(216g) Numerical Simulation On Unsteady Dispersion Characteristics of Particles in a Flow Between Coaxial Cylinders

Polymer composites including fine particles in polymer have been paid attention to because various functions are expected to appear in such materials.  The composites are made from suspensions, which disperse particles in molten polymer.   However, the dispersion of fine particles in polymer is not so easy. The fine particle has high specific surface area and then the inter-particle energy due to the surface energy becomes relatively large.  Thus, fine particles easily aggregate with each other and form larger clusters.  The functions of polymer composites severely depend on the particle dispersion.  Thus, it is of importance to control the dispersion characteristics of particles in molten polymer. 

In order to disperse particles in polymer, a kneading extruder with rotors is often used.  In such an extruder, high shear stresses are added to and break up the clusters with rotors.  However, stresses added vary in space and in time in the extruder.  Then, the dispersion of particles in polymer composites occurs non-uniformly in space and in time.  Due to this complexity, it is very difficult to know the dispersion characteristics in such an extruder with experimental studies.

               In order to calculate the dispersion characteristics in suspensions, Usui (1999) suggested a thixotropy model.  In the model, the average cluster size can simulate at a uniform shear rate by a balance of Brownian and shear coagulations and shear breakup and the viscosity characteristics can be also calculated with the model and a suitable viscosity model as Simha’s cell model.  In the previous studies (Hasegawa et al.,2009, Masuda et al., 2013), a new thixotropy model was developed for estimating the cluster size in an unsteady non-uniform flow.  With this model, the time-variation cluster size distributions can be locally obtained by a local balance of Brownian and shear coagulations and shear break-up of clusters.  In this study, this model has been applied to simulate the unsteady dispersion characteristics of fine particles in a flow between coaxial cylinders, when the inner cylinder rotates as a first step in order to investigate the dispersion characteristics in an extruder with rotors.  The unsteady start-up flow when the inner cylinder moves in one way with a constant angular velocity was focused on before the flow was fully developed as a steady Couette flow.  Under such a situation, fine particle clusters break up non-uniformly in space and in time.  From the results, the mechanism of particle dispersion under an unsteady flow can be basically clarified.

In this study, the effects of inter-particle bonding energy on dispersion characteristics have been investigated with the above situation.  The inter-particle bonding energy was changed in three steps; 5x10^-15, 5x10^-16 and 1x10^-16 J.  The computational domain was set in the region between coaxial cylinders, where the inner cylinder starts to rotate at the beginning of the computation.  The inner and outer cylinder diameters and the rotating angular velocity of the inner cylinder were set at 22.8 mm, 38 mm and ???.  Between coaxial cylinders, a Newtonian molten polymer with the viscosity of 195 Pas was filled and the fine particle with the diameter of 2.5 mm was included in the molten polymer with the solid volume fraction of 0.2.  As the initial condition, the cluster size was assumed to distribute uniformly at any position in the domain from 2 to 50 of particle number in a cluster.  Thus, the initial average particle number in a cluster was 25.5. 

The governing equations are two-dimensional continuity and momentum equations for the simulation on the fluid flow.  For the dispersion characteristics, the thixotropy model determining the time-variation of cluster size distribution affected by the flow shear rate was used.  The suspension viscosity was calculated by use of Simha’s model from the apparent solid fraction estimated with particle cluster size distributions.  This was used for the fluid viscosity in the momentum equations for fluid flow.  From the fluid flow, the local shear rate was obtained and leaded to local breakup of the clusters.  Thus, two-way coupling method was used for the flow and for the cluster distributions.

From the start-up of the inner cylinder rotation, the velocity and momentum fields were developed from the region near the inner cylinder to the region near the outer cylinder.  It was found from the results that the flow developing characteristics between coaxial cylinders was not severely affected by the inter-particle bonding energy under the present conditions.  On the other hand, the inter-particle bonding energy affects the cluster breakup behaviors.  The low inter-particle bonding energy accelerates breakup of clusters and high value decelerates the breakup rate.  The cluster breakup should obey the characteristics time determined by the inter-particle bonding energy divided by the viscosity, the square value of the shear rate and the cube value of the particle diameter.  It was found that the breakup behavior of clusters at each radius position can be rather well expressed with this characteristics time, though the difference due to the inter-particle bonding energy was not small in the outer region. 

On the other hand, the breakup rate decreases in the radius direction even on the time normalized by the characteristic time.  The shear rate is very high near the inner cylinder at the start-up while the shear rate is zero near the outer cylinder.  Thus, the time-dependency of the breakup was also affected by the shear history.  In order to take the shear history at each position into consideration, a normalized time was defined with the time multiplied by the integrated shear strain and divided by the characteristic time.  This is found to express rather well the time dependency of the cluster breakup at any flow positions between coaxial cylinders. 

However, the breakup of clusters was faster near the inner cylinder even on that normalized time.  From the detailed investigation, the solid volume fraction was found to decrease near the inner cylinder.  This occurs due to the fluid movement for the continuity at the start-up.  The particle moves with the fluid in the radius direction at the start-up and this causes the lower shear coagulation rate near the inner wall due to the low solid volume fraction.  Thus, it was concluded that the breakup of clusters quickly occurs in the region near the inner cylinder due to the fluid movement in the radius direction under the startup situation.

References

Hasegawa, E., H. Suzuki, K. Kameyama, Y. Komoda and H. Usui, 2009, “Model analysis on dispersion characteristics of fine particles in Newtonian molten polymer,” Advanced Powder Technology, 20-2, pp.139-144.

Masuda, K., H. Suzuki , Y. Komoda and R. Hidema, 2013, Particle dispersion/aggregation model in a non-uniform shear flow,” Nihon Reoroji Gakkaishi, 41-2, pp.69-75.

Usui, H., 1999, Rheology model of aggregable slurry on monodisperse silica particles,” Kagaku Kogaku Ronbunshu, 25-3, pp.459-465.