle ratio. As dispersive mixing is enhanced at negative throttle ratio, smaller size 100 parent agglomerates were placed initially in a fractions appear in the system and their entropy increases as small square in the middle section of the screw channel as well. Thus Renyi entropies can be used not only as a shown in Figure 1. The particles were tracked for 30 measure of distributive mixing, but also as an indicator for pitches of the extruder in each set of simulations. To the progress of dispersive mixing in the system. account for agglomerate erosion during its movement through the flow field, we need to consider the values for the shear rate and the flow strength along its trajectory at 3.4. A New Mixing Index each time step. Figure 2 shows particle size distributions at different There are two goals for a mixing process involving both cross sections of the extruder. As expected from the dispersive and distributive mechanisms: (i) reduce the size kinetic model employed, bimodal particle size of the minor component and (ii) distribute it homogeneously distributions were obtained. throughout the major one. In this section we define an index Figure 3 presents the reduction in size of parent that measures simultaneously the progress towards agglomerates (average value) along the axial length of the achieving both goals. To get some insight on how to develop such an index we start by considering the Shannon entropy of the whole c Scurati, A., Manas-Zloczower, I., Feke, D.: ACS, Rubber Div. system: Meeting, 52, GA (April 2002)  Stotal = −∑∑ pc , j ln pc , j , ∑f pc = 1 C M C (5) (10) = c 1 =j 1 c =1 c where pc,j is the probability to find a particle of size c in In our work we have explored a linear dependence of fc, bin #j. There are C different particle sizes present in the on the surface area of a particle of size c: f c= n + mrc2 system. Moreover, the probability to find a particle of size c in a bin j can be calculated by using the multiplicative (11) rule of the theory of probability: p= p j / c ⋅ pc , Here rc is the radius of a minor component particle, m is a c, j (6) slope coefficient (in this case negative in order to give more weight to smaller sizes), and n is calculated by using where pj/c is the probability to find a particle of size c in equations (10) and (11): ∑ ∑ the bin #j based on particles of size c only, i.e. the = + rc2 pc = conditional probability of occupying bin #j given that the C C size is c. In equation (6) pc is the probability to find a f p n m 1 (12) = = c c particle of size c in the whole system, irrespective of c 1 c 1 location. Combining equations (5) and (6), the Shannon entropy of the whole system can be written as: In our calculations we choose m = -1. ∑ ∑ ∑ M  C Figure 6 shows the evolution of the mixing index for Stotal = − pc ⋅  p j / c ln p j / c  − pc ln pc (7) C different designs and processing conditions. The highest   overall mixing efficiency was obtained for an extruder with = c 1= j 1 = c 1 helix angle of 17° operating at a negative throttle ratio. or 4. Conclusions = ∑ pc Sc + Ssize C Renyi entropies were employed to measure the degree Stotal (8) of distributive mixing in an extruder. However when Renyi c =1 entropies were calculated for different minor component sizes they could also point out to the degree of dispersive In equation (8), the first term on the right hand side is mixing in the system. A mixing index based on the an average of the fractional entropies associated to each calculation of Shannon entropy for different size fractions of of the sizes present in the system and weighed by their the minor component present in the system and giving concentrations. The second term is the entropy associated preference to the smaller sizes, can be used for simultaneous with the size distribution, which is independent of the characterization of both dispersive and distributive mixing. spatial distribution of the particles. Such an index may become a tool for design and process Inspired by equation (8), we propose a measure based optimization. Our calculations point out to an optimum in on the average of the fractional entropies, i.e. the first the helix angle of 17°. Also operating the extruder at term on the right hand side of equation (8). In the new negative throttle ratio improves mixing efficiency. mixing index we give preference, i.e. we weigh more heavily the smaller size minor component particles. Thus Acknowledgements we define the mixing index I: ∑ pc Sc K. Alemaskin acknowledges the financial support of I= C 1  this work through a fellowship from Dow Chemical (9) c =1 ln M Company. I. Manas-Zloczower and M. Kaufman acknowledge the financial support of this work by the  National Science Foundation through grant DMI-0140412. The weight pc is proportional to the concentration of particles of size c in the system, pc = f c pc . Division by  Key Words Numerical simulations, Renyi entropy, dispersive mixing, ln(M) is needed to normalize the index, i.e. the largest distributive mixing value of I is unity. be chosen so that 0 ≤ fcpc ≤ 1 and: The factor fc depends on the particle size and should  Fig. 1. Initial position of parent agglomerates Fig. 2. Particle size distributions at different cross sections of an extruder with 17° helix angle and Qp/Qd = 0 Fig. 3. Reduction in size of parent agglomerates along Fig. 4. Influence of design on the evolution of Shannon the axial length of the extruder entropy (Qp/Qd = 0) Fig. 5. Evolution of Renyi entropies for different size fractions Fig. 6. Overall mixing efficiency (based on Shannon entropy)