Powder Technology 253 (2014) 107–115
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A design heuristic for optimizing segregation avoidance practices in horizontal drum mixers Tathagata Bhattacharya, Suman K. Hajra, J.J. McCarthy ⁎ Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA
a r t i c l e
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Article history: Received 18 February 2013 Received in revised form 24 September 2013 Accepted 12 October 2013 Available online 29 October 2013 Keywords: Granular materials Rotating Tumbler Mixing Segregation DEM Simulation
a b s t r a c t Segregation is a major problem for many solid processing industries. Differences in particle size or density lead to flow-induced segregation within the surface layer. Here we examine methods of avoiding radial segregation in a horizontal drum mixer. Recently, it has been suggested that segregation in this type of particle mixer can be thwarted if the sheared (surface) regions of the bed are inverted at a rate above some critical frequency. Further, it has been hypothesized that the effectiveness of this technique can be linked to the probability distribution of the number of surface layer “passes” a particle takes per rotation of the drum. In this article, various baffle configurations are numerically and experimentally studied to investigate the efficacy of this measure as a design heuristic for the development of improved drum mixing devices. We choose the horizontal drum geometry as it represents the simplest possible example of a tumbler-type mixer, however, we expect the results found here regarding the efficacy of our design heuristic to be generic for any surface-dominated mixing device. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Granular materials are ubiquitous and it is hard to find a process industry which does not handle granular materials in some form. These materials are widely used in chemical, pharmaceutical, cement, food, construction and metallurgical industries. Mixing, heating and granulation of particulate materials are just a few of the many examples where a rotating drum or tumbler is used to achieve a desired objective. However, when a mixture of two different types of particles is tumbled in a rotating cylinder (a so called solid mixer) they tend to segregate or de-mix. This undesired phenomenon causes problems in maintaining uniform product quality and results in revenue losses across many industries. The flow behavior in a rotating cylinder can be complex and it is well established that mixing and segregation patterns are sensitive to the container geometry and fill level [1–3]. Henein and others [4,5] have identified multiple regimes of flow with increasing cylinder speed of rotation: slipping, avalanching, rolling, cascading and centrifuging. In industrial applications, rotating cylinders are typically operated in the rolling flow regimes. In the rolling regime, a thin layer of particles moves at high velocity in the free surface (called the shear or “surface” layer), whereas the rest of the bed rotates as a solid body (called the passive layer or static bed) [6]. In a half-filled tumbler with circular cross-section, the mixing and segregation of particles occur only in the
⁎ Corresponding author. Tel.: +1 412 624 7362; fax: +1 412 624 9639. E-mail address:
[email protected] (J.J. McCarthy). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.10.035
shear layer and, therefore, it is the shearing layer that needs particular attention if something is to be done to thwart segregation. One can observe that past studies that proposed solutions to combat segregation can be categorized into two groups, both of which are generally devised on an ad-hoc basis, and include: change the particles or change the process [7]. Samadani and Kudrolli [8] found that segregation could be reduced by adding a small volume fraction of fluid to a mixture of small and large glass beads. Li and McCarthy [9] found that segregation could be turned off or on by adding small amounts of moisture when using particles of various types, multiple sizes, and different surface characteristics. Hajra and Khakhar [10] showed that segregation could be eliminated by using a small (in comparison to the diameter of the cylinder) rotating impeller placed at the axis of rotation, that is, in the surface layer. Jain [11] and Thomas [12] performed experiments for binary mixtures composed of different sizes and different density particles and they found that mixing can occur instead of segregation if the denser beads are bigger (as the segregation tendencies will cancel), and also if the ratio of particle size is greater than the ratio of particle density. Recently, a more theory-based study by Shi et al. [13] has shown that periodic flow inversions via selective baffle placement – in a tumblertype mixer – can serve as a generic method for eliminating segregation regardless of particle properties. Beyond the work of Shi et al. [13], we find that very little is known from a theoretical point of view on the effect of baffles on solid mixing, even in simple cases limited to monodispersed systems [1,14]. In this paper, we use experiments and simulations to study the effects of various baffle designs and operating parameters on mixing
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of binary mixtures with different sizes or densities. We confirm that periodic flow inversion can be used to reduce segregation in a rotating cylinder and explore different baffle shapes and placements that are engineered based on Shi et al.'s [13] hypothesis of flow modulation. Attempts are made to link this technique to the probability distribution of the number of times a particle passes through the surface layer per rotation of the drum. Using this information, predictions are made as to which baffle configurations would produce better mixing, so that we might use this criterion as a design heuristic for screening future mixer geometries. 2. The hypothesis: time modulation and layer pass analysis Time-modulation in fluid mixing and other dynamical systems [15] is a fairly common practice, but has found only limited applications in granular processing [1,16,17]. As already mentioned in the Introduction section, Shi et al. [13] have shown that periodic flow inversions via selective baffle placement can serve as a generic method to thwart segregation. The key to adapting this idea to free-surface segregation lies in recognizing two important facts: that it takes a finite time for material to segregate and that there is always a preferred direction that particles tend to segregate. For example, in a free-surface flow, small particles need time to percolate through the flowing layer; thus, if the flow is interrupted before the small particles reach the bottom of the flowing layer, segregation can be prevented. This relatively simple observation can be employed to engineer systems that counteract segregation. In order to capitalize on the fact that this flow interruption can thwart segregation, one next needs to invert the flowing layer, prior to reinitiating the flow (and the segregation). One way to achieve this two-step process in a continuous flow is to invert the flowing layer at a sufficiently high frequency, f, f N t−1 s , where ts is the characteristic segregation time. A critical issue with this technique is that a full understanding of segregation kinetics – and therefore, the characteristic segregation time, ts – is still lacking. However, this hypothesis can be tested indirectly for many baffle configurations in a rotating cylinder by examining the probability distribution of the number of surface layer passes a particle takes per rotation of the drum, as described below. Consider an unbaffled, half-filled tumbler with circular cross section. In this geometry, one observes that the flow is composed of: first, a pass through the sheared/surface layer, then a pass through the static portion of the bed, and so on. It is important to note that the static portion of the bed in a rotating drum simply “stores” the material and returns it (after a full 180-degree change in orientation) for its next pass through the surface layer. Moreover, in an unbaffled tumbler, the surface layer itself also induces a 180-degree orientation change (due to the almost-linear shear profile within this layer). Thus, for a full rotation the effect of the static bed and surface layer on the segregation orientation cancels each other and a particle will return to the segregating region (the surface layer) always in the same orientation in which it left. If one were to quantify this orientation cancelation process, we could argue that matching a single surface layer pass to a single static bed pass is the worst-case scenario for segregation. In contrast, if the frequency of surface layer passes and static bed passes is decoupled, the orientation of segregation (for a tracer particle) would seemingly change for each iteration through the surface layer. Now if we place a baffle near the axis of rotation (thus, near to the shearing layer), we periodically alter the flowing layer so that we achieve both (i) a smaller average uninterrupted flow length, L, and (ii) significant variation in the time between layer passes. The fact that the static bed and the flowing layer(s) no longer produce related orientations is key to the results reported here. In order to test this hypothesis, the numerical tool DEM (Discrete Element Method) is used in the present study. The details of this technique along with its application to the present problem will be elaborated in the subsequent sections.
3. Modeling: Discrete Element Method (DEM) The Discrete Element Method (DEM) is also known as Particle Dynamics (PD) [18] where the trajectory of each and every particle is tracked via simultaneous integration of the interaction forces between individual pairs of particles [19,20]. While these forces typically include only contact forces – normal (Hertzian) repulsion and tangential (Mindlin) friction, see Ref. [21] – and gravity, additional particle interaction forces (such as surface adhesion [22,23] and van der Waals [24]) can be easily incorporated. In particular, this technique has been quite successful in simulating ensembles of granular materials, yielding insight into such diverse phenomena as force transmission [25], packing [26], wave propagation [27], agglomeration formation and breakage [22], cohesive mixing [28], bubble formation in fluidized beds [29], and segregation of free-flowing materials [30]. In a granular flow, the particles experience forces due to interactions between particles (e.g., collisions, contacts, or cohesive interactions) as well as interactions between the system and the particles (e.g., gravitational forces). In this work, the collisional forces are modeled after the work of Hertz and Mindlin [21]. A thorough description of the interaction laws from contact mechanics (collision forces) can be found in elsewhere [23,31,32]; therefore, they will not be reviewed here. Typically, a numerical experiment consists of a mono-disperse/bi-disperse system of perfectly smooth spheres bounded by a wall of immobile particles with periodic boundaries in the longitudinal z direction. The number of particles is determined by the fill level and the particle diameters. The wall of the drum is rotated at a constant angular velocity, ω. A typical initial condition for the rotating tumbler simulations is obtained by allowing a bed of particles, arranged in a randomly perturbed lattice, to settle under the action of gravity. From this initial configuration, a prescribed angular velocity is imposed and the simulation is allowed to proceed for approximately 20 revolutions based on the rotation rate prescribed. Then, the Intensity of Segregation (Eq. (1)), as defined in the next section, is determined as a function of time in order to measure the mixing (or lack thereof). Our experiments use cellulose acetate particles, however, in many of our simulations, the particle stiffness and other parameters used are reduced in order to decrease the required simulation time (using so-called “soft” particles; a practice shown to have essentially no impact on flow kinematics [20]). Table 1 lists the material properties used in the simulations. We should note that, when directly comparing simulation results with the experimental data, we use an elasto-plastic model with actual material properties (for cellulose acetate). On the other hand, “soft” material properties are used for performing other studies (e.g., effect of number of baffles) when comparison with experimental data is not the primary aim, as simply capturing correct kinematics should be sufficient in these cases. 4. Experiment Experiments are carried out in a quasi 2D rotating cylinder (1.5cm in length and 13.8 cm in diameter), which is mounted on a circular plate attached to a bigger rotating drum. The bigger drum is rotated using a computer controlled stepper motor at a fixed rate (6RPM). The cylinder is made up of two sets of transparent glass disks, that are fitted face-toface to close the cylinder. This arrangement also helps in dispensing
Table 1 Material properties used in the simulations. Parameter
Value
Young's modulus (E, GPa) Density (p, kg/m3) Coefficient of friction (μ) Poisson ratio (v) Yield stress (σ y, MPa)
1.5 (acetate), 0.03 (soft) 1300 (acetate), 1000 (soft) 0.30 0.43 (acetate), 0.33 (soft) 30.0 (acetate), 0.3 (soft)
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particles when a particular experiment is completed. We used nearly spherical cellulose acetate beads as the model particles. We mainly focus on size segregation experiments and binary mixtures (equal volume, 50:50 v/v) composed of 2 mm and 3 mm acetate beads are used. In a typical experimental run, particles are placed in the cylinder using a divider so that the initial particle bed is completely segregated. All experiments are carried out using 50% cylinder fill fraction. A digital camera (Nikon D200) with a resolution of 6 mega-pixels is used to capture the photographs as the experiment progresses and the evolution of the system from a completely segregated state to a mixed state is observed. Digitized images are also taken at low shutter speeds in a separate experiment to calculate the shearing or flowing layer thickness (δo) at the layer midpoint [33] when no baffle is used. Photographs are captured at every half cylinder rotation. An image analysis technique, which relies on the colors of the particles to identify differently sized particles, is used to compute the distribution of particles as a function of time. This procedure yields the information required to calculate the degree of mixedness for each experimental run. In this work, we quantify the degree of mixedness using the Intensity of Segregation (IS) (see Eq.( 1)) [30]. This measure is defined as the standard deviation of the compositions of the samples and, mathematically, it is defined as " IS ¼
N 1 X 2 ðC −hC iÞ N−1 i¼1 i
#1=2
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segregation (density/size) has no impact in developing arguments about why a special case (baffle configuration) is better. 5.1. Effect of baffle shape First, we examine four various baffle shapes: flat, S-shaped, reverse S-shaped, and C-shaped (Fig. 1). In each case, the effective length of the baffle remains constant. The effective length is defined as the length of the tightest bounding box enclosing the baffle in question. Fig. 2 shows the experimental and corresponding simulation images, which depict the asymptotic results for different shapes of baffles. The experimentally calculated IS is plotted in Figs. 3 (size segregation) and 4 (density segregation) as a function of time. For size segregation, the asymptotic average IS values (and standard deviation in parentheses) for the flat, reverse S, S and C-shaped baffles are 0.21 (0.01), 0.21 (0.01), 0.18 (0.01) and 0.19 (0.02), respectively. The corresponding IS values (and standard deviation) for the density segregation cases are 0.20 (0.02), 0.23 (0.02), 0.21 (0.01) and 0.20 (0.01), respectively. One can observe that there is no significant difference in mixing for the types of baffles considered. Since the asymptotic IS values are all well below 0.25, which is a typical threshold to determine if a system is mixed or segregated, this suggests that a flat baffle is comparable in performance to more complex/sophisticated designs like a C-shaped or an S-shaped baffle.
ð1Þ
where N is the number of uniformly distributed cells within the bed, Ci is the concentration at a specified cell, b CN is the mean value of the concentration of the mixture [30]. By way of interpretation, a low value of IS (say, below 0.25) corresponds to good mixing, while a “perfectly” segregated system will yield a value of 0.5 (for the equi-composition mixtures that we examine here). The concentration at a given point is calculated as the fraction of particles with a specific color (denoting a specific sized/density particle) in a box of specified size centered on the point. Also, as the IS is plotted over time and the mixing process evolves, a statistical steady-state (or the asymptotic mixing state) is defined as the point after which the Intensity of Segregation remains essentially constant. 5. Results In this section, we present the results obtained for different configurations of baffles for mixing in a tumbler. Both simulation and experimental results are presented, and they are also analyzed in light of the hypothesis that periodic flow inversion can thwart segregation. All modeling/experimental results are discussed by considering a constant baffle size (L/D = 0.5), which was experimentally examined previously by Hajra et al. [34] in the case of a flat baffle. We should note that the previous results were found to be largely insensitive to the length of the baffle, provided that it was long enough to interrupt the flow within the surface layer. The focus of discussion here is on size segregation, however, in what follows we will see that the type of
5.2. Effect of baffle placement In this section, we investigate how the placement of a baffle within the tumbler with respect to the axis of rotation can change the extent of mixing. We shift the baffle, both in the transverse and radial directions, and the baffle position becomes asymmetric with respect to the axis of rotation. A few baffle positions are considered for this investigation on baffle placement: a symmetric baffle placed at the free surface, a symmetric baffle placed within the shear layer, a baffle placed outside the flowing layer, a completely asymmetric placement and baffles with segments (two/three segments). Fig. 5 shows a schematic of the various baffle configurations studied in this work. The maximum shear layer thickness δ0 is calculated using a technique described by Orpe and Khakhar [33] so that we may properly place the baffles, as described. The shear layer thickness (δ0) is found to be approximately 15 mm for size segregation experiments. Fig. 6 shows a qualitative comparison between experiments and simulations for various baffle placements when the mixing process has reached an asymptotically unchanging state. We observe that DEM reproduces the asymptotic behavior quite well. Fig. 7 shows the calculated experimental IS values over time for various baffle configurations for 2 mm and 3 mm acetate beads. The asymptotic average IS values for the different cases are 0.25 (baffle placed in passive layer or outside shear layer), 0.23 (baffle placed in shear layer) and 0.21 (baffle placed on free surface). The corresponding standard deviations are 0.02, 0.01 and 0.01, respectively. Therefore, one can observe that the case where the baffle was placed on the free surface or within the shear layer, produces a low IS (less than 0.25) and yields good
Fig. 1. Different baffle shapes considered for this study: C-shaped, S-shaped, reverse S-shaped and flat baffle, respectively. Effective length of all types of baffles is constant. The cylinder is rotated at 6 RPM. Also, note that the initial bed is completely segregated with one quarter of the cylinder filled with one kind of particles and the other quarter with the second kind of particles.
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Intensity of Segregation (-)
0.5 Flat Reverse S shape S shape C shape
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Time (s) Fig. 3. Quantitative results for size segregation in a rotating cylinder with different baffle shapes as indicated in the legend. The cylinder is rotated at 6 RPM.
half filled under the given conditions. It is expected that this is due to the fact that the particles interact with the baffle more frequently – on a per layer-pass basis – in this geometry, but we should remind the reader that this does not imply that the mixing rate is fastest in this case (simply that the asymptoticly mixed state is more random). 5.4. Hypothesis testing via layer-pass simulations In the previous section, we have shown both experimentally and by way of simulations that segregation can be reduced if we consider changing baffle shapes or their placement in the tumbler. We also
0.5
mixing. Since the flowing layer is very thin, and the degree of dilation of the bed is somewhat variable, it is understandable that these two baffle placements are very similar.
5.3. Effect of tumbler filling fraction In this section, we discuss how the filling level of the tumbler affects the mixing behavior. We present only simulation results here. The filling level is varied between 20 and 70% and in all the cases, a flat baffle is axially placed on the free surface. Fig. 8 shows the asymptotic average value of the Intensity of Segregation (IS) obtained from DEM simulations plotted against different filling fractions. It is to be noted here that all of the fill levels produce reasonably good mixing results as the Intensity of Segregation (IS) is below 0.25 in all cases. However, within themselves, the best possible mixing is obtained when the tumbler is
Intensity of Segregation (-)
Fig. 2. Comparison of experimental and DEM results for various baffle shapes. The images show asymptotic (steady-state) mixing state of the rotating drum containing 2 mm and 3 mm acetate beads. Left column shows the experimental results and the right column shows the results obtained from DEM. A good agreement is observed between these two.
Reverse S shape S shape C shape Flat
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Time (s) Fig. 4. Quantitative results for density segregation in a rotating cylinder with different baffle shapes as indicated in the legend. The cylinder is rotated at 6 RPM.
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Fig. 5. Schematic showing various baffle configurations and baffle placements. Note that δ0 is the maximum depth of the shear layer.
recognize that this enhancement in mixing is due to the flow modulation introduced by the periodic alteration (in a Lagrangian sense) of the flowing layer and hence the effective direction of segregation (as discussed earlier). In this section, we attempt to quantify the extent of this flow modulation by numerically “counting” the number of layer passes per rotation that are obtained for various combinations of baffle configurations. To visualize what happens for different baffle configurations, we use non-segregating particles in a rotating drum for our simulations. In this way, by using identical (same size and density) particles we can focus our attention only on a single variable, i.e., the effect of the baffles (not the effect of particle properties themselves) on the flowing layer and flow inversions, allowing us to gauge the effect of the baffles on the global pattern of flowing layers. The results obtained through DEM are post-processed in the following way to obtain meaningful images and plots. Individual particles are tracked to determine whether it is in a shearing (flowing) layer or not. In order to do this, we first calculate the time averaged velocity field from identical configurations of the tumbler for a number of revolutions. Then, using the time averaged velocities, the vorticity (i.e., the ! curl of velocity, ∇ V ) is calculated and stored in a lookup table as a function of both tumbler orientation and position within the bed. Since the particles within the static bed do not move relative to one another (hence they yield a local vorticity of zero), obtaining a non-zero value for the vorticity implies that the particles are found within a shearing/flowing layer. We should note that the introduction of baffles will lead to potentially more than the one flowing layer that is obtained in an unbaffled tumbler. Also, the baffles occasionally induce “shear bands” within the bulk of the material. We do not distinguish between the shear bands and flowing layers, since both can induce local segregation and instead count each instance of non-zero vorticity as a single “layer pass”. The vorticity is obtained in the following way. Only the x and y components of the velocity field are considered as we are interested in the vorticity in the z direction. Using a suitable grid, the average velocities
Fig. 6. Images showing qualitative comparison between experiments and DEM simulations of the asymptotic state of mixing in rotating tumblers with various novel baffle configurations. An unbaffled case (top) has also been shown for comparison. The emphasis here is on different kinds of baffle placements within the tumbler. Two different sized particles are considered: 2 and 3 mm acetate beads. The effective length of the baffles is fixed and the cylinder is rotated at 6 RPM.
of each of the discrete particles can be mapped onto the grid and the curl can be calculated. By doing this, we obtain a reference lookup table of vorticity values on the grid points at different time instances for one complete revolution of the tumbler. In the next step, we consider a number of revolutions of the tumbler and in each instance of time, we re-map the reference curl values (from the grids) onto discrete particles using the lookup table corresponding to the same tumbler configuration. Re-mapping is done using a suitable interpolation scheme. A cut-off curl value is used, above which, a particle would be thought to be in the shearing layer. In the present study, a cut-off curl of 0.3 is used, which gives accurate flowing layer boundaries as verified by the video obtained from temporal images. Finally, the number of times a particle passes through a shearing layer for certain revolutions of the tumbler is counted and a distribution is obtained. Eight revolutions (3rd to 10th, as the flow becomes periodically reproducible from the 3rd
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Intensity of Segregation (-)
0.5
Passive layer Shear layer Free surface
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0.1
0
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Time (s) Fig. 7. Quantitative results for size segregation for a rotating cylinder with a flat baffle fixed at different locations as indicated in the legend. The cylinder is rotated at 6 RPM.
revolution onward) of the tumbler are found to be suitable for obtaining the distribution and it is observed that the distribution does not change significantly if more revolutions (more averaging time) are considered. A single pass through the shearing layer would re-orient the particles by 180°; therefore, the segregation direction (orientation) will change during the mixing process if the particles pass through the layer, on average, in fewer than half a rotation. This requirement, that proper segregation avoidance requires something other than a one-to-one matching of layer passes and bed passes, is a generic consequence of the fact that the rolling regime results in closed streamlines. Thus, the orientation change from a layer pass is always recovered by a bed pass (in the absence of baffles).
5.5. Application of layer pass to other cases
0.2
In this section, we attempt to apply the previously built concept of layer pass simulations to two other cases: optimization of the shape of S-shaped baffles and examining the effect of the number of baffles on mixing (an industrially-relevant issue).
0.18
Intensity of Segregation (-)
Fig. 9 shows the shearing layers for different baffle configurations. Two different colors are used to demarcate “flowing” and static layers. Fig. 10 shows the probability distribution of the number of layer passes a particle takes per rotation of the drum for many different baffle configurations. On examining this plot, we observe that, as expected, the unbaffled case exhibits a very narrow distribution centered on about 0.5 rotations. This result implies that, for an unbaffled tumbler, essentially all of the particles pass through the layer in half of a revolution (so that the solid body rotation and layer pass cancel each other and effectively yield a constant segregation orientation). In contrast, the promising other geometries (such as those that include baffles that are flat, symmetric, and S-shaped) result in much broader distributions suggesting that the orientation of a particle will become essentially uncorrelated to its previous orientation as the layer passes and solid body rotation will not be commensurate. Therefore, these candidates are expected to produce better mixing results as per our hypothesis. We should note that the critical attribute of the layer pass distribution is its breadth, rather than the location of the peak center. The location of the peak center, in fact, is strongly dependent on the filling level, but does not have a strong bearing on the re-orientation potential of the flow. Now, we take a look at the real experimental data to assess the true mixing performance. Fig. 11 shows the comparison of experimental IS values for various promising baffle configurations. This plot shows that, experimentally, a flat baffle or an S-shaped baffle both yield good mixing in comparison to the other cases. In order to correlate these results with our layer pass analysis we plot (in Fig. 12) the layer pass peak heights (from Fig. 10) against the asymptotic IS values (obtained from both experiments in Fig. 11 and simulations in Figs. 3 and 7). We choose to plot the peak heights of the layer pass distributions as a simple measure of broadening, assuming that a higher peak height means lower broadening; thus, as the height of the layer pass distribution peak decreases we expect better mixing. As expected, a flat baffle fitted symmetrically at the free surface or an S-shaped baffle produces better mixing than other types of baffles. We believe that this characterization tool built on the hypothesis mentioned earlier can be effectively used to test different baffle configurations and predict their performance, even for actual industrial solid mixers.
0.16
0.14
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0.1 20
30
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Fill Level (%) Fig. 8. Asymptotic average Intensity of Segregation (IS) is plotted against different filling levels in a rotating tumbler. In all cases, a flat baffle is placed on the free surface of particle bed and satisfactory mixing is exhibited (i.e., none of the cases showed significant segregation).
5.5.1. Optimizing S-shaped baffles We have seen, experimentally and computationally, that the S-shaped baffle performs better in terms of mixing. Therefore, here we take a closer look at the shape of the S-shaped baffle, and attempt to explore the parameters of the S shape in the hope of obtaining better mixing. Different S shapes are formed by varying the amplitude of a sinusoidal curve. The S shapes are generated from y ¼ A sinð2π xLÞ, and we vary the amplitude A at five different levels (refer to Fig. 13). We perform similar layer pass simulations on these five S shapes, and it turns out that the mixing performance does not change significantly if the shape is altered (although is does differ from the flat case, where A = 0). This is evident from Fig. 14 where it is clear that all of the nonzero distributions overlap with each other. 5.5.2. Effect of the number of baffles As we mentioned earlier, conventionally, baffles are placed at the periphery (wall) of tumblers in industrial applications. Little is known about the effect of the number of baffles on asymptotic segregation in a solid mixer (although a recent paper has examined their impact on
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Fig. 9. Snapshots showing the evolution of the shearing layer at different times (time increases from left to right) for a few baffle configurations obtained from DEM. The baffle configurations (top to bottom) are S-shaped, flat, three-segmented and two-segmented baffles. Dark particles are in the shearing layer and the lighter particles constitute the static bed in these pictures. Note that in some images a shear band is evident.
mixing rate [35]). Therefore, starting with a base case of no baffle attached to the wall, we systematically vary the number of baffles in a tumbler up to eight in total. These different cases are shown in Fig. 15.
0.5
These radial baffles are oriented in such a way that they have equal angular distance. For example, for a case with 6 baffles, the baffles make an angle of 60° with each other if extended up to the center. The baffle
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No Baffle Three segmented Two segmented Symmetric central S-shaped
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Rotations per Layer Pass (-) Fig. 10. Probability distribution of the number of layer passes a particle takes per rotation of the drum for many different baffle configurations.
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Time (s) Fig. 11. Quantitative results obtained from size segregation experiments for a rotating cylinder with various promising designs of baffles. The cylinder is rotated at 6 RPM.
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Experiment Model unbaffled
IS (Asymptotic Average)
0.35 3-segmented
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asymmetric
3-segmented in-layer on free-surface
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0.2 Fig. 15. Tumblers fitted with various numbers of radial baffles at the wall. Up to eight baffles have been considered.
S-shaped
0.15 S-shaped in-layer
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Fig. 12. Correlation between layer pass peak height and the asymptotic IS obtained from experiments and DEM modeling. Each point corresponds to a particular baffle configuration as labeled, and for each configuration, symbols ● (filled circle) and ♦ (filled diamond) represent experiment and simulation, respectively. Note that the error bars are very small (short white lines inside symbols) as average IS values are calculated when the asymptotic state is reached.
Fraction of Particles (-)
Layer Pass Peak Height
Three Baffles Four Baffles
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Five Baffles Six Baffles Seven Baffles
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0 0 Fig. 13. Various S-shaped baffles have been shown here. The amplitude of the S shape A is varied from 5% to 20% of L with an increment of 5%. L is the effective length of the baffle.
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1
Rotations per Layer Pass (-) Fig. 16. Probability distribution of the number of layer passes a particle takes per rotation of the drum for various numbers of radial baffles fitted to the wall of a tumbler.
length is the same for all the cases (L ≈ D/4). As usual, other parameters like the number of particles remain unchanged for all the cases to single out the effect of only one parameter (i.e., the effect of number of baffles) on the mixing. We perform similar layer pass simulations as discussed in Section 5.4 and plot the results in Fig. 16. We observe that, except in the cases with no baffle or one baffle, all other distributions are centered around 0.5 rotations per layer pass (the range of peak positions for all cases is 0.44–0.53). Surprisingly, a wider distribution is obtained when there is no baffle or just a single baffle fitted to the wall of the tumbler, than for larger numbers of baffles. As more baffles are added, the distributions become narrower, suggesting a poor mixing outcome. Fig. 16 suggests that even an unbaffled tumbler has a better mixing (antisegregation) performance compared to a tumbler with a number of baffles fitted radially to the wall (as normally used in industrial practices).
0.5
A=0.0 A=0.05L A=0.10L
Fraction of Particles (-)
0.4
A=0.15L A=0.20L
0.3
0.2
6. Conclusions
0.1
0 0
0.2
0.4
0.6
0.8
1
Rotations per Layer Pass (-) Fig. 14. Probability distribution of the number of layer passes a particle takes per rotation of the drum for many different S-shaped baffles. A is varied from 0% to 20% of L with an increment of 5%. L is the effective length of the baffle.
Segregation in granular materials has been studied for a long time but its theoretical understanding even in the most simple cases is not yet complete. When particles differ in almost any mechanical property, a small agitation leads to flow-induced segregation. Controlling or minimizing segregation continues to be a complicated problem. In the present study, it has been demonstrated that segregation in a rotating drum can be dramatically reduced by introducing periodic flow inversions within the drum by employing novel baffle designs. Both
T. Bhattacharya et al. / Powder Technology 253 (2014) 107–115
experimental observations and simulation results agree qualitatively and the simulation tool is further used to test the hypothesis which states that time modulation in the shearing layer is the key to thwarting segregation. Segregation can be minimized if the particle flow is inverted at a rate above a critical forcing frequency. For a rotating drum, we have presented here evidence that this can be translated to the width/height of the probability distribution of the number of times a particle passes through the flowing layer per rotation of the drum. A broader probability distribution signifies that the orientation of a mass of particles with respect to the segregation direction will become essentially uncorrelated to its previous orientation as it iterates through the flowing layer. Therefore, the baffle designs that produce a broader distribution are expected to yield better mixing results. It has been shown that the peak height of the layer pass distribution correlates strongly with the experimentally obtained Intensity of Segregation. This observation actually demonstrates that the hypothesis of flow inversion can be used for designing new baffles and examining the effectiveness of a new design. Moreover, the characterization tool (layer-pass simulations) that is developed to test the hypothesis can easily be used to examine different baffle configurations and predict their performances. References [1] J.J. McCarthy, T. Shinbrot, G. Metcalfe, J.E. Wolf, J.M. Ottino, Mixing of granular materials in slowly rotated containers, AIChE J 42 (1996) 3351–3363. [2] D.V. Khakhar, J.J. McCarthy, J.F. Gilchrist, J.M. Ottino, Chaotic mixing of granular materials in two-dimensional tumbling mixers, Chaos 9 (1999) 195–205. [3] S.E. Cisar, P.B. Umbanhowar, J.M. Ottino, Radial granular segregation under chaotic flow in two-dimensional tumblers, Phys. Rev. E. 73 (2006) 031304. [4] H. Henein, J. Brimacombe, A. Watkinson, Experimental study of transverse bed motion in rotary kilns, Metall. Mater. Trans. B 14 (2) (June 1983) 191–205. [5] J. Mellmann, The transverse motion of solids in rotating cylinders — forms of motion and transition behavior, Powder Technol. 118 (3) (AUG 28 2001) 251–270. [6] A.A. Boateng, P.V. Barr, Granular flow behaviour in the transverse plane of a partially filled rotating cylinder, J. Fluid Mech. 330 (1997) 233–249. [7] J.M. Ottino, R.M. Lueptow, Materials science — on mixing and demixing, Science 319 (2008) 912–913. [8] A. Samadani, A. Kudrolli, Segregation transitions in wet granular matter, Phys. Rev. Lett. 85 (2000) 5102–5105. [9] H. Li, J.J. McCarthy, Controlling cohesive particle mixing and segregation, Phys. Rev. Lett. 90 (2003) 184301. [10] [10] Suman K. Hajra and D. V. Khakhar. Improved tumbling mixers and rotary kilns. Indian Patent 213856, 2003.
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