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Axial Mixing of Binary Mixture in Horizontal Rotating Cylinder

Parag H. Malode and D. V. Khakhar. Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai - 400076, India

Granular materials occur quite extensively in nature as well as industry. The phenomenon of segregation is unique to granular materials. Various factors are known to affect the mixing and segregation of the granular materials. These include particle characteristics and interactions, container shape, degree of fill, speed of rotation or vibration, etc. (Ottino and Khakhar, 2000). Considerable volume of granular material is processed in rotating cylinders. Depending on the speed of rotation, different modes of motion result which are categorized as slipping, slumping, rolling, cascading, cataracting and centrifuging (Henein et al., 1983; Melmann, 2001). The rolling mode is the most employed in industrial processes using rotary equipments such as kilns, drum mixers etc.
In the rolling mode, the granular bed can be divided into two regions, a flowing layer, wherein the particles cascade down along the free surface; and a larger fixed bed, wherein the particles are carried up the drum wall. Mixing is predominant within the flowing layer and is the focus of interest. The objective of the work is to study the axial mixing of a binary mixture consisting of two different sizes in a horizontal rotating cylinder operated within the rolling regime.
Theory Mixing in the axial direction is considered to be purely diffusive caused by the random collisions of particles in the flowing layer. The axial dispersion model is the most common method of modeling axial mixing in both flowing and non-flowing, horizontal, rotating drums wherein the bed of particles is treated as a continuum. The mixing along the axis is represented by Fick's law (Sherritt et al., 2003),
\begin{displaymath}  \frac{\partial C}{\partial t} = D_{a}\frac{\partial ^{2}C}{\partial z^{2}}-u_{a}\frac{\partial C}{\partial z}  \end{displaymath} (1)

where, $D_{a}$ is the axial dispersion coefficient, $C$ is the concentration fraction of particles varying with the position $z$ along the axis, $u_{a}$ is axial velocity and $t$ the instantaneous time. The axial dispersion model has the advantage that a single parameter is used to quantify the amount of mixing without any limit on the number of particles.

In batch experiments, there is no net axial flow and hence no net axial velocity. Thus, with $u_{a}$=0, equation 1 reduces to the one-dimensional equation 2 (Hogg et al., 1966).

\begin{displaymath}  \frac{\partial C}{\partial t} = D_{a}\frac{\partial ^{2}C}{\partial z^{2}}  \end{displaymath} (2)

The solution of equation (2) for long times, gives the approximation (Rao et al., 1991),
\begin{displaymath}  C(z,t) = \frac{1}{2} + \frac{2}{\pi}{\mathrm{exp}}\bigg( -\f...  ...}D_{a}t}{L^{2}}\bigg){\mathrm{sin}}\bigg(\frac{\pi z}{L}\bigg)  \end{displaymath} (3)

where, L is the length of the cylinder along the axis. A plot of the concentration fraction against $sin(\pi Z/L)$ yields a slope from which $D_{a}$ can be calculated.
Experimental and analysis methodology
Experiments consist of tracer studies using a binary mixture of identical glass beads. Equal quantities of the two different-colored tracers are placed in a horizontal cylinder adjacent to each other so that the cylinder is exactly half-filled. This degree of fill is used for all experiments. The cylinder is then rotated at different rotational speeds and the surface concentration is captured through digital photography at specific intervals of time. Each digital image thus obtained constitutes of pixels, which is the unit of measurement in a digital format. The images thus obtained are then thresholded using a custom software program to identify the two colors at the pixel level. The total cylinder length along the axis is divided into equal vertical bins. The concentration fraction of one-colored pixels in each bin is determined by averaging the number fraction of all those one-colored pixels in that particular bin. This concentration fraction for each of the bins is plotted against $sin(\pi z/L)$ which corresponds to the axial position along the cylinder length, as shown in Figure 1.  The diffusivity, in terms of the axial dispersion coefficient is then calculated from the slope of this plot. The intercept is found to be close to 0.5.
Figure 1: Concentration fraction of red particles plotted against its axial position
Image cfracSin_3mm17-7577
Similar experiments with mixtures of two different sizes are being carried out. Axial segregation leading to band formation is observed due to size difference in the mixture of particles. This however occurs only beyond a critical rotational speed. Below this speed, the different size particles mix and do not segregate. Mixing experiments with different sized particles shall be carried out at the lower rotational speeds. These shall yield the individual and collective diffusivity of the particles.
Results
The effect of change of rotational speed of the cylinder and particle size on diffusivity is studied. The range of cylinder rotational speeds studied include 5,10,15,20 & 25 RPMs whereas the particle sizes are 1, 2 & 3 mm mean size. Figure 2  is the plot of the variation of concentration fraction with time for both 2mm and 3mm particles, by image analysis. The progression of the concentration fraction curve from a step profile to a straight line justifies the application of the theory. It is observed that the slope of the concentration fraction profile obtained from image analysis is always greater than that obtained from sampling. As a result, image analysis results in a lower axial dispersion coefficient as that compared to sampling. Figure  3  illustrates the comparison of $D_{a}$ obtained from image analysis with that from sampling. Both the methods compare well qualitatively. The diffusivity of particles increases with increase in rotational speed of the cylinder for a given particle size. The diffusivity of particles decreases with increase in size of particles for a given rotational speed. It is proposed to correlate the results to the flow and diffusivity in the layer by Hajra and Khakhar, 2005.
Figure 2: Comparison of concentration profiles obtained theoretically and experimentally for all cases
Image rfracT23_xfig

Figure 3: Diffusivity obtained by image analysis and sampling
Image da_rpmSize23
Bibliography
Hajra, S. K. and D. V. Khakhar, ``Radial mixing of granular materials in a rotating cylinder: Experimental determination of particle self-diffusivity,'' Phys. Fluids, 17, 013101 (2005).
Henein, H., J. K. Brimacombe, and A. P. Watkinson, ``Experimental study of transverse bed motion in rotary kilns,'' Metall. Trans. B, 14B, 191-205 (1983).
Hogg, R., D. S. Cahn, and D. W. Fuerstenau, ``Diffusional mixing in an ideal system,'' Chem. Eng. Sci., 21, 1025-1038 (1966).
Melmann, J., ``The transverse motion of solids in rotating cylinders - forms of motion and transition behavior,'' Powder Technol., 118, 251-270 (2001).
Ottino, J. M. and D. V. Khakhar, ``Mixing and segregation of granular materials,'' Annu. Rev. Fluid Mech., 32, 55-91 (2000).
Rao, S. J., S. K. Bhatia, and D. V. Khakhar, ``Axial transport of granular solids in rotating cylinders. Part 2: Experiments in a non-flow system,'' Powder Technol., 67, 153-162 (1991).
Sherritt, R. G., J. Chaouki, A. K. Mehrotra, and L. A. Behie, ``Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor,'' Chem. Eng. Sci., 58, 401-415 (2003).