(116d) Mixing of Granular Materials in Non-Circular Geometries

Introduction: Mixing of granular materials is a common unit operation in many of the chemical, metallurgical, pharmaceutical, fertilizer industries, just to name a few. Several processes are carried out in a horizontal rotary drum rotating about its own axis (e.g. rotary kilns, driers, mixers etc.). A variation in the rotational speed of the mixer leads to different regimes of flow within a mixer. The rolling regime, where there is a steady flow of material within a thin cascading layer at the free surface of the rotating bed (Henein et al. (1983)), is most commonly used in industrial processes. Mixing in horizontal drums occur in both transverse and axial directions. Although mixing in transverse direction is much faster as compared to that in the axial direction, in most continuous flow applications the ideal system comprises perfect transverse mixing and no axial dispersion. Thus improvements in the rate of transverse mixing are of considerable practical importance. In the rolling regime, the transverse plane can be divided into two distinct regions (Lehmberg et al. (1977)), namely - the flowing layer, where particles cascade down and the rotating fixed bed, where the particles are carried up by the drum wall. In the flowing layer the flow along the free surface is responsible for nearly all the mixing in the transverse direction. In spite of its wide range of applications, the understanding of the mixing of granular materials still remains incomplete. The presence of particles with different size, shape, density or surface property adds to the complexity and leads to segregation of particles. Several models have been developed in the past to understand the flow of granular material in the cascading layer (Khakhar et al. (1997), Elperin et al. (1998)). Conventional rotary drums are circular in cross section, where the motion of particles is regular. Recently it was shown that chaotic advection can be obtained in granular flows when rotated in non-circular cylinders (see Fig. 1). This concept was explored both experimentally and computationally in quasi-two-dimensional rotating containers of non-circular cross-sections (Khakhar et al. (1999)) that are symmetric about an angle of rotation. The focus of the current work is to study mixing in different cross section cylinders with the objective of determining mixer shapes that give best mixing rates obtained experimentally for different non-circular cross-sections and comparing with that of the conventional circular geometry.

Experimental Procedure: Mixing experiments are carried out in quasi 2-D geometries, Circle, Square, Tri2, Tri4 and Star (Fig. 2 ), that are half-filled. Glass beads of same size and density, colored in red and green are placed on either side of the center of the container. The particles are 3 mm in diameter and the thickness of the mixer is 10 mm (above 3 particle diameter) in all the cases studied. The mixers are rotated at 2 rpm using a computer controlled stepper motor. Digital photographs are captured using a digital camera (Nikon Coolpix 990) at high shutter speeds when the free flowing surface is parallel to one of the edges of the square geometry or when the surface touches the corners of the triangular projections in the case of other shapes. Each experiment is repeated thrice. The images captured from the experiments are analyzed using image analysis program. Pixels having the fractional green value beyond a particular threshold are converted to green and fractional red value beyond another threshold value to red. The thresholded images thus obtained are analyzed using two methods - intensity of segregation and centroid methods. The filled portion of the geometry in the thresholded images is divided into square bins of the size of 4 particle diameter and the concentration of green particles is calculated in each bin. The mean value and the standard deviation are calculated. The standard deviations thus calculated for each image (after every quarter or half rotation) of the mixer are normalized to 1. An average over the three sets of experiments is taken. An alternate method used for quantitative comparison is - tracking of the positions of the centroids of the two groups of differently colored particles (Metcalfe et al. (1995)). The centroid positions of the two groups are taken relative to that of the whole material and are normalized to one initially. When the material is perfectly mixed, the centroids of the two groups coincide with that of the whole material. The centroids are calculated for all the digital images, taken after every quarter or half rotation of mixer.

Results and discussion: The results, obtained in terms of intensity of segregation, are plotted against the number of rotations on a linear-log plot which is shown in Fig. 3. The initial mixing rates in these mixers is given by the slope of these curves. From the figure one can see that circular cylinders have best initial mixing rates as compared to any non-circular mixer, which is not apparent from visual observations. This is because the measure is sensitive only to local mixedness (4x4 particle diameters) and hence not useful in the initial stages when global mixedness is reducing rapidly. However, the measure is useful for the later stages when finer scale mixing occurs. At the late stages, it is evident that the circular mixer has high value of intensity of segregation as compared to other geometries. Hence the non-circular mixers lead to better equilibrium mixed state as compared to circular mixers. A comparison among the non-circular geometries show that Tri4, Star and Square have best equilibrium mixed state. Fig. 4 shows a comparison among the geometries using centroid method. As the mixer rotates, the centroid orbits oscillate, whereas the mixing of the particles due to flow causes these orbits to decay exponentially (see inset in Fig. 4). The decay time constants (mixing rates), as can be seen from Fig. 4, are low in non-circular mixers compared to circular mixers. Comparison among geometries suggest that Tri4 is the best mixer geometry, followed by Square and Star geometries. From Fig. 3 it can also be seen that Tri4 has best equilibrium mixed state. Hence, it might be concluded that both in terms of mixing rate and final mixed state Tri4 geometry is best among the geometries studied.

Conclusions: Mixing of granular solids is a complex phenomenon and is determined by both the kinematics of the mean flow and local diffusive and segregation fluxes arising from random motions of particles and from differences in properties of the particles being mixed. If the cross section of the rotating cylinder is circular, the mean flow is time independent and hence it is regular. In non-circular tumblers the flow is time periodic and hence exhibit chaotic advection. Presence of chaos in granular mixing increases the mixing rate. Mixing of same sized and density particles have been studied experimentally. Non-circular geometries mix the material faster compared to circular geometries. It was found that the circular geometry with four triangular projections protruded inside has the best mixing rate as well as best final mixed state. Square and Star geometries also have good mixing rates.

References: 1. J. Lehmberg, M. Hehl, K. Schugerl, Transverse mixing and heat transfer in horizontal rotary drum reactors, Powder Technology 18, 149-163 (1977). 2. H. Henein, J. K. Brimacombe, A. P. Watkinson, Experimental studies of transverse bed motion in rotary kilns, Metall. Trans. B 14B, 191-205 (1983). 3. Guy Metcalfe, Troy Shinbrot, J. J. McCarthy, J. M. Ottino, Avalanche mixing of granular solids, Nature 374, 39-41 (1995). 4. T. Elperin, A. Vikhansky, Kinematics of the mixing of granular materials in slowly rotating containers, Europhys. Lett. 43, 17-22 (1998). 5. D. V. Khakhar, J. J. Mccarthy, T. Shinbrot, J. M. Ottino, Transverse flow and mixing of granular materials in a rotating cylinder, Phys. Fluids. 9, 31-43 (1997). 6. D. V. Khakhar, J. J. Mccarthy, J. F. Gilchrist, J. M. Ottino, Chaotic mixing of granular materials in 2D tumbling mixers, CHAOS 9, 195-205 (1999).