(170a) Mixing of Cohesive Particles in a Shear Cell

Mixing of granular materials has economical importance in different industries, such as foodstuffs, pharmaceutical products, detergents, chemicals, plastics etc. In many cases a better mixing process could tremendously increase the quality and the value of product. Thus mixing is regarded as a key process and needs further study. However, mixing of granular materials has received less attention than fluids, althoughthe subject of granular mixing in several kinds of mixers has drawn the interest of several researchers in recent years. The convective and dispersive mixing mechanisms in a rotating tumbler, in a V-blender and in a tote blender had been examined by both experimental tests and computer simulations. From several studies, it was demonstrated that the mixing process occurred through a diffusion mechanism in different transport devices. In past decade, the interesting effect of cohesion between grains has intrigued researchers to investigate the mixing of cohesive particles. In many instances, the cohesive nature of a powder sample is a prime factor causing difficulties in powder flowability resulting in, for example, channeling and defluidization in combustion/feeder systems.

In this study, experiments of granular motion are performed in a shear cell, which is composed of an upper disk, a bottom disk and a motor. The bottom disk, with outside diameter of 45.00 cm, is driven by a 3 hp server motor. The rotation speed is controlled and can be measured by the server motor. The bottom disk is made of plexiglass to permit observation. An annular trough with inside diameter of 31.67 cm, outside diameter of 42.02 cm and depth of 4.5 cm was cut in the bottom disk. A stationary upper disk could be inserted into the trough where the granular materials are put in the test section. The height of the test section h could be adjusted by moving the upper disk and be measured by a dial indicator. The bottom wall velocity u₀ was chosen as 1.32 m/s in all experimental tests. Soda lime beads with an average diameter d_p of 2 mm (standard deviation of 0.09 mm and particle density of 2476 kg/m³) were used as granular materials in the experiments.

This paper intends to investigate the effect of adding little amount of liquid on the mixing behavior. We use silicoin oil as adding liquid with density of 950 kg/m³, surface tension of 20.4 mN/m and viscosity of 50 cs (centi-stoke, cm²/s). The dimensionless liquid volume V* is defined by the volume of adding silicoin oil divided by the summation of the volumes of adding silicoin oil and of the total glass beads. The current study only uses dimensionless liquid volume as the controlling parameter and V* = 0, 1.3*10^-4, 2.5*10^-4, 6.5*10^-4, 1.3*10^-3and 2.5*10^-3.

For mixing experiments, the same particles but with different colors are initially arranged in the top-bottom organization to investigate the mixing behavior of granular materials. The mixing process was recorded by a high-speed camera. The test section was divided into 10 regions along the transverse direction. Using image processing, the concentrations of white particles C(t) in each region were determined. The mixing layer thickness

 is defined from the width with concentrations ranging from 0.05 to 0.95. The symbols
 and
 denote the thicknesses of the mixing layers in the upper and the lower parts, respectively.

To investigate the diffusion process, the velocities, fluctuation velocities and self-diffusion coefficients were also determined from experiments by different combinations of colored particles: The black particles served as tracer particles and about 15% of black particles were mixed uniformly with 85% of white particles. All the experimental settings were the same as the mixing experiment, except for the arrangement of colored particles. The autocorrelation technique was employed to process the stored images and to decide the shift of each tracer particle in every two consecutive images.

Every test in this study was done for a total granular mass of 2.0 kg and a fixed channel height (2h) of 2.76 cm. Hence the average solid fraction of the channel was a constant, 0.6178. The results of experiments were compared with calculations from the diffusion equation. Meanwhile, the properties concerned with the development of mixing layer, for example, the mixing growing rate, the apparent self-diffusion coefficient, the granular temperature and the shear rate, were studied to explore the characteristics of mixing phenomena in the shear cell.

From the results of average distributions in the shear cell, the channel can be divided into two flow regimes: the ¡°ìfluid-like regime¡" in the upper section and the ¡°ìsolid-like regime¡" in the lower section, due to the scale of the shear rate.  In this paper, the top-bottom initial loading pattern is used to investigate the particle mixing in the shear cell. Thus, the test section is also divided into two regions, the upper channel (y>0) and the lower channel (y<0). To examine the influence of V* on the development of the mixing layers, the symbols in Figure 1 shows the developments of

Figure 1. The developments of the mixing layer thicknesses with time.

the mixing layer thickness with time for the six tests with different liquid volumes. It is clear that the mixing layers expand with time and develop faster in the initial stages. The mixing layer thicknesses of the upper channel,
, are slightly greater than those of the lower channel,
. Moreover, the influence of the adding liquid amount on the development of mixing layer is significant. From this figure, the mixing layer thicknesses are greater for the dryer cases.

Granular mixing mainly resulted from the diffusive motions of granular materials in the shear cell. Because of the top-bottom organization of different-colored particles, the particle mixing occurred in the vertical direction. Thus the diffusion equation can be written as

where C is the concentration of white particles, and D_yy is the transverse self-diffusion coefficient. The initial conditions are C = 1 when t = 0, y > 0; and C = 0 when t = 0, y < 0. The mixing layer thicknesses almost stop growing after a certain time (60 seconds), even with longer observation of the mixing process up to a half hour, and the mixing layer can not reach the walls. Thus it is reasonable to assume the boundary conditions for the above equation as:
as
; and
as
. The transverse self-diffusion coefficients D_yy is actually varied with the height of the channel and will be investigated later. However, in order to get the analytical solution, D_yy in Eq. (1) can be assumed as constants: apparent self-diffusion coefficients D_app,1 and D_app,2, to explore the bulk mixing behavior in the upper and lower channels respectively. In fact, the apparent self-diffusion coefficients denote the ¡°ìaveraged¡" self-diffusion in the upper and lower channels. Thus, we solve Eq. (1) for the regions of
(upper channel) and
(lower channel) with one additional boundary condition by assuming
 at y = 0. After solving the equations of C(t, y), according to the definition of mixing layer thickness in the experiments, C = 0.95 at y =
and C = 0.05 at y =
, that gives

The fitted curves in Figure 2 are the results of least-squares fits using the forms of Eqs. (2) and (3), indicating close correspondence with the experimental data. The deviations are due to the assumption of constant self-diffusion coefficients in the upper and lower channels in the diffusion equation.

The values of the apparent diffusion coefficients can be determined from the fitted curves (Eqs. (2) and (3)) in Figure 1. The results demonstrate that D_app decreases with the increasing liquid volume. Besides, the apparent self-diffusion coefficients in the upper channel are greater than those in the lower channel, indicating a better mixing in the upper channel. In addition, the mixing layer thicknesses of the upper part are greater since the apparent self-diffusion coefficients are greater. Therefore, the apparent self-diffusion coefficient D_app is an appropriable index to describe the ¡°ìaveraged¡" granular mixing condition in the shear cell.

The above mixing result is consistent with our earlier studies and verifies that in a granular flow system, adding more viscous liquid causes the system to become more cohesive and the flow behavior to become more solid-like. Thus, in many industrial applications, the granular flow system should be kept dry to avoid difficulties in transportation.

In this study, we also measure the distributions of diffusion coefficients. The semi-theoretical mixing layer thickness, calculated from the diffusion equation, Eq. (1), by substituting the measured self-diffusion coefficients, agreed well with the experimental data, indicating that the particle mixing in the shear cell was governed by the diffusive mechanism.