Ind. Eng. Chem. Res. 2004, 43, 5521-5528
5521
Application of the Discrete Approach to the Simulation of Size Segregation in Granular Chute Flow Jiayuan Zhang,† Ziguo Hu,‡ Wei Ge,*,† Yongjie Zhang,‡ Tinghua Li,† and Jinghai Li*,† Multi-phase Reaction Laboratory, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100080, People’s Republic of China, and Technology Center, Baoshan Iron & Steel Company, Ltd., Shanghai 201900, People’s Republic of China
Size segregation is commonly observed in flows of particles with different physical and mechanical properties. It can be favorable or undesirable, depending on different industrial processes. In our case, the size segregation of ore particles in a chute flow is utilized to prepare a well-bedded feedstock for high sintering quality, and magnets are installed below the chute to enhance size segregation. Discrete element method (DEM) simulations are carried out to explore the optimal arrangement of the magnets. The system is reduced to two dimensions, and the irregular ore particles of continuous size distribution are represented by rough disks of four diameters, with their properties reasonably sampled from the sintering materials. The magnetic fields are calculated using Ansoft in advance, which excludes the influence of the ore layer itself because it is weakly magnetic only. The simulations show that the proper magnetic field can significantly enhance the segregation, and it seems that the component normal to the chute has the dominant effect on segregation but the tangential component is also critical. The simulation results are in good agreement with our experiences in the industrial systems designed with consultation to these results. We suggest that despite major simplifications, the DEM is still an effective tool for dynamic studies and the design of complex granular flows in mineral processing. 1. Introduction Size segregation is commonly observed in flows of particles differing in physical and mechanical properties.1 The most common form of granular movement accompanied by intensive interparticle displacement is shear flow. In many industrial processes, such as the manufacture of fertilizers, detergents, pharmaceuticals, plastics, ceramics, and animal feeds, etc., the goal is to blend two or more particulate materials uniformly, where size segregation is undesired. However, in some cases in mining and agricultural engineering,2 size segregation is intended. This work deals with such a practical process where segregation is utilized to prepare a well-bedded feedstock for sintering of iron ore. In this process, as shown in Figure 1, sintering feed materials are poured down from a supplying hopper by means of a drum feeder and supplied to a sloping chute. Generally, the coarse particles tend to aggregate in the upper layer of the flow and the fine in its lower layer due to their size difference. As shown in Figure 1, when they are finally charged to a pallet moving continuously in a direction opposite to their movement on the chute, an inverse segregation appears, with the coarse particles in the lower portions and the fine on the top. Such a particle size distribution is necessary for high-quality sintering. Many methods are used to improve this distribution. For instance, if a vibrating sieve is installed a certain vertical distance above the sloping chute, the coarse particles will roll down upon the sieve directly, occupying the lower layer on the pallet, while the fine * To whom correspondence should be addressed. Tel.: +86-10-6255-8318. Fax: +86-10-6255-8065. E-mail: wge@ home.ipe.ac.cn (W.G.), jhli@ home.ipe.ac.cn (J.L.). † Chinese Academy of Sciences. ‡ Baoshan Iron & Steel Co., Ltd.
Figure 1. Sketch of the simulated system: 1, supplying hopper; 2, drum feeder; 3, sloping chute; 4, magnets; 5, moving pallet; 6, sampling cells.
particles will drop through the sieve, segregate on the chute further, and then cover the coarse layer. However, it is difficult to operate such equipment. In this paper, considering the nature of sintering materials, which mainly consist of ironstone, returned ore, and mill scale, we come to the idea of placing some magnets (see Figure 1) beneath the chute to enhance the size segregation. The idea is developed in theoretical analysis first, and three possible designs are proposed later, which are then examined by simulations with the discrete element method (DEM) to find the optimal design.
10.1021/ie034254f CCC: $27.50 © 2004 American Chemical Society Published on Web 04/30/2004
5522 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 2. Sketches of the three permanent magnet configurations.
2. Size Segregation Mechanisms and Proposed Magnet Arrangements Percolation, either spontaneous or shear-induced, is believed to be the primary mechanics for size segregation in granular flows.1,3,4 In our case, the size difference between most particles (see Table 2) is less than the spontaneous percolation threshold.5 Therefore, the effect of spontaneous percolation on segregation will be very weak if not negligible, and the shear-induced percolation should work as the chief segregation mechanism. This mechanism arises from the shearing between the particles, induced by pouring or stirring the particles into a heap or slope. It also occurs when a mixture is shaken or a particle bed is vibrated.6,7 The smaller particles tend to drain through the larger ones under the joint influence of stress and gravity so that the attraction of the magnets to magnetic particles can likely enhance the segregation by increasing the stress between the particles. On the other hand, segregation is balanced by the diffusion process, which is more significant in fast flow, where momentum transfer is dominated by particle collisions rather than by frictional rubbing.8 Because the magnetic attraction will slow the flow, the magnets are helpful to reduce diffusion also. A higher intensity of the magnets and hence stronger magnet forces are good for the enhancement of segregation in general, but if it is to high, the ore materials will conglutinate on the chute, which is unfavorable for continuous operation in the industrial process, so there is an optimal value for it. The distribution of the magnetic forces is another key factor. Though the component of the magnetic force normal to the chute is always downward, its tangential component may be upward or downward at different locations along the chute, depending on the configuration of the magnets. Obviously, an upward tangential component is favorable because it can enhance the shearing between the bottom and top layers of the particles; otherwise, it tends to reduce this shearing while increasing the velocity and hence diffusion of the flow. On the basis of the analyses above, three magnet configurations are proposed and tested, each using NdFe-B permanent magnets of the same properties but different numbers and dimensions. The retentivity Br of the permanent magnet used here is 1.23 T, and its relative permeability µ is 1.04. As sketched in Figure 2, the magnets are collocated symmetrically in all of these cases. The corresponding magnetic force distributions are shown in Figure 6. Case I consists of six narrow and relatively thick magnets with the same size and pitch, but their N pole is upward or downward alternatively. Such an arrangement results in a tan-
Figure 3. Equivalent diameter distributions under different magnetic fields.
gential magnetic force distribution similar to a sine wave (see Figure 6). Our original intention is to mimic a vibration effect that leads to segregation in purely horizontal sieving. Case II consists of six different magnets; the second and fifth are very wide and thin, with their poles on the sides. This arrangement results in tangential magnetic forces basically upward in half and downward in the other half. Case III is relatively simpler; there are only two magnets, which are much wider and thicker, with their poles on the surface or the bottom. It is an extreme version of case II because the tangential components of the magnetic forces are just the opposite for each half. However, the magnitude of the forces is increased. 3. DEM Simulations DEM has become a powerful tool for simulating granular flows since first proposed by Cundall in 1971.9 Also, its industrial application has been carried out over the past several years with simulations of ball mill dragline excavators and mineral sampling from conveyor belts,10,11 etc. Especially, DEM has already been used to simulate the radial and axial size segregation of granular materials in rotating drums and cylinders.12-14 However, only two types of particles are considered in these simulations. In this study, a modified DEM is carried out to simulate the granular flow for the three designs proposed above. Four types of particles with different diameters are considered, which complicates not only the computational algorithm but also the segregation mechanisms, as has been shown in Thomas’ experiments.15 For cost considerations, the study is limited to a two-dimensional system because the flow along the lateral direction of the chute is almost identical, and the particles are treated as circular disks because the particle shapes have comparatively less effect on segregation than their sizes.16 The simulations were performed based on the DEM model originally proposed by Cundall and Strack.17 The calculation alternates between the application of Newton’s second law to each particle and a forcedisplacement correlation at the contact point. A physical model consisting of an elastic spring and a viscous dashpot between particles is introduced to estimate the force on a given particle produced by the contact with another particle. In the shearing direction, a friction
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5523
Figure 4. Distributions of mass flow rates under different cases.
with neighboring particles or boundaries, can be formulated by the following equations:
mi
dVi dt
n
)
(Fcn,ij + Fdn,ij + Fct,ij + Fdt,ij) + ∑ j)1
Fmag,i + mg (1)
Ii
Figure 5. Equivalent diameter distributions under different magnetic force distributions: T, tangential; N, normal.
slider is also introduced to estimate the slip at the contact point. In this study, the method proposed by Tsuji et al.18,19 is adopted for calculation of the stiffness and damping coefficient, and a rolling friction model is further introduced for particle rotation.20 According to this model, the translational and rotational motions of particle i in a system at time t, caused by its interactions
dωi dt
n
)
(Tij + Mij) ∑ j)1
(2)
where mi, Ii, Vi, and ωi are the mass, moment of inertia, and translational and rotational velocities of particle i, respectively. The forces involved here are the gravitational force, mig, magnetic force Fmag,i, and interparticle forces between particles i and j, which include the normal and tangential contact forces Fcn,ij and Fct,ij and viscous damping forces Fdn,ij and Fdt,ij. The interparticle forces are summed over the n particles in contact with particle i and are dependent on the normal and tangential deformation, δn and δt. Torques Tij are generated by tangential forces (Fct,ij + Fdt,ij). Mij is the rolling friction torque arising from the elastic hysteresis loss and time-dependent deformation.21 The formulations used to calculate the forces and torques in eqs 1 and 2 are listed in Table 1. Equations 1 and 2 are solved numerically by an explicit finite
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Figure 6. Snapshots from the simulations under different conditions: N, normal; T, tangential.
difference method. The magnetic force H‚gradH is calculated in advance by the commercial software Ansoft, which excludes the influence of the ore layer itself. It is acceptable because hematite, the main component of the sintering material, is a weakly magnetic material only. The value of the magnetic force at any point is interpolated from grid values with an area-weighted averaging technique.22 The major parameters for simulation are listed in Table 2, where the continuous size distribution is represented by four particle sizes. The magnetic sus-
ceptibility of the particles is measured by an oscillating sample magnetometer at a magnetic intensity of 2000 Oe. The measurements show that the susceptibility varies with the particle diameter, with larger particles having smaller susceptibility, likely owing to a previous magnetic separation process. In our simulations, the susceptibilities of different size particles are taken as constant, though they are physically variable with the magnetic intensity. The sliding friction coefficient is measured according to the Coulomb friction law τ ) σ tan φ + C with a powder shearing
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5525 Table 1. Forces and Torques Acting on Particle i forces and torques
symbol
normal forces tangential forces rolling global
where kn )
(
contact
Fcn,ij
damping contact10 damping torque friction torque magnetic forces gravity
Fdn,ij Fct,ij Fdt,ij Tij Mij Fmag,j Gi
-ηnVn,ij -µs|Fcn,ij|[1 - (1 - min{|δt|/δt,max,1})3/2t -ηtVt,ij, ηt ) 2ηn/7 Ri × (Fct,ij + Fdt,ij) -µr|Fcn,ij| ω ˆ χH‚gradH mi g
)(
2 1 - νj2 4 1 - νi + 3 Ei Ej
equation -knδ3/2 n n
-1
)
RiRj Ri + R j
1/2
, ηn ) 2xmijkn
ln(1/e)
, mij )
xπ + [ln(1/e)] δn,ij Vt,ij 2-ν n) , t) , δt,max ) µs δ , Vij ) Vj - Vi + ωj × Rj - ωi × Rj, |δn,ij| |Vt,ij| 2(1 - ν) n 2
2
mimj , ω ˆ ) ωi/|ωi|, mi + m j
Vn,ij ) (Vij‚n)‚n, Vt,ij ) Vij - Vn,ij
Table 2. Major Simulation Parameters parameter
chute
particles
diameter, mm
Young’s modulus E, GPa Poisson ratio ν restitution coefficient e sliding friction coefficient µs rolling friction coefficient µr, mm thickness in the Y direction, mm width in the X direction, mm inclined angle R height above the pallet Hc, mm
200 0.33 0.8 0.8 0.05 40 700 50 150
10.9 0.27 0.60 1.00 0.01
0.75 (d1) 1.50 (d2) 2.50 (d3) 4.00 (d4)
apparatus, whereas Young’s modulus and the Poisson ratio of hematite are obtained by a combination of a digital spackle correlation method and resistance strain gauges. However, the rolling friction coefficient is taken from the literature.19 4. Results and Discussion To characterize the segregation quantitatively, we first introduce a parameter equivalent diameter de for the particles falling to a certain cell under the chute, 4
de )
∑ i)1
4
(nidi)/
ni ∑ i)1
(3)
where ni is the number of particles with the diameter of di. If all particles are mixed homogeneously, de is 0.981 mm, as shown in Table 2. Apparently, a higher value of de in the right cells means a better segregation effect. In addition, distributions of the mass flow rate of the four kinds of particles are also calculated, as plotted in Figure 4. To describe the separation of these curves quantitatively, another parameter R is also defined as
R)
2∆w W1 + W2
(4)
where w is the distance between the peaks of the two curves and W1 and W2 are the widths of these two curves. This idea is illustrated in Figure 4 for case I. In this work, the value for the two curves of the smallest and largest particles, R14, is taken as the major indicator. Obviously, larger R14 indicates a stronger size segregation. A combination of de and R14 is believed to give a more precise expression of the segregation degree. In our simulations, the two parameters are measured at a height of Hc ) 150 mm below the lower tip of the chute, equal to the value used in industrial experiments,
susceptibility λ/10-4 g/cm3
mass fraction, %
2.25 2.00 1.75 1.50
35.0 15.2 27.5 22.3
equivalent diameter, mm density, kg/m3 mass flow rate, kg/m‚s
0.981 1870 39.2
and in all simulations, the feeding mass flow rate is 39.2 kg/s with respect to the unit width (m) of the chute. 4.1. Effect of the Magnetic Field. The equivalent diameter de distributions in different cases are plotted in Figure 3, where an increase of de toward the right side is apparently observed, demonstrating the effect of the inclined chute even in the absence of a magnetic field. Experimental results for case I and the case with no magnets agree well with simulation results. Obviously, the magnetic field is very effective for the enhancement of size segregation. In all three cases with magnetic fields, the equivalent diameter de increases sharply to the right of a certain cell, and in case III, the surface of the particle layer even consists of large particles only. The mass flow rate distributions in Figure 4 and the snapshots of the simulation in Figure 6, where the particles are drawn as circles with radii proportional to their diameters, illustrate this effect in more detail. The total mass flow rate integrated from the four curves in Figure 4 is denoted as S and listed in each inset, which is very close, if not equal, to the feeding mass flow rate of 39.2 kg/m‚s. Therefore, the curves are valid as a breakdown of the mass balance of the flow. The difference comes from the leaking of a few particles out of the left boundary of the sampling cells and the temporal fluctuation of the flow rate during the sampling period. From Figures 3 and 4, it can be deduced that case III has the best effect on the segregation among the three cases because of its largest de ) 4 mm and highest separation degree R14 ) 0.32. It is interesting that the simplest arrangement of the magnets produces the best results. In fact, it supports our analysis regarding the effect of tangential forces on percolation, but that of frequent variation seems different from a real vibration of the chute. 4.2. Effect of the Magnetic Force Distribution. Obviously, the difference among the effects of magnetic
5526 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 7. Distributions of (a) equivalent diameter de, (b) segregation degree R14, and (c) mass flow rate with different distances D between the magnets and the chute surface in case III.
fields on the size segregation quality lies in their different distributions of magnetic forces on the particles. Figure 5 gives results of simulations similar to
those of the three cases above; only one of the normal or tangential components of the magnetic forces are exerted, however.
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Figure 8. Experimental snapshots and corresponding velocity vectors from a simulation of case I.
Obviously, segregation is mainly produced by the normal component of the magnetic force, because in all cases, the effect of the normal component on the size segregation is more significant than that of the tangential component. However, it is never as good as that for the case when both components act together. Though the tangential component of the magnetic force alone seems to have little influence on the size segregation, it gives important assistance to the normal component as expected, so it is also necessary. Figure 6 shows snapshots of the particle layer in these cases. Curves in Figure 6b-d represent the normal (red) and tangential (blue) magnetic forces acting on the particles adjacent to the chute surface in the three cases. Apparently, case III has the strongest magnetic intensity and hence magnetic force. It can be seen clearly that, when the magnetic force is exerted, the particle layer becomes thicker and hilly, especially in case I, and the hunches always occur at positions where the tangential magnetic force changes its direction, while the normal component is mainly responsible for the thickness of the flow because its attraction to the particles will increase the interparticle friction. Therefore, with a stronger normal force, the granular flow becomes thicker.
As discussed in section 2, when the tangential component in the lower part of the chute acts upward along the chute, size segregation becomes stronger. Because case III has a much wider area with upward tangential force, it leads to increasing segregation degrees in the order of cases I-III. Considering that the normal component on the particles is everywhere attractive in each case, the direction of the tangential component will be decisive on the segregation degrees. However, alternation of the tangential component in the direction along the chute has no apparent benefits for segregation, because the tangential component can only accelerate or decelerate (see Figure 8d) all of the particles but cannot change their movement directions because the chute is static, which is not the case for a real vibrating bed. 4.3. Effect of the Magnetic Intensity. The magnetic intensity should be another factor affecting segregation under similar distributions of the magnetic field. In this section, for convenience, different magnetic intensities are realized by changing the distance of the magnets to the inclined chute. Physically, the magnetic intensity, and therewith the magnetic force on the particles, decreases with the increase of distance D.
5528 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Results from simulations using the magnet configuration of case III are shown in Figure 7. It is obvious that, with the increase of distance D from 0, 5, 10, 20, to 30 mm, the segregation degree R is enhanced first and then weakened, giving an optimal distance of about 10 mm, as shown in Figure 7b, though the equivalent diameter distributions are very similar, with D not more than 10 mm. Actually, besides conglutination on the chute, which is impossible for a sustained operation, too high a magnetic intensity may also result in severe compression of the sintering material and hence the increase of its bulk density on the pallet, which is disadvantageous to sintering productivity.23 Therefore, besides the magnetic force distribution, a suitable intensity also needs to be optimized in the industrial process. This can be achieved by adjusting the distance D between the magnets and the chute or directly the properties of the permanent magnets. Nevertheless, the first method should be more flexible because the magnetism of the sintering material is not always the same and the second only works if the magnets are strong enough. Our simulation agrees well with the experiments, as can be found in the snapshots from experiments and the simulated particle velocity vector distributions shown in Figure 8. The overall particle velocity is decreased much because of the attraction of the magnetic force, resulting in less dispersion, and in the hunched parts, where the direction of the tangential magnetic force is upward, the flow is slowed further. 5. Conclusions A granular flow of sintering materials down a rough chute is simulated with four kinds of representative particles of different diameter and magnetism using a DEM. Simulations are carried out for three different arrangements of permanent magnets beneath the chute. The results show that the favorable influence of the magnetic field on size segregation is very significant, which suggests a reliable and convenient way to improve the sintering quality in practice. Magnetic force enhances the segregation of such a granular chute flow not only by strengthening the shearinduced percolation but also by weakening the diffusion effect simultaneously. Simulation results show that the magnetic force normal to the chute has a dominant effect on segregation, while the role of the tangential component is also critical when it works together with the normal component. Because the direction of the normal component is always downward, variation of the direction of the tangential component can affect the segregation strongly. Besides the magnetic field distribution, the magnetic intensity needs to be optimized also. The simulations also suggest that, despite major simplifications, DEM is still an effective tool for dynamic studies of such complex granular flows in mineral processing. Acknowledgment We thank Dr. Guoqiang Liu (Institute of Electrical Engineering, Chinese Academy of Sciences) for his help in the magnetic field computation and vice Prof. Liya Cui (Functional Materials Research Institute, Central Iron & Steel Research Institute of China) for the
measurement of susceptibility. The financial support from Natural Science Foundation of China (Grants 20336040 and 20221603) and the National Key Program for Developing Basic Sciences of China (Grant G1999022103) is also gratefully acknowledged. Literature Cited (1) Williams, J. C. The Segregation of Particulate Materials: A Review. Powder Technol. 1976, 15, 245-251. (2) Savage, S. B.; Lun, K. K. Particle Size Segregation in Inclined Chute Flow of Dry Cohensionless Granular Solids. J. Fluid Mech. 1988, 189, 311-335. (3) Savage, S. B. Interparticle percolation and segregatiom in granular materials: A review. Developments in Engineering Mechanics; Elsevier: New York, 1987; pp 347-363. (4) Bridgwater, J.; Sharpe, N. W.; Stocker, D. C. Particle mixing by percolation. Trans. Inst. Chem. Eng. 1969, 47, T114-T119. (5) Bridgwater, J.; Ingram, N. D. Rate of Spontaneous Interparticle Percolation. Trans. Inst. Chem. Eng. 1971, 49, 163-169. (6) Williams, J. C.; Shields, G. The Segregation of Granules in a Vibrated Bed. Powder Technol. 1967, 1, 134-142. (7) Harwood, C. F. Powder Segregation Due to Vibration. Powder Technol. 1977, 16, 51-57. (8) Vallance, J. W.; Savage, S. B. Particle segregation in granular flows down chutes. In IUTAM Symposium on Segregation in Granular Flows; Rosato, A. D., Blackmore, D. L., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999; Vol. 81, pp 31-51. (9) Cundall, P. A. A Computer Model for Simulating Progressive Large Scale Movements in Blocky Rock Systems. Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy, France, 1971; Vol. 1, pp 132-150. (10) Langston, P. A.; Tuzun, U.; Heyes, D. M. Discrete Elements Simulation of Granular Flow in 2D and 3D Hoppers: Dependence of Discharge Rate and Wall Stress on Particle Interactions. Chem. Eng. Sci. 1995, 50, 967-987. (11) Cleary, P. W. Modelling Comminution Devices Using DEM. Int. J. Numer. Anal. Methods Geomech. 2001, 25, 83-105. (12) Dury, C. M.; Risow, G. H. Radial Segregation in a TwoDimensional Rotating Drum. J. Phys. I 1997, 7, 737-745. (13) Rapaport, D. C. Simulational Studies of Axial Granular Segregation in a Rotating Cylinder. Phys. Rev. E 2002, 65, 0613061-061306-11. (14) Mishra, B. K.; Thornton, C.; Bhimji, D. A preliminary numerical investigation of agglomeration in a rotary drum. Miner. Eng. 2002, 15 (1 and 2), 27-33. (15) Thomas, N. Reverse and Intermediate Segregation of Large Beads in Dry Granular Media. Phys. Rev. E 2000, 62, 961-974. (16) Lawrence, L. R.; Beddow, J. K. Powder Segregation During Die Filling. Powder Technol. 1968, 2, 253-259. (17) Cundall, P. A.; Strack, O. D. L. A Discrete Numerical Model for Granular Assemblies. Geotechnique 1979, 29, 47-65. (18) Tsuji, Y.; Tanaka, T.; Ishida, T. Lagrangian Numerical Simulation of Plug Flow of Cohesionless Particles in a Horizonal Pipe. Powder Technol. 1992, 71, 239-250. (19) Tsuji, Y.; Kawaguchi, T.; Tanaka, T. Discrete Particle Simulation of Two-Dimensional Fluidized Bed. Powder Technol. 1993, 77, 79-87. (20) Zhou, Y. C.; Wright, B. D.; Yang, R. Y.; Xu, B. H.; Yu, A. B. Rolling Friction in the Dynamic Simulation of Sandpile Formation. Phys. A 1999, 536-553. (21) Tabor, D. Mechanics of rolling friction II. Proceedings of the Royal Society of London, Ser. A: Mathematical, Physical and Engineering Sciences 1955, 229, 198. (22) Hoomans, B. P. B.; Kuipers, J. A. M.; Briels, W. J.; van Swaaij, W. P. M. Discrete Particle Simulation of Bubble and Slug Formation in a Two-Dimensional Gas-Fluidised Bed: a Hard Sphere Approach. Chem. Eng. Sci. 1996, 51, 99-118. (23) Cappel, F.; Wendeborn, H. Sintern Von Eisenerzen; Verlag Stahleisen MBH: Dusseldorf, Germany, 1973; Chaper IV.
Received for review November 17, 2003 Revised manuscript received February 11, 2004 Accepted February 13, 2004 IE034254F
