DISPERSIVE AND DISTRIBUTIVE MIXING CHARACTERIZATION IN EXTRUSION EQUIPMENT

Transcription

1 DISPERSIVE AND DISTRIBUTIVE MIXING CHARACTERIZATION IN EXTRUSION EQUIPMENT Winston Wang and Ica Manas-Zloczower Department of Macromolecular Science Case Western Reserve University Abstract Mixing is a key step in almost every polymer processing operation. The traditional methodology for improving machinery performance has relied more on users' experience and trial and error experiments. Recently, the fast development of advanced computing resources has enabled the use of numerical modeling as an alternative and more efficient approach in studying the influence of design and processing conditions on equipment mixing performance. In this work using numerical simulations we record the flow history of a number of tracers in the equipment and use them in conjunction with dispersion kinetics models to evaluate minor component size and concentration distributions. Background One of the principal roles played by polymer processing equipment such as extruders is that of mixing. In order to measure the effectiveness of a given extruder at mixing, a benchmark criterion is necessary. There are two types of mixing that need to be evaluated: distributive mixing and dispersive mixing. In dispersive mixing, solid agglomerates or liquid droplets held together by interfacial tension must be subjected to mechanical stress in order to reduce their length scale. Therefore, the most important flow characteristics determining dispersive mixing efficiency are the magnitude of shear stresses generated and the quality/strength of the flow field (elongational flow components). In distributive mixing, repeated rearrangement of the minor component enhances system homogeneity. In continuous mixing processes, composition uniformity at the emerging stream is directly related to the material residence time distribution. Numerical simulations of polymer processing equipment provide an effective means of analyzing the capability of a given machine. Description of Method Flow Field Simulations: In our group we carried out three dimensional, isothermal flow simulations for various batch and continuous mixing equipment (1-14). In this project, we first looked at the flow patterns in a twin-flight, single screw extruder. A fluid dynamics analysis package-fidap, using the finite element method was employed to solve the 3D, isothermal flow of a Newtonian fluid. No slip boundary conditions on the screw surfaces and barrel walls were used. The operating conditions were selected such that 1 clockwise revolution of the screw was made per unit time, and a pressure difference applied across the inlet and outlet surfaces satisfied the condition of Q p /Q d = -1/2. The equations of continuity and motion for the steady state, isothermal flow of an incompressible Newtonian fluid were solved: V = 0 (1) ρvv = V τ [ ] (2) The Cray T-90 at the Ohio Supercomputer Center running FISOLV was used to solve the field equations. Flow Field Characterization: To evaluate the flow field quality we looked at the shear stresses generated and elongational flow components. One simple way to quantify the elongational flow components is to compare the relative magnitudes of the rate of deformation, D, and the vorticity, ω, tensors. The parameter λ old, defined as: λ old = D D + ω (3) can be used as a basic measure of mixing efficiency for the machine design and/or operating conditions. λ old is equal to one for pure elongation, 0.5 for simple shear and zero for pure rotation. A more rigorous method to quantify the elongational flow components is by employing a flow strength parameter, S f, that is frame invariant: S f = 2trD ( 2 ) 2 tr D o 2 (4) where D o 2 is the Jaumann time derivative of D i.e. the time derivative of D with respect to a frame of reference rotating with the same angular velocity as the fluid element. The parameter S f ranges from zero for pure rotational flow to infinite for pure elongational flow. A simple shear flow corresponds to a S f value of one. For consistency we normalize the flow strength parameter and call it λ new :

2 λ new = S f 1+ S f (5) Like λ old, λ new is equal to one for pure elongation, 0.5 for simple shear and zero for pure rotation. Particle Tracking: In work previously done by our group(3, 15), an algorithm was developed for tracking massless points that affect neither the flow field nor other particles. Since the flow field is completely deterministic, the location of the particles can be found by integrating the velocity vectors: t 1 X(t 1 ) = X(t 0 ) + V(t)dt (6) t 0 where X(t 1 ) is the location of a particle at any time t 1, X(t 0 ) is the location of the same particle at initial time t 0, and V(t) is the corresponding velocity vector of the particle. The location of each particle is calculated every time step. We use a coordinate system that rotates with the same angular velocity as the screw, so that one finite element model could be used for all the calculations. We also use a periodic boundary condition to simulate an extruder longer than the finite element model. Due to the no-slip boundary conditions, a particle which runs into, or overshoots a wall, is considered stuck there and no longer moves. Parent Agglomerate Size Distribution: Recording particle flow histories allows one to calculate the minor component (agglomerate or droplet) size distribution at the exit from the extruder. From work previously done in our group, erosion kinetics for relatively sparse agglomerates in simple shear flows can be described by (16): dr dt = k( βf h F c ), (7) where R is the radius of the agglomerate, t is the time, β is a proportionality factor that reflects the fraction of the overall hydrodynamic force bearing on the fragment, K is a proportionality factor related to the structure of the agglomerate, F h is the hydrodynamic force inducing erosion and F c is the cohesive force resisting fragmentation. The hydrodynamic and cohesive forces can be written as: F h = 5 2 µ Ý γ πr 2 (8) F c =κr m (9) with µ the fluid viscosity and γ&the shear rate. Parameter κ is a scaling factor that gauges the strength of the individual bonds between the interacting particles and m is a measure of the agglomerate packing structure. Parameter m varies from 0 for sparse agglomerates to 2 for dense agglomerates (16). Integration of the rate of erosion equation for sparse agglomerates (m=0) gives: ( ) [ ] 1 R ( Θ 2 1)1 e 2ΘKF c t / R 0 = R 0 Θ ( 1+Θ)+ ( 1 Θ)e 2ΘKF c t / R 0 (10) where R o is the initial agglomerate size and the parameter Θ is given by: Θ 2 = 5πµ Ý γ βr 0 2 2F c. (11) For denser agglomerates (m = 1), the integration of the rate of erosion equation gives: [ ( ( )] (12) 1 RR 0 = 1 be bt b + ar 0 e bt 1 where, a = 5Kβµ Ý γ π 2 (13) b = Kk (14) Since the kinetic models in Equations (10) and (12) were developed by considering erosion in simple shear flow, we need to modify them to account for the different flow field strengths that a particular agglomerate may experience. Based on work by Bentley and Leal (17) and Kharkhar and Ottino (18), showing that the critical capillary number for droplet breakup in different types of flow ranging from simple shear to pure elongation, depends on the shear rate as well as on a flow strength parameter, we modify the expression for the hydrodynamic force on a spherical agglomerate in the principal strain direction to read: F h = 5λµ Ý γ πr 2 (15) where Ýγ is the shear rate and λ is the flow strength parameter as defined in Equation (3). Using Equation (15) for the hydrodynamic force will modify the parameters in the erosion kinetic models (Equations (10) and (12)) according to: Θ 2 = 5λπµ Ý γ βr 2 0 F c (16) and a = 5λkβµ Ý γ π. (17) Particle flow histories were used in conjunction with the erosion kinetic models to calculate the parent agglomerate size distributions at various axial cross sections of the extruder. To calculate the dynamics of particle size distribution, we account for the reduction in size of the parent agglomerate during each time step t, by taking into consideration the actual shear stress and flow strength experienced by the particle during that time step. Also, in the kinetic models for erosion, the initial agglomerate size R o is adjusted at each time step. Distributive Mixing Characterization: Distributive mixing involves movement of fluid elements, blobs of fluid or clumps of solid from one spatial location in a system to another. An exact description of distributive

3 mixing involves specification of the location of the interfaces as a function of space and time. Owing to the enormous number of minor component particles involved in a mixing process, only a statistical definition of the state of distributive mixing can be derived. As in work previously performed in our group (6, 7, 15), characterization of distributive mixing of the minor component was done by examining the spatial distribution of a number of massless tracers in the system. Pairwise correlation functions: To quantify the distributive mixing of a minor component, we use pairwise correlation functions. The basis of the pairwise correlation is quantifying the distance between pairs of particles. Since our particles are represented as points, we use the discrete version of the pairwise correlation function: f()= r 2 NN ( 1) δ ( r i + r)δ ( r i ) (18) i where f(r) is the coefficient of the correlation function for a separating distance between particles ranging from r- r/2 to r+ r/2; the summation is preformed over all possible but non-repeating pairs of particles, i.e. the number of pairs for a system containing N particles is N(N-1)/2, which is exactly the inverse of the normalizing factor in f(r) and δ(r) is defined as 1 if a particle is present and 0 if a particle is absent. The vector r originates from the vector r. The coefficient f(r) can therefore be interpreted as the probability of finding a neighbor particle at a displacement ranging from r- r/2 to r+ r/2. For small enough r, f(r) can be represented as: f(r) = C(r)*2πr r (19) where C(r) is the coefficient of the probability density function of the correlation function f(r). In addition the following identity is obtained: r max C(r)* 2πrdr = 1 (21) r= 0 where r max is the largest dimension in the system such that C(r) is always zero for r>r max ; i.e., there is no correlation between points further apart than r max. In this work, the pairwise correlation function was calculated using the cylindrical coordinates of the particle positions. This allows for evaluation in a domain that is unhindered by the existence of the screw. The effectiveness of particle spatial distribution at different axial cross sections was measured by a parameter ε, which defines the difference between the actual distribution C(r) and the corresponding ideal distribution C(r) ideal as follows: ε = r =r max Cr () Cr () ideal rπdr (22) r = 0 Parameter ε ranges between 0 and 1. Results and Discussion Numerical Model: The finite element mesh shown in Figure 1 contains node brick fluid elements, and node quadrilateral boundary elements, for a total of nodes. FIMESH, a mesh generator part of FIDAP, was used to create the computer model. The physical parameters of the extruder are given in consistent dimensionless length units. The radius of the barrel was , and the radius of the screw was , the flight clearance was , and the flights were 0.44 thick. The extruder is squared pitched, so one pitch corresponds to approximately eight length units. The flow field calculations required 395 user CPU seconds over a total elapsed time of 860 seconds and 11.8 MWord of memory. Particle Tracking: In this project, 655 tracers were grouped into 5 clusters of 131 particles and were initially placed in the center of the numerical model. Figure 2 shows the initial position of these tracers, which were followed for different time units with a time step of The tracers are given an initial size R o = 1.55 mm and their dispersion is recorded along the extruder. We assume a constant and continuous feed of particles at their initial positions, such that our results pertain to steady state conditions. Dispersive Mixing: Using data from previous experiments in our group studying dispersion of silica agglomerates having a density of 0.14 g/cm 3 (sparse agglomerates) in poly(dimethyl siloxane) of viscosity 60 Pa.s in a cone-and-plate device (simple shear flow) yielded the following parameters for the kinetic model described by Equation (10) : F c /β = N; KF c = 1.102x10-5 m/s (19). Results from similar studies with carbon black agglomerates of density 0.4 g/cm 3 (medium dense agglomerates) (20) were fitted to the following parameters of the kinetic model described by Equation (12): a = m -1 s -1 ; and b = s -1. Figure 3 shows agglomerate size distributions for the case of sparse agglomerates (m=0) and medium dense agglomerates (m=1) at axial positions corresponding to the exit planes of extruders 2 and 8 pitches long. The sparse agglomerates eroded quickly and 22% reached a radius of less than 0.3 mm within 2 pitches. After 8 pitches close to half of the agglomerates were that size. In this model, the minimum agglomerate size is largely dependent on the shear rate/stress experienced. The agglomerates were unable to erode smaller than mm because they did not travel through regions of high enough shear stress. As we would expect, due to the larger cohesive force the

4 denser agglomerates experienced slower erosion and after 8 pitches they were still fairly large. Figure 4 shows the evolution of the mean agglomerate size along the extruder length, as well as the evolution of the standard deviation. The medium dense agglomerates have initially a higher standard deviation than the sparse ones, indicating that they are more sensitive to variations in particle flow history. However, as they become smaller and progressively harder to erode, the standard deviation decreases. Distributive Mixing: Figure 5 shows that the position of the particles at the exit of an 8 pitch extruder. Visual inspection of these distributions can reveal some characteristics of the mixing process. Particles injected near the flights appear poorly distributed and tend to stay close together while cycling around the channel and exhibit very poor distributive mixing. Particles injected in the center of the channel appear to be better distributed in the cross section. For a meaningful and unbiased comparison among different operating conditions and/or different designs, a reference is needed to base the comparison. For this, as in our previous work (6, 7, 15) we define an ideal distribution to be one that consists of the same number of particles (655) randomly distributed over the extruder cross section. An example of an ideal distribution is shown Figure 6. Figure 7 illustrates the evolution of the parameter ε along the axial length of the extruder. As expected, distributive mixing is improving along the extruder length, however no significant changes can be observed beyond an axial length of 6 pitches. Conclusions Towards the goal of developing new mixing criteria for process control and equipment scale up, we have presented the use of particle tracking as a method of capturing the dynamics of the mixing process. By tracking the motion of particles in the flow field, we have obtained the flow history for each particle (flow strength and shear stress experienced.) When used along with a dispersion kinetic model, this allows for the calculation of agglomerate size distribution and average agglomerate size. This illustrates one possible avenue of developing a new dispersive mixing criterion. The spatial distribution of a minor component at various axial cross sections of the extruder, reported in terms of the probability density function of a pairwise correlation function, can form the basis for evaluating distributive mixing. Via the ε parameter, we compare the evolution of the pairwise correlation function along the extruder with an ideal case where the minor component particles are randomly distributed throughout the cross section. The deviation of the actual minor component distributions from the ideal case illustrates one possible way of quantifying distributive mixing. Acknowledgements The authors would like to express their gratitude to the National Science Foundation for supporting this research project under the grant DMI References 1. Manas-Zloczower, I., Z. Tadmor, Adv. Polym. Technol., 3, 213 (1983). 2. Yang, H.-H., I. Manas-Zloczower, Polym. Eng. Sci., 32, 1141 (1992). 3. Manas-Zloczower, I., Rubber Chem. Technol., 67, 504 (1994). 4. Yang, H.-H., I. Manas-Zloczower, Int. Polym. Proc., 9, 291 (1994). 5. Wang, C., I. Manas-Zloczower, Int. Polym. Proc., 9, 46 (1994). 6. Li, T., I. Manas-Zloczower, Int. Polym. Proc., 10, 314 (1995). 7. Li, T., I. Manas-Zloczower, Chem. Eng. Commun., 139, 223 (1995). 8. Wang, C., I. Manas-Zloczower, Int. Polym. Proc., 11, 115 (1996). 9. Cheng, H. F., I. Manas-Zloczower, Polym. Eng. Sci., 37, 1082 (1997). 10. Yao, C. H., I. Manas-Zloczower, Int. Polym. Proc., 12, 92 (1997). 11. Cheng, H., I. Manas-Zloczower, Int. Polym. Proc., 12, 83 (1997). 12. Manas-Zloczower, I., H. F. Cheng, J. Reinf. Plast. Compos., 17, 1076 (1998). 13. Cheng, H. F., I. Manas-Zloczower, Polym. Eng. Sci., 38, 926 (1998). 14. Yao, C. H., I. Manas-Zloczower, Polym. Eng. Sci., 38, 936 (1998). 15. Wong, T. H., I. Manas-Zloczower, Int. Polym. Proc., 9, 3 (1994). 16. Bohin, F., D. L. Feke, I. Manas-Zloczower, Rubber Chem. Technol., 69, 1 (1996). 17. Bentley, B., L. Leal, J. Fluid Mech., 167, 241 (1986). 18. Khakhar, D. V., J. M. Ottino, J. Fluid Mech., 166, 265 (1986). 19. Bohin, F., I. Manas-Zloczower, D. L. Feke, Chem. Eng. Sci., 51, 5193 (1996). 20. Li, Q., D. L. Feke, I. Manas-Zloczower, Rubber Chem. Technol.,68,836(1995). Figure 1: Finite element model for a twin-flight single screw extruder

5 Figure 2: Initial positions of the 5 clusters of 131 tracers each. Figure 5: Evolution of particle spatial distribution calculated for 655 particles after 8 pitches. Figure 3: Parent agglomerate size distributions for 655 agglomerates after 2 and 8 L s (pitches) of the extruder Figure 6: Ideal particle spatial distribution for 655 particles. Figure 4: Evolution of mean parent agglomerate size and the standard deviation calculated for 655 agglomerates along the extruder length. Figure 7: Evolution of parameter ε along the axial length of the extruder.

6 Key words: Dispersive Mixing characterization, distributive Mixing Characterization, Agglomerate size distribution, single screw xtruder.