Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 9, No. 8 (1999) 1467–1484
c World Scientific Publishing Company
MIXING OF GRANULAR MATERIALS: A TEST-BED DYNAMICAL SYSTEM FOR PATTERN FORMATION K. M. HILL, J. F. GILCHRIST and J. M. OTTINO∗ Department of Chemical Engineering, Northwestern University, Evanston, IL 60208, USA D. V. KHAKHAR Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India J. J. McCARTHY Department of Chemical and Petroleum Engineering, University of Pittsburgh, PA 15261, USA Received April 1, 1999 Mixing of granular materials provides fascinating examples of pattern formation and selforganization. More mixing action — for example, increasing the forcing with more vigorous shaking or faster tumbling — does not guarantee a better-mixed final system. This is because granular mixtures of just barely different materials segregate according to density and size; in fact, the very same forcing used to mix may unmix. Self-organization results from two competing effects: chaotic advection or chaotic mixing, as in the case of fluids, and flow-induced segregation, a phenomenon without parallel in fluids. The rich array of behaviors is ideally suited for nonlinear-dynamics-based inspection. Moreover, the interplay with experiments is immediate. In fact, these systems may constitute the simplest example of coexistence between chaos and self-organization that can be studied in the laboratory. We present a concise summary of the necessary theoretical background and central physical ideas accompanied by illustrative experimental results to aid the reader in exploring this fascinating new area.
1. Introduction Granular materials display a rich variety of dynamical phenomena that have far-reaching implications for industrial and natural processes. One of these phenomena is mixing, especially when it is tied to its counterpart, flow-induced segregation. Industrial examples of mixing and segregation appear in the pharmaceutical, food, chemical, ceramic, metallurgical, and construction industries [Bridgwater, 1995; Fan et al., 1990; Nienow et al., ∗
1985]. However, the understanding of the fundamentals of granular mixing remains incomplete, especially when compared with that of fluid mixing [Ottino, 1989a]. The picture is changing though. Granular materials have attracted considerable recent attention in the physics community [Jaeger & Nagel, 1992; Behringer, 1993; Jaeger et al., 1996a, 1996b]. Notably for granular mixing is that chaosinspired ideas — which have been central in advancing the understanding of fluid mixing [Aref,
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Fig. 1. The result of flow-induced segregation of mixtures of different-sized [(a) and (b)] and different density (c) granular materials placed in a rotating cylinder. The system quickly evolves, in a revolution or so, from a mixed state (a) to a segregated state [(b) and (c)]. (d) shows the equilibrium segregation structure corresponding to a mixture of particles with different densities using the computational model described in the text.
1990; Ottino, 1990] — have been found applicable to granular mixing as well [Khakhar et al., 1999a]. Granular materials, however, introduce additional difficulties. The main difference among these is flow-induced segregation (for a general review see [Rosato, 1999]). Often, granular systems evolve quickly through complex dynamics into a state of self-organization. For example, in a 2D (short) tumbled cylinder this may lead to radial segregation, a segregated core of smaller or denser particles, as shown in Fig. 1 [Ristow, 1994; Cantalaub & Bideau, 1995; Cl´ement et al., 1995; Khakhar et al., 1997b; Dury & Ristow, 1997]. In long cylinders axial banding may follow, according to certain particle properties [Fig. 2] [Nakagawa, 1994; Hill & Kakalios, 1995]. The 2D case is the focus of this paper.
Recently it was shown that chaotic advection, which can be used as a tool to improve fluid mixing, may also improve granular mixing [Khakhar et al., 1999a]. This concept was investigated in systems of granular materials in rotating pseudo 2D drum mixers of different shapes. Consider the system depicted in Fig. 3. Under suitable conditions — easy to achieve in the laboratory — the flow of noncohesive granular materials in 2D rotating circular containers achieves a continuous flow, the so-called rolling regime. A sketch of the flow is shown in Fig. 3. The flow is confined to the top free surface in the form of a thin shear-like layer whereas the rest of the material moves in solid-like rotation with the mixer walls. Material is fed into the flowing layer for x < 0 and leaves the layer for x > 0. When the
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Fig. 2.
Axial segregation of different-sized particles in a long rotating drum mixer.
Fig. 3. Schematic view of the continuous flow regime in a rotating cylinder. The dotted curve denotes the interface between the continuously flowing layer and the region of solid body rotation. The mixer is rotated with angular velocity, ω, and the velocity profile within the layer, vx , is nearly simple shear. The vy component is not shown. A typical particle trajectory is depicted.
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flow is steady (i.e. it is not a function of time) as is the case for a circular mixer the “streamlines” or circulation paths are closed and time-invariant. A two-dimensional flow can be derived from a streamfunction Ψ, such that vx = ∂Ψ/∂y, vy = −∂Ψ/∂x where Ψ = Ψ(x, y). The structure of the steady flow is Hamiltonian with one-degree of freedom, and therefore it cannot be chaotic (see [Ottino, 1990, pp. 215, 216]). However, if the container is noncircular the situation is very different; the flowing layer grows and shrinks in time with a frequency no less than twice that of the mixer rotation, and Ψ = Ψ(x, y, t). Adding time-periodicity — the system is now referred to as having one-and-a-half degrees of freedom — makes chaos possible (for details see [Ottino, 1989]). Thus, if chaotic advection improves mixing, noncircular mixers should mix more efficiently than circular mixers. Experimental studies using colored tracer particles in otherwise mono-disperse granular materials confirm that increased mixing rates occur in noncircular containers [Khakhar et al., 1999a]. Differences in particle properties make the system’s behavior even more interesting from a dynamical systems viewpoint. Small differences in size and/or density of the grains of the granular material lead to flow-induced demixing or segregation, a phenomenon without parallel in fluid systems. Granular flow is almost invariably accompanied by segregation, a major impediment to mixing that interacts nontrivially with chaotic advection. Much of the interesting dynamics for segregation and mixing of granular materials can be traced back to the flowing layer; it is only here that the grains are free to move relative to their neighbors. For the rest of the bed the grains are locked into position relative to each other in solid body rotation until they are fed into the layer. However, additional dynamical details governed by the shape of the container are also important, as will be discussed below. Consider now a model that can be used to investigate the behavior of these kinds of systems.
2. The Basic Model Figure 3 depicts the essential details of a model developed by Khakhar et al. [1997a]. This model describes the flow of identical particles in the flow regime where the flowing layer is steady and thin, and the free surface is nearly flat. The particles outside of this layer move in solid body rotation.
Metcalfe et al. [1995] addressed the case of slower flow, where this flow becomes time periodic and consists of discrete avalanches; the case corresponding to cohesive particles, leading also to a timeperiodic flow is considered by Shinbrot et al. [1999]. For the purposes of this tutorial description, we focus solely on the regime corresponding to steady flow. Our initial remarks are restricted to a halffull circular container. In this case the free surface has length 2L (constant) and the flowing layer has thickness δ(x) that varies along the length but which is otherwise independent of time. A theoretical analysis indicates that δ(x) can be approximated by a parabola [Khakhar et al., 1997a] δ = δ0 (1 − (x/L)2 ) ,
(1)
where δ0 is the thickness at x = 0. The velocity profile within the flowing layer can be approximated by dx = vx = 2u(1 + y/δ) dt
(2)
dy = vy = −ωx(y/δ)2 dt
(3)
where u is the average velocity at the center of the flowing layer and is given by u = ωL2 /(2δ0 ). The vy -component [Eq. (3)] is chosen to satisfy mass conservation. This model captures the essence of flow in the layer. Additional computations indicate that different velocity fields, for example vx ∼ y 3/2 , vy ∼ xy 5/2 , give essentially the same mixing patterns, demonstrating that macroscopic geometrical effects (i.e. the shape of the container) control the important details of the physics. This model, coupled with solid-body rotation in the bed, is sufficient to compute a Poincar´e section for the circular mixer, shown in Fig. 4(a). As is standard in fluid mechanics, the flow is interpreted in a continuum sense and collisional diffusion — the random-like motion of individual particles as they collide with other particles — is ignored in the particle trajectory calculations. The plot is trivial for the circular mixer; the system is regular; particles can cross streamlines only by collisional diffusion, and the system exhibits slow radial mixing. To adapt the model for noncircular mixers, one needs only to make L time dependent thus changing the model from a 2D model in x and y to a 3D model in x, y and t. This transforms the equations for δ(x), vx and vy , to interact nonlinearly. These equations, without further modification, are
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adequate for mixers that are rotationally 180◦ symmetric providing a stationary origin and convex boundaries to preserve the layer shape. For example, the dimensionless length of the flowing layer L(t) for a square mixer is expressed as
[Khakhar, 1997a]. This implies that u ∼ L(t); thus the speed u changes with mixer orientation; the longer the layer, the faster and deeper the flow.
3. Advection and Poincar´ e Sections
1 π π 7π if θ < or |π − θ| < or θ < | cos θ| 4 4 4 L(t) = 1 otherwise | sin θ| (4)
Figures 4(b) and 4(c) show the Poincar´e sections for a half-filled elliptical and square mixers respectively, illustrating both regular regions (KAM islands) where particles can be trapped near elliptic points (marked in red) and also regions where chaotic trajectories exist near hyperbolic points. The Poincar´e sections shown were generated by plotting the location of advected
Consider one final element to complete the model. Experiments show that the layer maintains geometric similarity, so that δ0 (t)/L(t) is a constant
(a)
(b)
(c) Fig. 4. Poincar´e sections corresponding to (a) a half-filled circle, (b) ellipse and (c) square. The positions of the particles are marked after every half revolution for the circle and ellipse and every quarter revolution for the square.
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particles every half rotation of the mixer, except those for the square which were plotted every quarter rotation. Figures 5(a) and 5(b) show the Poincar´e sections for the square mixer when filled 5% less and 5% greater than half, respectively, illustrating that the degree of filling is critical. The mixer is no longer 180◦ rotationally symmetric from the perspective of the rotating particles. The circulation time in solid body rotation increases with fill level, and thus the circulation of the particles is no longer synchronous with the rotation of the mixer. We focus primarily on the half-full case for purposes of this tutorial.
4. Collisional Diffusion Consider now the addition of collisional diffusion. Savage [1993] proposed the following scaling relation for the diffusion coefficient based on hard particle dynamics simulations Dcoll
dvx = f (η)d2 , dy
(5)
where dvx /dy is the velocity gradient across the layer, and d is the particle diameter. The prefactor f (η) is a function of the solids volume fraction, η; in our simulation we take f = 0.025.
< 1/2 full
How does collisional diffusion enter into the mixing picture? Consider what happens to a group of localized tracer particles in the flowing layer. Figure 6 shows a computational simulation of the time evolution of a blob of tracer particles during a typical mixing experiment. The left column shows the evolution without collisional diffusion, and the right column includes diffusion. Thus, in a typical experiment the blob is deformed into a filament by the shear flow and blurred by collisional diffusion until particles exit the layer. In mixers with circular cross-sections, this is the only form of mixing. Particles then execute a solid body rotation in the bed, re-enter the layer, and the process repeats. The diffusional mixing process can be simulated by adding a noise term to the particle advection Eqs. (2) and (3) to mimic diffusion, a Lagrangian approach. Thus the dynamical system is: dx = vx dt
(6)
dy = vy + S dt
(7)
where S is a white noise term which upon integration over a time interval (∆t) gives a Gaussian
> 1/2 full
Fig. 5. Illustration of the variation of the Poincar´e sections for the square when the filling level is changed about 50%. The locations of the elliptic and hyperbolic points, are very sensitive to the degree of filling about the half-full level. (a) When the fill level drops to 45%, both the hyperbolic points and the elliptical points move significantly away from the center of rotation radially, and there is a string of high period islands surrounding the elliptical points. By contrast, (b) when the fill level is increased to 55%, both the hyperbolic point and elliptical points approach the center and the overall circulation pattern becomes dominated by unbroken tori (not unlike that for the circular mixer).
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Fig. 6. Computational results depicting the time evolution of a “blob” of tracer particles as it moves through the flowing layer. The left-hand column shows the effect of advection; the right-hand column includes collisional diffusion.
(a)
(b)
(c)
Fig. 7. Computational results for a system of unmixed particles differing only in color. The initial conditions are shown in the inset of (a), and both advection and collisional diffusion are included during the calculation of the mixing process. The intensity of segregation is shown for large systems PeL = 104 and small systems PeS = 102 . The states of the larger systems after 20 rotations are shown in Figs. 7(b) and 7(c).
random number with variance 2Dcoll ∆t. Note that the diffusional effect is incorporated only in the ydirection. Diffusion along the layer (x-direction) is neglected, since diffusional effects are masked by convection (i.e. particles move much faster than they diffuse). This can be put on a quantitative basis by computing the P´eclet number, a dimensionless ratio that measures the importance of convection versus diffusion. The P´eclet number for flow in the x direction is Pe = uL/Dcoll , ranging from 102 in laboratory mixers to 104 in industrial mixers, whereas the P´eclet number in the y direction is much smaller (by a factor of (δ0 /L)2 ∼
0.0025). At this stage the model gives us a complete model of mixing of monodisperse spherical particles in a mixer of noncircular, 180◦ symmetric cross-sectional shape. While diffusion works approximately the same for all mixers, the combination of diffusion with advection introduced by unsteady flow increases mixing efficiency as shown in Fig. 7. For two systems of identical particles differing only in color with initial unmixed conditions shown in the inset of Fig. 7(a), the mixing rate is significantly higher for the square mixer than the circular mixer as shown in the graph in Fig. 7(a). The state of the systems after
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20 rotations are shown in Figs. 7(b) and 7(c) to further illustrate the difference in the mixing efficiency of the two mixers.
5. Segregation Consider now the effects of segregation. Granular mixtures of even slightly dissimilar materials will segregate in many situations, including when they are shaken or tumbled. One of the better-known examples of this behavior is the “Brazil-nut” effect [Williams, 1963; Rosato et al., 1987], where large particles rise to the top of a shaken container of mixed nuts. This may be due to a “shadowing” effect. That is, while all particles are jostled up during a shake, there is a greater probability that a space will be made for a smaller particle to slip below a larger one than vice versa. In this way the smaller particles gradually work their way down to the bottom of the container, and the larger particles drift up. In certain situations, this segregation effect has been shown to be driven by convective motion in the granular materials [Ehrichs et al., 1995]. Another example of segregation involves segregation driven by the flow of granular materials. When granular mixtures are poured in a thin space between two vertical walls (like a Hele–Shaw cell), the components may separate into stratified layers. The pattern resembles the stratification pattern found in sedimentary rocks and may be responsible for this phenomenon in certain situations [Makse et al., 1997; Koeppe et al., 1998]. Another example of flow-induced segregation is radial segregation in a rotating drum mixer, where the different components separate in the direction perpendicular to the axis of rotation [shown in Figs. 1(b)–1(d)]. In terms of the streamfunction, ∇Ψ, and the gradient of concentration, ∇c, are collinear. Thus, a radially segregated structure, in a circle, is an invariant structure. For a physical picture of flow-induced segregation, consider a mixture of particles of different sizes in a steady chute flow. As first observed over a century ago [Reynolds, 1895], when granular materials flow, they dilate and as the material dilates, voids are created. Small particles can squeeze into small voids below a large particle, but the reverse is much less likely to occur resulting in a net segregating flux of the smaller particles downward, away from the free surface. The experimental work of Nityanand et al. [1986] illustrates the typical behavior of systems with size segregation. Similarly, when mix-
tures of particles differing in density move through a shear layer, the denser particles are more likely to sink into a lower layer. This will be described in more detail below. In long rotating cylinders, radial segregation is often followed by axial segregation [Donald & Roseman, 1962; Das Gupta et al., 1991; Hill & Kakalios, 1995]. After continued rotation of granular materials in a long mixer, certain mixtures will further segregate into relatively pure, single component bands along the axis of rotation. [See Fig. 2] Here the mechanism for the case of different size particles is believed to be due to differences in angles of repose of the two materials which produce small differential axial flows for the two materials [Das Gupta et al., 1991; Hill & Kakalios, 1995]. This is a more complex phenomenon than simple radial segregation. Some mixtures do not axially segregate, whereas other systems of granular materials exhibit reversible phenomena when they segregate at higher speeds and remix at lower speeds of rotation. Some mixtures evolve from the simple initial banding pattern into further pattern development. This is discussed in more detail in other works [Hill et al., 1997; Choo et al., 1998]. Many of these issues remain imperfectly understood. Radial segregation in the cross-section of the circular mixer may play a role in the axial segregation, and thus the cross-sectional shape of the container may have a strong influence in the results [Hill et al., 1999]. How is segregation added to the advection picture? A model for segregation based on density difference can be based on an “effective buoyancy force” [Khakhar et al., 1997b]. The effects of segregation can be accounted for in terms of drift velocities with respect to the mean mass velocity. As usual the effects are significant only in the direction normal to the flow. The segregation velocity for the more dense particles (labeled “1”) can be written as vy1 = −
2β(1 − ρ)Dcoll (1 − f ) d
(8)
and for the less dense particles (labeled “2”) as vy2 =
2β(1 − ρ)Dcoll f d
(9)
Here, as before, Dcoll , is the collisional diffusion, β is the so-called dimensionless segregation velocity which is determined by fitting Monte Carlo and “soft-particle” computations; f is the number fraction of the more dense particles, ρ < 1 is the
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density ratio, and d is the particle diameter. Similar expressions can be obtained for the case of systems differing in size, at low solids volume fractions [Khakhar et al., 1999b]. In our simulations we take β = 2. To add this element into the advection model, assume first that the mean flow is still the same as if all particles were identical, so that Eqs. (6) and (7) still apply. Note also that the effects of segregation, as well as those of collisional diffusivity described earlier, are significant only in the direction normal to the flow (apparent, again, when the P´eclet numbers in each direction are considered) so Eq. (6) remains unchanged. Thus to compute the effect of segregation on the earlier models for the circulation patterns in the different mixers, one needs only to rewrite Eq. (7) with the segregation terms, Eqs. (8) and (9). Substituting the form for vy from Eq. (3), Eq. (8) takes the following form for the more dense particles:
y1 dy1 = −ωx dt δ
2
+S−
2β(1 − ρ)Dcoll (1 − f ) , d (10)
and for the less dense particles, Eq. (9) becomes:
dy2 y2 = −ωx dt δ
2
+S+
2β(1 − ρ)Dcoll f . (11) d
Thus we have a complete model for mixing and segregation for both circular and noncircular rotating 2D drum mixers. Results are presented in Figs. 8(g) and 8(h), and will be discussed in Sec. 7. The case corresponding to size segregation is more complex [Khakhar et al., 1999b].
6. Experiments Typical experiments in this area are fast, inexpensive and reproducible. They can be conducted using a variety of noncohesive particles from seeds to spherical beads. We caution the reader that when the particle size is less than 1 mm in diameter, effects such as clumping due to moisture or attractive and repulsive forces due to static electricity become more prominent. Thus, it is recommended that bead sizes not smaller than 0.5 mm in diameter are used. For reducing the effects of static electricity, antistatic spray is also recommended. The particles we used for the results presented below are spherical beads (Quackenbush Co.) with sizes ranging from 0.8 to 2 mm and densities of 2.5 and 7.8 g/cm3 (glass
and steel, respectively). To minimize axial segregation effects, it is recommended that the depth of the tumblers used be limited to a few particles, say 5–7. The design of the mixers is simple. The outer shape may be cut from anything from thick poster-board to Plexiglas and sandwiched between two plates. The results presented in this paper were obtained using three different quasi-2D mixers: a circular mixer, an elliptical mixer, and a square mixer, all of approximately the same depth (∼ 6 mm) and surface area (∼ 600 cm2 ). The faceplate of the mixers is made of Plexiglas for ease in observation and data acquisition while the rear plate is fashioned of aluminum and can be grounded to minimize electrostatic effects. Two types of experiments are possible for visualizing the chaotic flow, one that is more difficult, another that is surprisingly easy and efficient. In the first case, one might attempt to capture the circulation patterns in a mixer by seeding the bulk granular materials with, for example, darker but otherwise identical particles in certain regions of the mixer. This is what was done for the blob experiment shown in Fig. 9. In previous work we have shown this to be useful in determining the mixing efficiency for the systems of identical particles. One might also use this type of experiment to depict the general mixing pattern, although, in general it is not effective in identifying the details of the Poincar´e sections, or even distinguishing regular islands from the chaotic regions. Collisional diffusion effects in laboratory-size experiments are large and blur the details. Furthermore, this experiment is difficult to execute and its effectiveness is strongly dependent on the initial placement of the tracer particles. On the other hand, segregation experiments are easy to set up. In general, for these experiments, we start with a (roughly) well-mixed system of granular materials in the mixer and rotate the mixer at a constant rate until an equilibrium state is obtained for a specific orientation of the mixer. (The comparison should be made at equal angles of orientation; the pattern becomes a periodic function of time with a frequency no less than twice that of the mixer rotation.) The equilibrium segregation structure obtained is independent of the initial arrangement of the particles in the mixer and strongly resembles the Poincar´e section. As we are primarily interested in visualizing the rough details of the Poincar´e section, only the latter (segregation) experiments are discussed here.
1476 Fig. 8. Results from segregation experiments for quasi two-dimensional tumbling mixers filled halfway with mixtures of granular materials. “E” denotes experimental results and “C” denotes computational results. (Compare with the segregation results for the circular mixer in Fig. 1 and the Poincar´ e sections in Fig. 3.) Images are taken while the mixer is rotated, though the images of the square are rotated counter-clockwise by ∼ 30◦ to maximize the use of space. The competition between mixing and segregation was studied using ternary mixtures of particles differing in size (0.8 mm blue, 1.2 mm clear, and 2.0 mm red glass spheres) and binary mixtures of particles differing only in density (2 mm glass and steel spheres). The volume fraction of dense beads in the glass and steel bead system is 0.25, and the volume fraction of the smaller beads used in the mixture of different-sized glass beads is 0.25 — equal fractions of medium and larger beads are used. Computational results are shown only for binary mixtures of particles differing in density. Equilibrium segregation structures are shown for two different orientations for the systems of different-sized spheres (a)–(d), showing that, unlike the systems segregated in the circle, the shape of the segregated structures depend on the mixer orientation. Results from systems of granular materials differing only in density are similar, both in experiment (e) and (f) and computations (g) and (h).
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For the segregation experiments, it is helpful to start with a well-mixed system. Any imperfect initial mixing leads to a longer time for the system to resemble its final equilibrium state of segregation. The mixers described above are small enough to allow one to accomplish a thoroughly mixed initial condition without too much difficulty. We simply mix the materials first by hand, and then shake and rotate the mixers randomly once they are sealed to achieve a well-mixed system. Once the systems are mixed, we use a computer-controlled stepper motor (Compumotorr) to rotate the mixers at approximately 1 revolution per minute so that a steady flow with a flat free surface is developed. (The Froude number F = ω 2 L/g ≈ 2 × 10−4 for the circle.) Preliminary experiments are recorded by means of Polaroidr pictures until suitable conditions are identified. More detailed data was taken from images obtained using a Kodakr CCD camera, ideal for quantitative image analysis. Direct measurement of the concentration distribution of particles in a segregating system is also possible by injecting a setting fluid into the particle bed and freezing the particle configuration. Samples are then taken from different regions by cutting the bed and concentration may be obtained by separating the particles by sieving (different-sized particles) or using a magnet (if one of the components is magnetic).
7. Typical Results Consider now examples of the results produced by these systems. Equilibrium segregation structures for half-filled circular, elliptical, and square mixers are shown in Figs. 1 and 8 for systems of spheres differing in size [Figs. 1(b), 8(b) and 8(d)] and density [Figs. 1(c), 1(d), 8(e) and 8(f)]. Computational results based on the incorporation of all details (Secs. 4 and 5) into the basic model (Sec. 2) are shown in Figs. 1(d), 8(g) and 8(h). As seen in Fig. 8, the shape of the container not only has a strong influence on the Poincar´e sections (Fig. 4), but it plays a dominant role in determining the equilibrium segregation structures for the different systems as well. The segregated regions for half-filled systems are similar to the regular islands in the Poincar´e sections. Thus, segregation has an unanticipated benefit: Heavier particles tag the regular regions in the Poincar´e sections.
The classic radially segregated structure for the circular mixers is shown in Figs. 1(b)–1(d). In this case, the dynamics in the flowing layer cause the smaller (more dense) particles to move to lower levels in the layer leading to a segregated timeinvariant (rotationally symmetric) core region that coincides with the streamlines. However, the equilibrium segregation structures obtained in noncircular mixers do not follow this simple rule, as shown for the half-filled elliptical [Figs. 8(a), 8(c), 8(e) and 8(g)] and square [Figs. 8(b), 8(d), 8(f) and 8(h)] mixers. In this case the region with a high concentration of the small (or more dense) particles is located away from the center of rotation and is nearly separated into two distinct regions between the corners of the square and the center of rotation. Unlike the circular mixer, the precise shapes of the segregated regions are periodic in time and depend on the instantaneous orientation of the mixer, as illustrated by comparing Figs. 8(a) and 8(b) with Figs. 8(c) and 8(d), respectively. How do these complicated structures arise from initially well-mixed systems? In all cases shown, initially well-mixed systems segregate rather quickly — apparent within a half rotation, all systems segregate into a state resembling radial segregation. For the circular mixer, this pattern remains relatively stable and becomes the final state of segregation. Since the shape of the mixer is rotationally invariant, the important dynamics lie solely in the differential flow. While diffusion in the flowing layer may tend to jostle the particles away from the center of rotation, an equilibrium, radially segregated structure, is eventually reached. This initial radial structure may not be invariant though. Instead, in the case of the halffilled noncircular mixers, the initial radial segregation gradually evolves into the more complicated segregation patterns, as shown for the square mixers in Fig. 10. (The pattern development for the ellipse follows the same course.) The heavier or smaller particles that sink into the regular regions or islands are more likely to be mapped back to the same region than the particles in the chaotic region. Hence, as the system evolves, the segregated islands around the elliptical points grow and become more distinct. This argument is further supported by the strong agreement between experiment and computational predictions [Figs. 1(e), 8(g) and 8(h)] based only on the mixer shape, diffusion, and the density segregation model described above.
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Fig. 9. Images from mixing experiments and computations using tracer particles. Deformation and mixing of a “blob” due to chaotic advection and diffusion are clearly apparent, though due to the low P´eclet number for typical experimental mixers the mixing patterns are blurred due to collisional diffusion.
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Fig. 10. Evolution of binary mixture of 2 mm glass and steel beads as the system is rotated at 1 rpm in a half-filled square mixer. The number of rotations completed is noted in the upper right-hand corner of each image as the system evolves from an initially well mixed state (denoted as IC) to the final (steady) segregation pattern.
Note that both the Poincar´e section and the equilibrium segregation structures depend strongly on the degree of filling about the half-full level, as shown in Figs. 5 and 11, respectively. Figures 11(a) and 11(b) show that for mixtures of spheres differing only in density, and systems at just below (45%) and just above (55%) the half-full level, the complicated segregation patterns disappear. In these cases, the patterns revert to the simple radial structure one might expect from earlier experiments with the circular mixers. For mixtures of spheres differing in size, the results are the same for a mixer filled less than half full [Fig. 11(c)] but are more complicated when just over half full [Fig. 11(d)], where a striping pattern dominates. This phenomenon, unrelated to the Poincar´e sections, is outside the scope of this paper. For filling fractions of 25% and 75%, the circulation patterns (not shown) indicate a return of regular regions partway between the corners of the square and the center of the flowing layer. This is mirrored in the equilibrium segregation structures where the asymmetric islands return. For an example, see the cover of this issue.
8. Methods of Analysis The fact that the data are taken in the form of digital images lends itself to a range of digital analysis techniques, many of which may be known to readers of this journal. We restrict ourselves to two, though in fact, many others are possible. The most common measure of mixing is the standard deviation of the concentration fluctuations about the mean concentration — the socalled intensity of segregation [Dankwerts, 1952]. For the systems in this work, this measure of segregation captures the initial radial segregation but, unfortunately, indicates that all systems considered segregate nearly equally. Thus, this measure fails to capture any of the aspects arising from spatial structure. (See Fig. 12(a), for example.) Figure 12(b) demonstrates that a different method of analysis picks up these differences. Certain aspects of the structure can be quantified by measuring the perimeter length p and the area, a,
> 1/2 full
< 1/2 full
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Fig. 11. Variation in segregation patterns in the square mixer when the filling level is changed to about 50%. (Compare with the Poincar´e sections in Fig. 4.) Figures 11(a) and 11(b) show that for systems of different density (equal size) granular materials, the complicated segregation patterns disappear, and the equilibrium state of segregation resembles that of a circular mixer (simple radial segregation, Fig. 1). For different-sized particles (equal density), the 45% case results [Fig. 11(c)] are similar to those in Fig. 11(a). However, due to instabilities in the flow, the 55% system never reaches a final segregated pattern, but produces instead an ever changing pattern of streaks. The corresponding patterns in the circular mixer remain radially segregated state for all filling levels examined.
of the segregated region, where p corresponds to an iso-concentration line c(x, y) = K, where K is a suitable threshold. (We have done this for K = 0.75, i.e. for regions of segregation containing 75% or greater concentration of dense beads.) Thus, p2 /a is a nondimensional measure of this segregated structure. For a perfect semi-circle this value is p2 /a = (2/π)(2 + π)2 ≈ 17. Any deviation from straight radial segregation will increase the value of p2 /a. As shown in Fig. 12(b), this measure gives a dramatically different result for the
50% full and 45% full square. The result for the square filled slightly less than half full remains constant, between 20 and 25. On the other hand, the value for a half-filled square rises up slowly over the first 3–5 revolutions while the final stable structure is first forming (see Fig. 8), to nearly twice that value. It remains nearly constant for any particular orientation for over forty revolutions. The value of this measure oscillates depending on the orientation of the mixer; the measurements in this graph were done with the orientation of the mixer shown in
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9. Recommendations
Fig. 12. Measurements of the segregation pattern for different fill levels of the square mixer. In Fig. 12(a) the standard deviation from the mean concentration quickly rises with the initial radial segregation for the 7/16 full and the 1/2 full mixers. It then shows little change (if any) when the final more complicated pattern forms over the next four rotations for the half-full square mixer. In contrast, Fig. 12(b) demonstrates that a nondimensional measure of the pattern complexity — p2 /a described in the text — captures the changes. For the graph in Fig. 12(b), the measurements were taken for the region of the segregated structure of 75% or greater dense beads. The inset shows the change in this value as the threshold value is changed from 50% to 80%, demonstrating that the value is nearly constant, though it increases somewhat for increasing threshold value. The error bars in the main graph represent the variance of p2 /a from its mean value once the segregated structure had reached its equilibrium state (after five full rotations).
Fig. 10. The p2 /a measure captures the difference in segregation structure for the different fill levels of the square and the gradual evolution for the halffilled square mixer from one of radial segregation to a more complex pattern of petals.
The experiments described in this paper provide a test-bed for the study of nonlinear behavior. Undoubtedly, there are several avenues that may be pursued to fully elucidate the physics governing these systems; the experimental and computational phenomena are quite rich, and there are many questions that have yet to be answered regarding the universality of the experimental results. For example, we noted that the details of the patterns observed are sensitive to the fill level of the mixer, particularly when it is near 50%. The reasons for this behavior, however, are not completely understood though the underlying Poincar´e section seems to play an important role. To better understand how the structure of the Poincar´e section interplays with segregation phenomena one might need to consider variations in the type of mixture (studying segregation by size, density, etc.); the ratio of the relevant values for the components (size, density, etc.); the speed of rotation, and the mixer size. All these may need to be supplemented by companion computational studies; these may focus on effects hard to capture experimentally. A listing may include varying the depth of the flowing layer, the velocity profile, and changing the P´eclet number (effectively, the relative value of the diffusion coefficient). An alternative to physical experimentation for probing these systems is Particle Dynamic Simulations, and a few words in this regard should be mentioned here. These techniques are based on the same premise as Molecular Dynamic Simulations — i.e. particles (molecules) are treated as distinct entities. The macroscopic behavior of the particles is developed from either conservation of linear and angular momentum for each binary collision (hard particle approach) [Campbell, 1989] or the simultaneous solution of Newton’s equation of motion for each particle (soft particle approach) [Cundall & Strack, 1979; Walton, 1993]. These two methods are primarily used for low-density, fast-flow or high-density, slow-flow, respectively. The flow in a tumbler consists of a spectrum which spans these extremes — low-density, fast-flow in the surface layer, and high-density, slow-flow or in some case nearly “stagnant” regions in the bulk of the bed. For this reason, “hybrid” techniques have been developed to model this type of device [McCarthy & Ottino, 1998a]. The advantage of these techniques
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is that they allow measurements of quantities that would be difficult or impossible to measure in a physical experiment — such as velocity and concentration profiles, segregation rates and other details. Also, computational experiments allow close control of particle properties (size, shape, density, frictional characteristics), experimental conditions (gravity, wall friction, etc.), and additional interaction forces like liquid-bridging and van der Waals forces [Thornton & Yin, 1991]. Using these techniques, therefore, one might more easily probe such complexities as: 3D effects, the role of friction, size segregation, the role of cohesive forces [Shinbrot et al., 1999; McCarthy & Ottino, 1998b], etc. on the current model. In the context of the ideas presented here, Particle Dynamic Simulations should be regarded as a tool that allows for the elucidation of constitutive equations (size and density segregation) that can be incorporated into continuum descriptions of mixing and segregation in granular flows. (For a complete review see [Ottino & Khakhar, 2000]). In fact, the track advocated here — one focusing on a dynamical system viewpoint — considers the continuum description as the starting point. In its simplest form the continuum model outlined in the earlier sections gives a good description of mixing and segregation of granular materials in rotating cylinders of different cross-sectional shapes and for equal sized particles [Khakhar et al., 1999a]. Accepting this base-model as essentially correct, there are several questions that need to be investigated as additional physical effects are added to it. Equations (6) and (7) define a conservative chaotic dynamical system with stochastic forcing due to particle diffusion. An analysis of how diffusion destroys the invariant structures (KAM surfaces, etc.) present in the unforced system, i.e. without diffusion, and to what extent diffuse versions of these structures remain, is important for understanding mixing in the system. The issue of diffusion is more important in the case of granular materials than in the case of fluids. (See Fig. 7.) The inclusion of segregation fluxes in the model [Eqs. (10) and (11)] changes the character of the system. The system is now dissipative, and particle trajectories of a component in the mixture converge to an attractor for that component in the absence of diffusion. In the case of a circular mixer, the boundary of the attractor corresponds to a streamline of the flow. When the shape of the containers is changed from the base-case of a circle
the situation becomes more complex; the system is now chaotic. The results presented above indicate that attractors survive in the presence of diffusion and equilibrium segregation structures correspond roughly with the Poincar´e sections for the base-flow (i.e. with no diffusion or segregation). These aspects need to be studied in more detail and some degree of mathematical investigation should be possible. The situation resembles the case of a fluid with suspended solid particles but in many respects is simpler. Finally consider extensions of the model that may bring new concepts from a dynamical systems viewpoint. For example consider the extension of the model for the case of different-sized particles. In this case the velocity gradient must become composition dependent — higher gradients for a larger fraction of the smaller particles. While a rigorous approach to this problem appears formidable and would require simultaneous solution of the mass and momentum balance equations together with the convective diffusion equation, a simpler option, acceptable on physical grounds, may be to consider the average velocity (u) and layer thickness (δ) to be functions of composition chosen so as to satisfy continuity and mimic the actual flow. Such a model could reveal the implications of the coupling between velocity and composition on the mixing and segregation and pattern formation in the system. It is apparent that granular mixing and segregation experiments can benefit from an infusion of dynamical systems thinking. The enrichment is not entirely one-sided though; granular flow experiments serve to pictorially illustrate important concepts in nonlinear physics and dynamical systems and the balance between disorder and selforganization.
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