DEM simulations were performed over a wide range of angular drum velocities Ω for all six particle mixtures of Table 2. The flow profiles of the granular beds at a discrete set of angular velocities were characterized, by visual inspection of the simulations, following the classification scheme by Mellmann. The flow profiles were found to be insensitive to the granular properties, and fully determined by the angular velocity of the drum. Previous studies suggest that the relevant dimensionless number for the flow profile is the Froude number, , that is the ratio of centrifugal and gravitational forces.[3, 11, 57] Hence, we will frequently convert drum angular velocities into Froude numbers for the reader's convenience. The flow profiles in simulations at Ω = π rad/s or 30 revolutions per minute (rpm), corresponding to a Froude number of 0.035, as well as in slower revolving drums, were characterized as being in the rolling regime. A cascading profile was observed in simulations at 2π rad/s or 60 rpm (Fr = 0.14) and 10 rad/s (Fr = 0.36), a cataracting profile for Ω = 15 rad/s (Fr = 0.80) and 16 rad/s (Fr = 0.91), and a cataracting-centrifuging profile for simulations at 18 rad/s (Fr = 1.16) and 19 rad/s (Fr = 1.29). We also verified the existence of a centrifuging profile, in simulations with Ω = 25 rad/s (Fr = 2.23). The simulations at angular velocities between the aforementioned ranges displayed smooth transitions between the flow profiles of the bracketing regimes, hence the aforementioned numbers are not to be interpreted as sharp phase boundaries.
The rotating drum empowers the four segregation mechanisms, as described in the introduction, to collectively create a mixed or segregated steady state by working in unison or in discord. The final configuration and its degree of mixing, therefore, reflect the relative effectiveness of the four segregation mechanisms within the limitations posed by the flow profile and the particle properties. Figure 2 shows the mixing curves for five systems as a function of angular velocity/Froude number. Each data point represents the average of at least two simulations, including one starting with a randomly filled drum and one initially block-wise segregated drum, typically running for 30 revolutions. From a set of simulations in the flowing regime, differing only in their initial configurations, a standard deviation in the degree of mixing of about 0.03 was established. The mixing curves are remarkably similar for all simulated systems. Segregation predominates at low drum velocities, as illustrated by the snapshots in Figure 3. The degree of mixing ϕ rises with increasing angular velocity, passes through a well-mixed range centered around Fr = 0.56, and then decreases with a further increase in the drum velocity. Movies and snapshots of the simulations confirm this trend of a high degree of mixing at intermediate drum angular velocities, which appears remarkably insensitive to the granular properties, while segregation prevails in both tails of the plot. Inspection of the segregated beds reveals the formation of cylindrical cores running the entire length between the two vertical walls bounding the drum. Interestingly, the radially segregated patterns at low Fr are consistently inverted at high Fr. We exploit this property and for clarity henceforth systematically assign the label a to the particle type that accumulates in the center at low Fr and in the periphery at high Froude numbers. These simulation results will be discussed below in detail, to analyze which segregation mechanism dominates under specific conditions, where for clarity we have separated the low rotational velocities (flowing and cascading regimes, upto and including optimum mixing) from the high rotational velocities (cataracting and centrifuging regimes, beyond optimum mixing). Note that the simulations of Arntz et al., which include the current reference system, indicate that the peak of optimal mixing shifts with increasing (decreasing) fill levels to higher (lower) rotational velocities.
Figure 2. The mixing parameters ϕ of steady-state half-filled drums as functions of the Froude number.
The five bidisperse granular mixtures differ in mass, radius and/or density of the particles. Systems 1 (open triangles), 2 (open diamonds), and 5 (open circles) represent mixtures with identical densities, radii and masses, respectively. System 3 is drawn as solid triangles and System 4 as solid stars, while System 6 is left out as its graph closely follows that of System 5. Further details on the particle properties are provided in Table 2. The arrows at the top of this figure mark the ranges of the rolling (RO), cascading (CS), cataracting (CT), and cataracting-centrifuging (CT/CF) flow regimes.
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Low rotational velocities
For low Fr values, well below the mixing maximum, the bed is in the rolling regime. Figures 4 and 5, subfigures A and B, show a densely packed passive bulk that slowly rotates with the drum, with a flowing layer on top.[39, 58] The reduced number density, or increased porosity, of this mobile layer is due to the frequent collisions between the relatively fast moving particles under a high local velocity gradient and thereby creates a productive environment for the segregation and mixing mechanisms. With increasing drum velocity, the excited layer grows thicker and less dense, as can be seen from Figures 4 and 5 by comparing sub A and B with sub E and F. Vector plots of the local velocity difference between the two particle types, see the third columns in Figures 4 and 5, indicate where segregation and mixing occur, and thereby permit an interpretation in terms of the four mechanisms. Figures 4C and 5C reveal that in the rolling regime, segregation occurs throughout the flowing layer, while in the mixing regime around Fr = 0.56 the segregation is mainly concentrated in the bottom-right region of the flowing layer, see Figures 4G and 5G. Such a shift of locus is not without consequences, even if the same segregation mechanism remains dominant, for the overall segregation behavior of the bed. Note that subtracting two averages to arrive at a difference one order-of-magnitude smaller is well known to produce a high noise level, unless the two individual averages can be established to a high degree of accuracy. Subdividing the drum into small cells to obtain a spatial distribution reduces the number of particles averaged over and thereby enhances the noise in all plotted distributions. Similar considerations hold true for the noise level in the velocity standard deviations plotted in the fourth columns of Figures 4 and 5, with an accurate standard deviation requiring far more extensive sampling than needed for an accurate average.
Figure 4. Analysis of the granular bed with particles of equal radius (System 2).
The horizontal rows represent, from top to bottom, beds in the rolling regime (Ω = π/2 rad/s, Fr = 0.01), near optimum mixing (Ω = 4π rad/s, Fr = 0.56) and in the cataracting regime (Ω = 16 rad/s, Fr = 0.91). The first vertical column shows the local occupied volume fraction. The second column shows the particle velocities, where we have plotted the relative particle velocity with respect to the uniform rotation of the drum, , rather than the absolute velocity, for clarity of the plot and for easy identification of the active region(s). The reference arrows in the top right corners represent 0.2, 0.9, and 1.1 m/s, respectively, from top to bottom. The third column illustrates the velocity difference between the two particle types, va−vb, with reference arrows of 0.013, 0.05, and 0.14 m/s, respectively. The fourth column shows the width of the local velocity distribution, where the horizontal (vertical) components of the plotted vectors denote the standard deviations along the horizontal (vertical) direction, with reference arrows measuring 0.013, 0.03, and 0.05 m/s, respectively.
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Figure 5. Analysis of the granular bed with particles of nearly equal mass (System 4).
The conditions and set-ups of the subgraphs are identical to those used in Figure 4. Note the strong similarities with System 2, see Figure 4, despite the marked differences in the radius and mass of the b-type particles in Systems 2 and 4 (the a-type particles are identical). The deviations are largely limited to the velocity differences and velocity standard deviations at the higher Froude numbers.
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The particles in System 1 differ in size and mass, but are of equal density and consequently impervious to the buoyancy mechanism. Our discussion of this system will be brief, because segregation in the absence of buoyancy has been discussed in a number of studies[13, 18, 40, 59] and because a detailed analysis of this particular system has appeared elsewhere. It was argued there that the moderate segregation (ϕ ≈ 0.6) results from the downward percolative motions of the small particles in the flowing layer, culminating in a radial core of the smaller a-type granules and a periphery of predominantly the larger b-type particles, see Figure 3. The comparison with System 2, to be discussed below, confirms that the mass difference is not responsible for segregation at these low Froude numbers. With increasing angular velocity, that is toward and throughout the cascading regime, the flowing layer becomes more porous, hence less selective to percolation of small particles, and mixing by random collisions gains prominence, as illustrated by Figure 4 of Arntz et al., achieving a well-mixed state around Fr = 0.56. The latter mixing process is further supported by an increase of the particles' velocities with Fr, which also enhances the inertia mechanism.
The particles in System 2 have distinct densities and masses, but equal radii to eliminate the percolation mechanism. Again, the reduced number density and relatively high velocities of particles in the flowing layer, see Figures 4A, B, E, and F, promote reshuffles of the granules. At low Froude numbers, segregation occurs along the entire length of the flowing layer, see Figure 4C, but most prominently where the layer is at its thickest. The gradual formation of a radial core of the denser a-type particles, see Figure 3, is attributed to the buoyancy mechanism, that is the denser particles are sinking down in the flowing layer till they settle on a more closely packed region. Although the a- and b-type particles are comparable in mass to the b- and a-type particles of System 1, respectively, the heavier particles accumulate in the core in System 2 but in the periphery in System 1. This indicates that at these low Froude numbers the segregation process is more sensitive to radius and density differences than to mass differences. It then follows that the equal density particles of System 1 segregate by percolation, as was mentioned above, while the equal radius particles of System 2 segregate by buoyancy. With increasing Froude number, the random collision and inertia mechanisms become more important, as illustrated by the large mixing region in Figure 4G, thus reducing and near Fr = 0.56 even suppressing the segregation process in System 2.
In System 3, the radii and densities are chosen such that their associated segregation effects are acting in opposite directions: the percolation mechanism drives the small particles (of low density) to the radial core, whereas the buoyancy mechanism strives for a core of the high-density particles (of large radius). Figures 2 and 3 show that this system remains well-mixed for angular drum velocities in the rolling and cascading regimes.[12, 19, 24] The absence of segregation, despite the order-of-magnitude difference in the particles's masses, provides further support for the aforementioned observation that inertia is of little importance at these low Froude numbers.
If, in contrast, the particles of one type are both smaller and denser than the particles of the other type, as in Systems4, 5 and 6, then buoyancy and percolation co-operate in driving the smaller and denser a-type particles to the radial core, see Figure 3. The mixing parameter, see Figure 2, indicates that the resulting segregation is indeed more intense than in Systems 1 and 2, as is also clear from Figure 3, where only either one of these two mechanisms is active. Here again, the segregation becomes less intense and eventually vanishes with the Froude number rising to 0.56. A comparison of Systems 4 and 6 shows that an increase in the density ratio, from 3 to 4, does not significantly enhance segregation, suggesting that the buoyancy mechanism has already reached its optimal performance at the former ratio.
High rotational velocities
At high angular velocities beyond the optimum mixing regime, that is for Froude numbers exceeding 0.56, the bed is in the cataracting, cataracting-centrifuging, or centrifuging regime. For the cataracting regime, Figures 4 and 5, subfigures I and J show that particles with high mobilities are thinly distributed over a large volume of the box, thus offering a high potential for intermixing. The plots in the second row of Figure 6, however, indicate that the particles following ballistic trajectories through the sparsely populated volume above the bed collide only infrequently and hence hardly contribute to (de)mixing. The particles rolling down the diffuse surface of the bed, see Figures 4J and 5J, are susceptible to segregation by percolation and buoyancy, as visible in the second row of Figure 6. Both particle flows are reunited at the lower end of the flowing layer, which appears in the second row of Figure 6 as the dominant region of segregation. In the cataracting-centrifuging and centrifuging regimes, that is for Froude numbers above 1.0, one or more layers of centrifuging particles cover the entire drum well. These layers may show buoyancy and percolation effects, with the denser or smaller particles moving radially outward, respectively. At higher drum velocities, the bed becomes too closely packed to permit relative particle motions, and airborne particles impinging on the inner centrifuging layer make a small contribution to the (de)mixing process.
Figure 6. Snapshots and velocity-difference plots at high drum angular velocities.
Shown are Systems 1–5 (from left to right) at drum angular velocities of 16 rad/s (Fr = 0.91, cataracting) in the top two rows and at 18 rad/s (Fr = 1.16, cataracting-centrifuging) in the bottom two rows. The a-type particles are depicted in light-green, the b-type particles in dark red. The reference arrows in the second row correspond with 0.04, 0.14, 0.10, 0.10, and 0.14 m/s, and those in the fourth row with 0.08, 0.17, 0.07, 0.20, and 0.22 m/s. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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The reference system, with particles bidisperse in size and mass, shows a moderate to intense segregation at high Froude numbers. At equal particle densities, this segregation most likely results from percolation in the flowing layer, as is further supported by the observations for the other systems to be discussed below. Arntz et al.  argued that the shift of the main percolation region from the center of the flowing layer in the flowing regime to the tail of the flowing layer in the cataracting regime is responsible for inverting the net effect of the percolation mechanism, which thus for Froude numbers approaching unity gives rise to a radial core of large particles (“inverse radial segregation”, see the snapshots in the first column of Figure 6) while a radial core of small particles forms at low Fr (regular radial segregation, see Figure 3). The mixing parameter of System 1 shows a short plateau for Froude numbers from 0.9 to 1.1, and smaller fluctuations are visible for the other systems near the same range of Froude numbers. As this plateau is unexpected, we performed supplementary simulations, both at the originally selected angular velocities and at additional intermediate values (see Figure 2), which confirm that the shoulder reproduces. Most likely, this plateau reflects that the bed is still slowly evolving when the simulations were discontinued after 30 revolutions. A selected number of simulations were continued for many more revolutions, but this hardly changed the degrees of mixing in these drums, indicating that the final steady state is being approached only very slowly. This slight indeterminacy in the steady-state degree of mixing does, of course, not alter the main observation that the particles tend to segregate at high drum velocities.
The equal radius particles of System 2 are well mixed in the cataracting regime, thereby suggesting that a mass or density difference is insufficient to induce segregation in this Froude regime and confirming that percolation is the driving mechanism for System 1 in the cataracting regime. The velocity difference plots in the second column of Figure 6 show outward pointing arrows at the lower right side, indicative of inward motion of the lighter b-type particles relative to the denser and heavier a-type particles. A more detailed analysis and visual inspection of movies reveal that in this region the airborne particles impinge on the bed surface, often bouncing back before being taken up by the bed. The inertia mechanism, that is lighter particles rebound more strongly than heavier particles, then gives rise to the velocity differences, see the second column of Figure 6, and large standard deviations, see Figure 4L, in this region. The random nature of the collisions, with light and heavy particles bouncing back in various directions, appears to promote mixing and effectively suppresses possible segregation mechanisms. A gradual decrease of the order parameter is observed for Froude numbers exceeding 0.9, where the buoyancy effect induced by the centrifugal pseudo force drives the denser a-type particles to the drum wall (recall that the particles in System 2 are of equal size, thereby eliminating the percolation mechanism).
The properties of the particles in System 3—small low-density particles and large high-density particles—were specifically chosen to balance the percolation and buoyancy mechanisms in the rolling and cascading regimes. Figure 2 and the third column of Figure 6 show that this balance does not extend to the cataracting and centrifuging regimes. The accumulation of the small low-density particles at the drum wall, be it with a higher degree of mixing than in System 1, indicates that percolation has gained the upper hand over buoyancy.
The radius and density combination in System 4 yields two particle types with comparable masses and thus largely eliminates the inertia mechanism. The standard deviations in the velocities are considerably reduced relative to those of the unequal mass particles in System 2, as can be seen by comparing Figure 5L with Figure 4L. The percolation and buoyancy mechanisms are seen to collaborate in driving the denser and smaller a-type particle to the periphery, see the fourth column in Figure 6. A similar cooperative effect dominates in System 5, whose particles of exactly equal mass are seen to segregate in the fifth column of Figure 6, and in System 6, where the smaller particles are heavier than the larger particles.
Other particle properties
Besides the radius, mass, and density, we have also systematically varied all other particle parameters appearing in the force Eqs. (1)-(3) and assessed their respective influences on the segregation process. These simulations were mainly carried out using the reference system, setting all parameters to the default values listed in the second column of Table 1. The properties of all a-type or all b-type particles were altered, one by one, over the ranges indicated in the third column of Table 1. Most simulations were confined to the rolling regime, to keep the required computer time manageable, with brief excursions to other systems and higher drum angular velocities to confirm the general validity of our findings. While changes in the radius (System 2) or density (Systems 3–6) relative to the reference system notably affect the degree of mixing, the explored variations of the dynamic interparticle friction coefficients between a-type particles , between b-type particles , and between mixed particle pairs , the particle–wall friction coefficients , with p = a or p = b, the particle–particle and particle–wall tangential damping coefficients and , the particle–particle elastic stiffness and the normal damping coefficients and (which are related to the tabulated restitution coefficients and ) hardly affect the mixing behavior. Only at certain extreme values are differences detectable, like a bed that remains immobile at very low particle–wall friction coefficients or an increased degree of mixing for fully elastic particle–particle collisions.
Of particular interest is the roughness of the particles, which is represented in the current simulation model by the dynamic friction coefficient and the damping coefficient , because some simulation models are built on the assumption that differences in roughness give rise to segregation.[1, 60] We find no evidence for this assumption in the reference system, in line with the experimental observations by Pohlman et al. Only at low friction coefficients, that is for below about 0.25, are there noticeable deviations. In mixtures of rough and very smooth particles, we observe that a thin layer of predominantly smooth particles forms at the vertical drum walls while the middle of the drum remains similar to that in the reference simulation. Furthermore, mixtures of smooth particles appear to segregate less well, with more small particles at the periphery and more large particles in the core, than otherwise equal mixtures of rough particles.