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Effect of bend angle on granular size segregation in the chute flow under periodic flow inversion
Powder Technology 360 (2020) 177e192
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Effect of bend angle on granular size segregation in the chute flow under periodic flow inversion Bhargav Mantravadi*, Danielle S. Tan Department of Mechanical Engineering, National University of Singapore, Singapore, 117575
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 January 2019 Received in revised form 1 October 2019 Accepted 4 October 2019 Available online 11 October 2019
Segregation of granular homogeneous mixtures is a common problem faced by many industries. In recent years, periodic flow inversions have shown to mitigate both size and density segregation. However, they are not fully explored. In this paper, we investigate the effect of bend angle (q) in the chute flow using three-dimensional Discrete Element Method (DEM) simulations. We observed a significant difference in size segregation when changing the bend angle (q) in terms of qualitative behavior and extent of segregation. For moderate and dense regimes, minimum segregation is observed for bend angle range of 120 e150 . Based on our results, we propose to use chutes with periodic 150 bends instead of vertical chutes, to reduce segregation and thus improve mixing. © 2019 Elsevier B.V. All rights reserved.
Keywords: Periodic flow inversion Segregation Granular flows DEM simulations
1. Introduction Granular mixture handling is commonly encountered in chemical and pharmaceutical industries in chutes, drums and silos, for instance Ref. [1]. The mixture is usually subjected to various industrial processes. As a consequence of the excitation, under some conditions the mixture segregates [2,3]. Even though particle segregation is of prime importance in mineral processing industries, it is a serious problem in pharmaceutical and agricultural industries where uniform blend is desired [4]. Segregation of granular homogeneous mixtures is a common problem faced by many industries. In recent years, periodic flow inversions have shown to mitigate both size and density segregation. However, they are not fully explored. In this paper, we investigate the effect of bend angle in the chute flow using threedimensional Discrete Element Method (DEM) simulations. We observed a significant difference in size segregation when changing the bend angle (q) in terms of qualitative behavior and extent of segregation. For moderate and dense regimes, minimum segregation is observed for bend angle range of 120 e150 . Based on our results, we propose to use chutes with periodic 150 bends instead of vertical chutes, to reduce segregation and thus improve mixing. Amongst all the industrial processes, chute flow is of practical
* Corresponding author. E-mail addresses: [email protected] (B. Mantravadi), mpe_dtan@nus. edu.sg (D.S. Tan). https://doi.org/10.1016/j.powtec.2019.10.005 0032-5910/© 2019 Elsevier B.V. All rights reserved.
importance in granular transport and one of the easiest to model in experiments. Our prime interest in this paper is to study the effect of chute geometry on segregation, in particular some of the macro and microscopic properties that affect the flow dynamics. Works that focused on the effect of external factors on segregation are less numerous. Manson and Levy [5] studied particle paths after a bend in pneumatic transport of granular materials. GDR MiDi [6] presented an extensive collection of experimental and numerical results on steady uniform granular flows for six different geometries and highlighted that constitutive laws are dependent on geometry. Shi et al. [7] proposed that by altering the direction of flow, granular segregation can be minimized. A theoretical framework for perturbation frequency was developed for density segregation in free surface flows. Later, Hajra et al. [8] studied the effect of flow perturbations on size segregation. Experimental and numerical simulations were performed to estimate the critical length of the chute to avoid segregation. From the works of GDR MiDi [6] and Hajra et al. [8], it is evident that external effects like chute geometry play a significant role in segregation. As our ultimate goal is towards reducing segregation to obtain homogeneous end products, we want to investigate the reduction in segregation caused by periodic flow inversion i.e periodic perturbations in the granular flow direction by employing bends in the chute [7e9]. Specifically, we are interested in studying the extent of segregation in chute bend by varying bend angle. This is unlike Hajra et al.’s work [8] which focused on the frequency of bends in periodic flow inversions. Additionally, we will consider flows in the dilute, moderate and dense regimes, as previous researchers (e.g.
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Fan and Hill [10]) have observed that segregation is significantly affected by the mixture concentration. Since it is quite difficult to establish theoretical framework for complex geometries with different flow regimes, Discrete Element Method (DEM) simulations were performed to model the bulk movement of the binary mixture consisting of equal volume concentrations of 2 mm and 3 mm spherical particles. Section 2 contains a description of the chute geometry, while details regarding DEM simulations are described in Section 3. The numerical results are verified and validated using experimental results in Section 4.
Table 1 Parameters used for numerical simulation. Diameter of particles (d1 ; d2 )
2 mm, 3 mm
Particle density (r) Young's modulus (E) Particle-Particle restitution coefficient ðeÞ Particle-Particle friction coefficient (m) Poisson's Ratio (v) Simulation time step (△t)
2000 kg/m3 5 106 Pa 0.9 0.3 0.45 5 106 s
Fig. 1. Sketch of the chute used to study size segregation in periodic ow inversion. (a) Three-dimensional sketch of the chute; (b) Front face of the chute (in the z ¼ 0 plane) where the region AOCD is shaded blue; (c) Back face of the chute (in the z ¼ 0.05 plane) where the region EFGH is shaded yellow; (d) Side view of the chute showing the thickness. The coordinate system is shown with origin at point O. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 2. (a)e(e) Snapshots of smaller (light) and larger (dark) particles at t ¼ 0, 2, 5, 10 and 20 s for 4 ¼ 0.43, q ¼ 90 .
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The extent of segregation is quantified by plotting the local solids volume fraction profile and the results are presented in Section 5. 2. Chute geometry Fig. 1 illustrates the chute considered in this study. The chute consists of a bend with length h ¼ 0.02 m. Chute width w is 0.05 m and thickness t is 0.05 m in X and Z dimensions respectively. The chute consists of two inclined legs, each of length l ¼ 0.15 m. This value was chosen to be less than the critical length for segregation, 5.4 m, as described in Ref. [11]. Bend angle q is the angle between the inclined legs. Gravity acts along -Y direction. The boundaries are periodic in Y direction (vertical). There is a pair of walls along Z direction (front and back faces) which are highlighted in Fig. 1. The length of the leg is less than critical length, where the critical length is the minimum length required for the mixture flowing down the inclined plane to get segregated [11]. 60 , 90 , 120 , 150 and 170 bend angles were investigated for global solids volume fraction of 0.25, 0.43 and 0.54 (dilute to dense regime). The local solids volume fraction of the particles were extracted from the cuboidal domain AOCD EFGH having dimensions of 0.05 0.01 (m2) in XY plane and 0.05 m along Z dimension space.The simulation parameters are detailed in Table 1.
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there was a need to model such systems differently. One such technique, proposed by Cundall and Strack (1979) [15], is the Discrete Element Method (DEM). In recent years, DEM proved to be a promising tool for simulating multiple sphere particle dynamics as there is more freedom to specify particle-particle interactions, particle shape and material properties [10,16e19]. In this approach, particles are tracked in Lagrangian frame and interact in pair-wise manner. The discrete nature of the output of such simulations gives a better idea of the granular motion, mixing and segregation patterns of mixtures. Hence, DEM is employed in this study to describe the flow dynamics. The individual particle motion is governed by Newton's Laws and integrated using suitable time step [15]. Here, the contact force we use for particle-particle interactions is a non-linear soft sphere contact force model based on Hertzian and Mindlin contact theories [19,20], and the Coulomb law of friction. The time step used in the simulations is taken as 0.2 times the Rayleigh critical time step [21,22] and the overlap/diameter varies from 0.001 to 0.005. The soft particle approach is used here, where the particle stiffness in the simulations is artificially reduced to increase the numerical time step without significant effect on the kinematics [23]. In this paper, simulations are run using the open-source code LIGGGHTS [24] with Velocity Verlet scheme for time integration.
3. Discrete Element Method (DEM) simulations In dilute flows, the granular system is dominated by binary collisions, hence Kinetic Theory of Granular Flow (KTGF) which is derived from molecular gas dynamics predicts the particle dynamics well [12,13]. KTGF assumes chaotic motion and uses a statistical approach where the particle velocities are decomposed into time-averaged and fluctuating velocity components and the energy related with the random motions is represented by granular temperature corresponding to thermodynamic temperature. However, in moderate and dense flows, the particles usually remain in contact with several neighbors for longer periods of time and hence kinetic theory fails to predict the dynamics accurately [14]. Thus
Table 2 Total number of particles in each simulation case. Bend angle/Regime
60+
90+
120+
150+
170+
Dilute Moderate Dense
18374 31604 39689
24383 41939 52668
28994 49869 62627
31892 54854 68887
32770 56365 70784
Fig. 4. 4S profile along X direction at t ¼ 10 s, 4 ¼ 0.54. The three simulation cases differ only in the random number generator seed (prime number greater than 10000) which is responsible for the particle insertion.
Fig. 3. (a) Evolution of 4S profile; (b) Average vertical velocity variation with time for 4 ¼ 0.43, q ¼ 90 .
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The local truncation errors for position and velocity are Oð△t 4 Þ and Oð△t 2 Þ respectively and global truncation error for both position and velocity is Oð△t 2 Þ [25]. Simulations are run systematically for 4 ¼ 0.25, 0.43 and 0.54 for different bend angle and bend length. In all the results described here, the mixture consists of equal global solids volume fraction of 2 mm and 3 mm spherical particles.
In order to quantify the extent of segregation, local solids volume fraction (4) is extracted from the cuboidal volume AOCD EFGH as shown in Fig. 1. For this purpose, bins of size 0.01 0.01 0.0166 (m3) are created throughout the region ABCD. The reason for choosing this bin size is to have reliable packing fraction. Sufficient bins are required along X direction in order to quantify the segregation and also this is the direction in which the focus of the paper is. To ensure more particles for bin, the bin size along Z direction is increased to 0.0166 m. The number density of 2 mm and 3 mm particles is extracted from the bins, which is used for calculating the local solids volume fraction 4 (equations (1) and (2)).
4 ¼ 4S þ 4L 4S ¼
NS V S N VL ; 4L ¼ L VABCD VABCD
(1)
(2)
where V is volume, N is number of particles and subscripts S and L denote 2 mm and 3 mm particles respectively. In order to quantify the extent of mixing, intensity of segregation (S) of each species (i) is calculated using the following expression:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP u Nbin ci ci 2 t n¼1 n S¼ ðNbin 1Þ Fig. 5. Segregation flux in vertical chute (180 bend) for dense regime, 0e50 s.
(3)
where Nbin is the number of bins, cin ¼ 4i =4 is concentration of
Fig. 6. The experiment used to approximate the periodic flow.
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species i in bin n and ci is mean concentration of species i in the system. Since global solids volume fraction 4s ¼ 4l in all simulation cases, cs ¼ cl ¼ 0:5 and csn þ cln ¼ 1. S greater than 0.25 implies a poorly mixed system, while lower values of S imply a better mixed
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system [26]. In this study, we are more interested in the relative changes in S, rather than the absolute value of S. To better understand the dynamics that result in mixed and segregated states, normalized segregation flux (Sf ) given by equation (4) is also
Fig. 7. The top most section of the experimental setup, where W ¼ 0.07 m, L ¼ 0.15 m and h ¼ 0.02 m.
Fig. 8. (a) Snapshot at steady state (data extracted at 15000 fps) of 2 mm (light) and 3 mm (dark) particles; (b) comparison of local solid volume fraction at steady state for 4 ¼ 0.25, q ¼ 90 , h ¼ 0.02 m.
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plotted.
Sf ¼ 4i Dui ¼
4i ðui uÞ CvD
(4)
where CvD is steady-state spatially averaged vertical velocity. The sign of the flux indicates the relative segregation direction. Also, Lacey Mixing Index (as shown by equation (5)) is plotted for moderate and dense regimes.
M¼
s2o s2 s2o s2r
(5)
where s2 ¼ S2 is the variance of concentration in each bin, s2o ¼
iÞ ci ð1 ci Þ is the maximum value of the mixture variance, s2r ¼ ci ð1c n is the minimum variance in the bin, n is the possible number of particles in the bin.
First, the 2 mm sized particles are filled in the chute followed by the 3 mm sized particles. A virtual wall is present along the plane OCGF during the particle filling process. Once the filling is finished, the virtual wall is removed at time t ¼ 0. In all the simulations, particles are packed in the chute as shown in Fig. 2(a). Initially, smaller particles are inserted and allowed to settle under gravity, followed by larger particles. The number of particles inserted depends on the global solids volume fraction and the geometric specifications. The total number of particles in the entire system for all the simulation cases is summarized in Table 2. Fig. 2(a)e(e) show snapshots of the mixture for 4 ¼ 0.43, q ¼ 90 , h ¼ 0.02 m at t ¼ 0, 2, 5, 10 and 20 s respectively. Fig. 3(a) shows the temporal variation of 4S profile for q ¼ 90 , 4 ¼ 0.43 and h ¼ 0.02 m. At t ¼ 0 s, volume AOCD EFGH is completely filled with 2 mm sized particles and hence 4S is maximum. As time progresses, larger particles mix with small particles and 4S decreases in region AOCD EFGH. Eventually, the system reaches a mixed state and 4S
Fig. 9. (a)e(e) Snapshots of the dilute mixture (4 ¼ 0.25) at steady state for different bend angles; smaller (light) and larger (dark) particles.
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attains a steady profile. From Fig. 3(a), it is observed that there is no significant difference in 4S profile at t ¼ 10 s and t ¼ 20 s and hence we consider the system to have reached a steady state by t ¼ 10 s. We confirmed that the kinematics had also reached a steady state by considering the particle-averaged vertical velocities plotted in Fig. 3(b). There is an initial transience due to the sudden removal of the virtual wall at t ¼ 0 and it is observed that the averaged velocity reaches a steady state quite quickly. It is essential to ensure that the simulations are repeatable and the initial randomness in the particle positions does not influence
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the steady state results. The particle insertion depends on the random number generator seed which is a prime number greater than 10000. For this purpose, three test cases with different particle insertion seeds were simulated for 90 bend with 4 ¼ 0:54 and h ¼ 0.02 m. Fig. 4 shows 4S variation along X direction for the three cases. All three curves show similar qualitative trend with negligible differences in peaks. For the volume AOCD EFGH, the space averaged local solids volume fractions for the three cases are 0.298, 0.311 and 0.295 respectively with a standard deviation of 0.006. Hence, we believe that the steady state results are relatively
Fig. 10. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.25 of smaller (light) and larger (dark) particles.
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independent of the initial particle positions. 4. Verification and validation In this section we describe how our simulation results were validated. For the dense flows we consider the work of Fan and Hill [10]; for the dilute flows we use a small-scale physical experiment. 4.1. Verification of dense flow regime Fan and Hill [10] numerically studied the size segregation in vertical chute and reported in the dense regime, larger particles segregate to the region of higher shear rate gradients and smaller particles segregate to the region of lower shear rate gradients. We thus checked our results for 180 bend angle i.e. vertical chute, to see if the same phenomenon is observed. Fig. 5 shows the segregation flux profile of the particles. We note that even though the different chute dimensions and data extraction times led to a difference in magnitude, our simulation results exhibit the same segregation flux direction as that of Fan and Hill, and therefore the same segregation phenomenon. 4.2. Validation of dilute flow regime In order to validate the numerical results in dilute regime,
experiments are performed for 90 bend. Fig. 6 shows the schematic representation of the experimental setup. The complete setup (Fig. 6) is composed of 4 sections (8 legs). Acrylic is used for fabricating the four sections. Glass particles of 2 mm and 3 mm in diameter are used, where the density of material is 2500 kg=m3 . We interchange the legs to periodically invert the flow. Initially, the mixture is inserted in Section 1, closed by a stopper at the bottom (stopper 1). The insertion is done similar to that of the numerical simulations. Stopper 1 is then opened to allow the mixture to flow through Sections 2, 3 and 4. The mixture then settles in Section 4 as stopper 2 is in closed condition. The positions of Section 1 and Section 4 are then interchanged. At this point, stopper 2 is opened to allow the mixture to flow through the remaining three sections, where stopper 1 is in closed condition. Again Sections 4 and 1 are interchanged. This process is continued till a mixed state is reached. Fig. 7 shows Section 1 of the experimental setup. Since all four sections of the experimental setup are identical, only one section has been shown. The 2 mm and 3 mm particles are yellow and red, respectively. There is slight change of settled particle positions for dilute regime as most of the flow is a free surface flow. The local solids volume fraction of the particles close to the wall is measured in the bend region of Section 3 using high-speed video camera, when the mixture is in dynamic state. Since the data is measured in the dynamic state, the small change in the previous settled particle positions in dilute regime will not have any influence on results.
Fig. 11. (a) Variation of segregation intensity (S) with bend angle; (b)e(d) Profile of segregation flux (Sf), along chute width, for different bend angles for 4 ¼ 0.25.
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Initially, the particles in Section 1 are inserted such that the global solids volume fraction in Section 3 is close to 0.25 in dynamic state. We used ImageJ (an image processing software [27]) to determine the local solids volume fraction close to the chute wall. Fig. 8(a) shows the snapshot of the particles at steady state and Fig. 8(b) shows comparison of 4S profile obtained from experiments and numerical simulations. The brightness of the particles is used as threshold parameter to distinguish the particles at the wall. Since acrylic is used for fabricating the chute wall in experiments, the friction coefficient of the wall is slightly different from that used in the simulations. Accumulation of small particles close to the bend wall is observed, similar to that of the simulations. Hence, 4S is higher close to the bend wall. The experimental and numerical results have a decent agreement. This serves as a benchmark validation for dilute regime. From the results in verification and validation cases, we believe that the numerical results obtained in this study are credible in both dilute and dense regimes. 5. Results and discussion Granular flows are divided into three categories [28] based on the local inertial number given by I ¼ g_ dðr=PÞ0:5 where d is average particle diameter, P is pressure, r is particle density and g_ is shear rate.
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First, the dense regime, where the flow has considerable inertia, while the contact network still prevails and percolates through the particles. Second, the quasi-static regime where the inertia of the granular particles is negligible. Third, the inertial regime where inertial number is high enough and the flow is dilute and rapid. Our simulations span both the inertial and dense regimes, the results of which are described in Sections 5.1, 5.2 and 5.3 respectively. As described in Section 3, we observe both the rate of segregation as well as the change in kinematics to determine whether the flows have reached steady state. The kinematics reach a steady state more quickly than segregation intensity, so the time taken to reach steady state is determined based on segregation intensity (S). For our simulations, steady state for bend angles of 60 , 90 , 120 and 150 is reached in 20 s; for 170 , 300 s and for 180 , 800 s. 5.1. Dilute regime Fig. 9 shows snapshots of the dilute mixture (4 ¼ 0:25) at steady state for different bend angles. The flow is mostly a free surface flow as the regime is dilute (4 ¼ 0:25) and symmetric in both the legs. Fig. 10 (a)e10(e) show the plot of 4 along X direction. For 60 , 90 and 120 small particles accumulate near bend, hence 4S is higher near the bend wall AO (Fig. 1) and decreases with increasing X. Even though periodic inversion is employed, at the downstream of the bend, the mixture is expected to segregate in conventional manner
Fig. 12. (a)e(e) Snapshots of the moderate mixture (4 ¼ 0.43) at steady state for different bend angles; smaller (light) and larger (dark) particles.
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(with less segregation intensity) where smaller particles segregate to the bottom of the flow while large particles segregate to the top of the flow. However, for 60 , 90 and 120 bends, 4S shows a different trend altogether. This is primarily because of accumulation of small particles near the bend wall AO (Fig. 1). At 150 , accumulation no longer occurs and almost half of the bins close to wall AO remain empty. Hence, 4 is zero in the left half of the bins. At 170 , the mixture again occupies all bins. Fig. 11(a) shows the variation of segregation intensity with bend
angle. There is no variation in segregation intensity with change in bend angle except for 150 . At 150 , half of the bins are empty, therefore segregation intensity is larger. This is because segregation intensity (S) also considers how uniform the mixture is distributed in all bins. Fig. 11(b)e11(d) show the variation of segregation flux of small particles with bend angle. The direction of segregation flux (Sf ) for both particle species is similar, except that the peaks are of different magnitudes. Since the direction is of primary interest, Sf for only one particle species is plotted. With increasing bend angle,
Fig. 13. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.43 of smaller (light) and larger (dark) particles.
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the segregation flux magnitude is found to decrease and a change in segregation flux is observed at 170 bend. As shown in Section 5.3, flow inversion is effective only till 150 . For 170 bend, the flow phenomenon is similar to that of a vertical chute and the segregation flux direction is inline with Fan and Hill results [10]. Even though the magnitude of Sf decreases, there is no pronounced effect on S. This might be due to the high diffusive nature of the particles. 5.2. Moderate regime Fig. 12(a) -12(e) show snapshots of the mixture at moderate solids concentration (4 ¼ 0:43). The flows are again mostly free surface flows and symmetric in both legs. However, the flow is confined i.e. non-free surface flow near the bends for 60 , 90 and 120 . Fig. 13(a)e13(e) show the variation of 4 along X direction. For 60 bend, inverse grading occurs and smaller particles segregate towards the bottom of the flow while large particles segregate towards the top of the flow. Hence 4S peak is formed at high value of X/W and 4L peak is formed at low value of X/W. It is observed that small particles accumulate near the bend wall AB and thus another peak of 4S is observed near to the wall AO. With increasing bend angle, accumulation of small particles near the bend is observed to be less. For 90 , the peak due to small particles shifts slightly towards the intermediate values of X/W, while large particles
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segregate towards the top of the flow and form a peak in 4L near the bend wall AO. Khakhar et al. [29] and Wiederseiner et al. [30] reported S shaped 4S profile downstream of an inclined plane. Excluding the bins close to the wall because of accumulation, similar S shaped profile of 4S is observed for 60 bend. Fig. 14(a) shows the variation of intensity of segregation (S) with bend angle for 4 ¼ 0:43. Segregation intensity decreases with bend angle till q ¼ 150 and then increases, resulting a minimum at q ¼ 150 . Fig. 14(b) shows the Lacey mixing index for all the bend angles; the 150 case exhibits the highest mixed state. This behavior can be better explained with the help of segregation flux (Sf ), plotted in Fig. 14(c) and (d) when the system is about to reach a steady state. The magnitude of segregation flux reduces as the bend angle is increased until q ¼ 150 . At q ¼ 150 , the mixture shows a slight tendency of segregation reversal. At q ¼ 170 , a complete change in segregation direction is observed. Since the mixture is in the state of segregation reversal at 150 , a severe drop in segregation intensity is observed at q ¼ 150 . 5.3. Dense regime Fig. 15(a)e(e) show snapshots of the dense mixture (4 ¼ 0:54) at steady state for the five bend angles, with corresponding solid volume fraction profiles in Fig. 16(a)e(e). Unlike dilute and moderate regimes, the mixture is in contact with all the walls and the flow is generally not a free surface flow. A free surface with very
Fig. 14. (a) Variation of segregation intensity (S) with bend angle; (b) Variation of Lacey mixing index (M) with bend angle; (c)e(d) Profile of segregation flux (Sf), along chute width, for different bend angles for 4 ¼ 0.43.
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Fig. 15. (a)e(e) Snapshots of the dense mixture (4 ¼ 0.54) at steady state for different bend angles; smaller (light) and larger (dark) particles.
small surface area is formed in both legs for 90 and 120 , which can be considered almost negligible. At 150 , the flow is not symmetric and a free surface of larger area compared to that of 90 and 120 is formed in second leg. The flow no longer has a free surface for 170 and 180 . Kinetic sieving and squeeze expulsion are the dominant segregation mechanisms in dense regime due to the high packing fraction. This phenomenon is clearly observed for 60 bend (Fig. 16(a)) where the small particles are concentrated at the bottom of the flow and large particles at the top of the flow. However, segregation is not strong when flow inversion is employed. As the bend angle is increased, the mean shear rate increases. The small particles tend to move to the region of low shear rate gradient i.e. towards the center (intermediate X/W values). The peak of small particles is thus slightly shifted towards center (intermediate X/W values) as seen for 90 bend (Fig. 16(b)). The large particles tend to move to the regions of high shear rate. Since the shear zone thickness is larger at the top of the flow and smaller at the bottom of the flow, there is more tendency for the large particles to accumulate at the top of the flow. For 120 , the shear rate increases further. So, the small particles move further towards the center (intermediate X/W values) and large particles towards walls. This phenomenon continues with increasing bend angle. For 170 , no free surface is observed and the flow is almost equivalent to a vertical flow. In this
case, a clear segregation is observed, where large particles segregate towards the walls (region of high shear rate gradient) and small particles towards center (region of low shear rate gradient). The cross sectional view of the chute for 170 is shown in Fig. 18. Since the mixture is influenced by the four walls, larger particles segregate to the four shear zones close to the walls. Fan and Hill [10] have reported similar observations for vertical chute in dense regime. The shear zones that form near walls DJMH (bottom) and AKNE (top) are of primary interest as we are dealing with the change in bend angle. Based on the velocity gradients, we can estimate shear zone thickness [31]. Higher velocity gradients imply larger shear zone thickness and vice versa. For confined flows i.e. non-free surface flows in inclined chute, the shear zone thickness is not same on both sides of the flow. Shear zone thickness is found to be larger at the top of the flow and maximum for 60 bend. With increasing bend angle, the shear zone thickness decreases on both sides, but significantly on the top side of the flow. At 170 , there is a slight asymmetry in the shear zone thickness and finally at 180 bend angle, the shear zone thickness is found to be minimum with equal thickness on both sides and velocity profile is symmetric as shown in Fig. 17(a) and (b), where the vertical velocity (v) is nondimensionalized by the maximum vertical velocity (vm ). The flow exhibits a plug flow i.e. compact flow close to center (X/W ¼ 0.6)
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Fig. 16. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.54 of smaller (light) and larger (dark) particles.
while the mixture is less compact in the shear zones, where fluctuations in solid concentration occur. Due to the limitation in the number of bins, the fluctuations could not be visualized in this work. GDR MiDi [6] presented velocity profiles for dense granular chute flows with bend angle (inclined with the vertical) 0 , 33 and 59 . The normalized velocity profiles are in good qualitative agreement with the present results. Fig. 19(a) shows the variation of segregation intensity (S) with bend angle. S initially increases till q ¼ 120 and then decreases till q ¼ 150 , resulting in a lower segregation intensity. S again
increases for q > 150 . Fig. 19(b) shows the variation of Lacey mixing index with bend angle and the highly mixed state at 150 bend angle can be observed. This behavior can be explained by segregation flux plotted in Fig. 19(c) and (d) for large particles. The trend of segregation flux (Sf ) in dense regime is found to be different from that of dilute (inertial) regime. The magnitude of Sf increases from q ¼ 60 to q ¼ 120 and then decreases. Hence, S also initially increases and then decreases. The Sf profile at 90 and 150 overlap in the left half of the bins. Even though the magnitude of Sf is minimum at 170 , it is observed that flow inversion does not
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Fig. 17. (a) and (b) Normalized vertical velocity profile along X direction (chute width) for 4 ¼ 0.54 at bend angles of (a) 60 e150 , and (b) 170 and 180 .
experiment for free surface flows and tube length of 0.46 m. In a crude comparison, similar segregation intensity values close to 0.35 are observed in the dilute regime (free surface flow) in our simulations. Bhattacharya and McCarthy [9] studied the mixing mechanism in periodic chute flow and a mixed state with a Lacey mixing index close to 1 is obtained after a characteristic segregation time, which is the similar behavior observed in our study. Khakhar et al. [32] reported temporal variation of segregation intensity in half filled mixers (approximately moderate regime); the asymptotic segregation intensities in the range of 0.1e0.2, which is similar to the values reported in the present work. 6. Conclusion
Fig. 18. Cross-sectional view (along CD) of the chute for 170 bend angles at steady state (t ¼ 300 s) for 4 ¼ 0.54.
occur beyond 150 . Hence, segregation flux alone is not sufficient to comment on the state of the system. It has to be complemented by the evolution of segregation intensity as well. Fig. 20(a) and (b) show the evolution of S for 150 and 170 respectively. The time taken for the intensity of segregation to reach an asymptotic value is the characteristic segregation time (tcs ) [9]. For 150 , the characteristic segregation time is approximately 15 s. The order of magnitude of tcs in our simulations compare well with that of Bhattacharya and McCarthy's work. This comparison reiterates the credibility of our numerical results. And, it is obvious that for any q 150+, tcs would be less than 15 s. However, for 170 , tcs is observed to be 200 s. It is quite clear that the flow inversion is effective only up to 150 as the tcs is bounded to a small value. Therefore, tcs in a way dictates whether the flow inversion is effective or not. Fig. 21 shows the segregation intensity as a function of bend angle, for the three different flow regimes. Overall, the segregation intensity values are found to be maximum for dilute regime followed by moderate and dense regimes. Hajra et al. [8] reported segregation intensity values close to 0.3 during the rock and rotate
In this work we have shown that bend angle plays a vital role in controlling size segregation in periodic flow inversion. Segregation intensities were found to be maximum in dilute regime followed by moderate and dense regimes. This is due to the fact that the overall segregation flux magnitudes are maximum for dilute flows, followed by moderate flows and least for dense flows. In addition, when flow inversion is employed, the segregation intensities assume very low values for dense flows followed by moderate and dilute flows. Dilute flows were least affected by the change in bend angle. However, moderate and dense flows have shown significant differences in flow behavior and segregation with varying bend angle; in particular minimum segregation is observed for bend angles in the range of 120 to 150 . From the temporal evolution of segregation intensity, q greater than 150 is found to be ineffective for flow inversion. With increasing bend angle, streamwise velocity increases for all the regimes. In the inertial regime, the magnitude of segregation flux decreased with increasing bend angle. Also, a change in the direction of segregation flux is observed at 170 . Unlike inertial regime, a different phenomenon is observed in dense regime. Regardless of increase in streamwise velocity, segregation flux magnitude initially increased from 60 to 120 and then decreased for q > 120 . No change in the direction of segregation flux is observed. This emphasizes that segregation phenomenon is specific to the granular flow regime. It is well known that the size segregation occurs in the vertical chute flows under gravity. We propose that, instead of vertical chute, introducing 150 degree bend periodic flow inversion will help in reducing the segregation considerably. We recognize that the chute and mixture simulated here are very simple, and do not contain all the complexities of a realistic industrial system, such as a non-binary granular mixture with
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Fig. 19. (a) Variation of segregation intensity (S) with bend angle; (b)e(d) Variation of segregation flux (Sf) with bend angle for 4 ¼ 0.54.
Fig. 20. Evolution of segregation intensity (S) for (a) 150 and (b) 170 , 4 ¼ 0.54.
different composition or chutes with non-rectangular cross-sections. However we believe that our results are still generally applicable e since such granular flows will be largely similar, the trends we observed here should remain qualitatively the same. We would expect perhaps a difference in the segregation rate, that is, the time to reach steady state may differ. There is also the
possibility that the cross-sectional geometry of some chutes may lead to a minor shift in the optimal bend angle. For such systems, our results may perhaps be useful as a starting point from which to begin a more detailed investigation.
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Fig. 21. Segregation intensity as a function of bend angle, for the three different flow regimes.
Acknowledgement The physical experiments in this work were funded by the Singapore Ministry of Education Tier 1 grant, WBS no. R-265-000591-114. We would like to thank fellow research group members Dr. Ho-Minh Dao, Tong Jian and Ong Yin Xian for their helpful discussions. References [1] J.K. Prescott, J.W. Carson, Analyzing and overcoming industrial blending and segregation problems. IUTAM Symposium on Segregation in Granular Flows, Springer, 2000, pp. 89e101. [2] T. Shinbrot, Non-equilibrium patterns in granular mixing and segregation, Phys. Today (2000) 25e30. €ter, Focus on granular segregation, New J. Phys. 15 (3) [3] K.E. Daniels, M. Schro (2013), 035017. URL, http://stacks.iop.org/1367-2630/15/i¼3/a¼035017. [4] C.-C. Liao, S.-S. Hsiau, Transport properties and segregation phenomena in vibrating granular beds, KONA Powder Part. J. 33 (0) (2016) 109e126. [5] A. Levy, D.J. Mason, The effect of a bend on the particle cross-section concentration and segregation in pneumatic conveying systems, Powder Technol. 98 (2) (1998) 95e103. [6] G. MiDi, On dense granular flows, Eur. Phys. J. E 14 (4) (2004) 341e365. [7] D. Shi, A.A. Abatan, W.L. Vargas, J. McCarthy, Eliminating segregation in freesurface flows of particles, Phys. Rev. Lett. 99 (14) (2007) 148001.
[8] S.K. Hajra, D. Shi, J. McCarthy, Granular mixing and segregation in zigzag chute flow, Phys. Rev. E 86 (6) (2012), 061318. [9] T. Bhattacharya, Controlling Mixing and Segregation in Time Periodic Granular Flows, Ph.D. thesis, University of Pittsburgh, 2011. [10] Y. Fan, K. Hill, Phase transitions in shear-induced segregation of granular materials, Phys. Rev. Lett. 106 (21) (2011) 218301. [11] T. Bhattacharya, J. McCarthy, Chute flow as a means of segregation characterization, Powder Technol. 256 (2014) 126e139. [12] G. Wang, J. Ni, The kinetic theory for dilute solid/liquid two-phase flow, Int. J. Multiph. Flow 17 (2) (1991) 273e281. [13] X. Zhao, M. Massoudi, Flow of granular materials with slip boundary condition: a continuumekinetic theory approach, Appl. Math. Comput. 242 (2014) 518e527. [14] K. Hill, D.S. Tan, Segregation in dense sheared flows: gravity, temperature gradients, and stress partitioning, J. Fluid Mech. 756 (2014) 54e88. [15] P.A. Cundall, O.D. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47e65. [16] T. Shinbrot, F.J. Muzzio, Noise to order, Nature 410 (6825) (2001) 251. [17] J. McCarthy, J. Ottino, Particle dynamics simulation: a hybrid technique applied to granular mixing, Powder Technol. 97 (2) (1998) 91e99. [18] P.W. Cleary, M.L. Sawley, Dem modelling of industrial granular flows: 3d case studies and the effect of particle shape on hopper discharge, Appl. Math. Model. 26 (2) (2002) 89e111. [19] A. Di Renzo, F.P. Di Maio, An improved integral non-linear model for the contact of particles in distinct element simulations, Chem. Eng. Sci. 60 (5) (2005) 1303e1312. [20] A. Di Renzo, F.P. Di Maio, Comparison of contact-force models for the simulation of collisions in dem-based granular flow codes, Chem. Eng. Sci. 59 (3) (2004) 525e541. [21] J.M. Boac, R.K. Ambrose, M.E. Casada, R.G. Maghirang, D.E. Maier, Applications of discrete element method in modeling of grain postharvest operations, Food Eng. Rev. 6 (4) (2014) 128e149. [22] A. Jensen, K. Fraser, G. Laird, Improving the precision of discrete element simulations through calibration models, in: 13th International LS-DYNA Conference, Detroit, 2014. [23] S. Lommen, D. Schott, G. Lodewijks, Dem speedup: stiffness effects on behavior of bulk material, Particuology 12 (2014) 107e112. [24] C. Kloss, C. Goniva, A. Hager, S. Amberger, S. Pirker, Models, algorithms and validation for opensource dem and cfdedem, Prog. Comput. Fluid Dyn. Int. J. 12 (2e3) (2012) 140e152. [25] Q. Sun, G. Wang, Mechanics of Granular Matter, WIT Press, 2013. [26] J. McCarthy, T. Shinbrot, G. Metcalfe, J.E. Wolf, J.M. Ottino, Mixing of granular materials in slowly rotated containers, AIChE J. 42 (12) (1996) 3351e3363. [27] C.A. Schneider, W.S. Rasband, K.W. Eliceiri, Nih image to imagej: 25 years of image analysis, Nat. Methods 9 (7) (2012) 671e675. [28] J. Gaume, G. Chambon, M. Naaim, Quasistatic to inertial transition in granular materials and the role of fluctuations, Phys. Rev. E 84 (5) (2011), 051304. [29] D. Khakhar, J. McCarthy, J. Ottino, Mixing and segregation of granular materials in chute flows, Chaos, Interdiscip. J. Nonlinear Sci. 9 (3) (1999) 594e610. [30] S. Wiederseiner, N. Andreini, G. Epely-Chauvin, G. Moser, M. Monnereau, J. Gray, C. Ancey, Experimental investigation into segregating granular flows down chutes, Phys. Fluids 23 (1) (2011), 013301. [31] O. Pouliquen, R. Gutfraind, Stress fluctuations and shear zones in quasistatic granular chute flows, Phys. Rev. E 53 (1) (1996) 552. [32] D. Khakhar, J. McCarthy, J. Gilchrist, J.M. Ottino, Chaotic mixing of granular materials in two-dimensional tumbling mixers, Chaos, Interdiscip. J. Nonlinear Sci. 9 (1) (1999) 195e205.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Effect of bend angle on granular size segregation in the chute flow under periodic flow inversion Bhargav Mantravadi*, Danielle S. Tan Department of Mechanical Engineering, National University of Singapore, Singapore, 117575
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 January 2019 Received in revised form 1 October 2019 Accepted 4 October 2019 Available online 11 October 2019
Segregation of granular homogeneous mixtures is a common problem faced by many industries. In recent years, periodic flow inversions have shown to mitigate both size and density segregation. However, they are not fully explored. In this paper, we investigate the effect of bend angle (q) in the chute flow using three-dimensional Discrete Element Method (DEM) simulations. We observed a significant difference in size segregation when changing the bend angle (q) in terms of qualitative behavior and extent of segregation. For moderate and dense regimes, minimum segregation is observed for bend angle range of 120 e150 . Based on our results, we propose to use chutes with periodic 150 bends instead of vertical chutes, to reduce segregation and thus improve mixing. © 2019 Elsevier B.V. All rights reserved.
Keywords: Periodic flow inversion Segregation Granular flows DEM simulations
1. Introduction Granular mixture handling is commonly encountered in chemical and pharmaceutical industries in chutes, drums and silos, for instance Ref. [1]. The mixture is usually subjected to various industrial processes. As a consequence of the excitation, under some conditions the mixture segregates [2,3]. Even though particle segregation is of prime importance in mineral processing industries, it is a serious problem in pharmaceutical and agricultural industries where uniform blend is desired [4]. Segregation of granular homogeneous mixtures is a common problem faced by many industries. In recent years, periodic flow inversions have shown to mitigate both size and density segregation. However, they are not fully explored. In this paper, we investigate the effect of bend angle in the chute flow using threedimensional Discrete Element Method (DEM) simulations. We observed a significant difference in size segregation when changing the bend angle (q) in terms of qualitative behavior and extent of segregation. For moderate and dense regimes, minimum segregation is observed for bend angle range of 120 e150 . Based on our results, we propose to use chutes with periodic 150 bends instead of vertical chutes, to reduce segregation and thus improve mixing. Amongst all the industrial processes, chute flow is of practical
* Corresponding author. E-mail addresses: [email protected] (B. Mantravadi), mpe_dtan@nus. edu.sg (D.S. Tan). https://doi.org/10.1016/j.powtec.2019.10.005 0032-5910/© 2019 Elsevier B.V. All rights reserved.
importance in granular transport and one of the easiest to model in experiments. Our prime interest in this paper is to study the effect of chute geometry on segregation, in particular some of the macro and microscopic properties that affect the flow dynamics. Works that focused on the effect of external factors on segregation are less numerous. Manson and Levy [5] studied particle paths after a bend in pneumatic transport of granular materials. GDR MiDi [6] presented an extensive collection of experimental and numerical results on steady uniform granular flows for six different geometries and highlighted that constitutive laws are dependent on geometry. Shi et al. [7] proposed that by altering the direction of flow, granular segregation can be minimized. A theoretical framework for perturbation frequency was developed for density segregation in free surface flows. Later, Hajra et al. [8] studied the effect of flow perturbations on size segregation. Experimental and numerical simulations were performed to estimate the critical length of the chute to avoid segregation. From the works of GDR MiDi [6] and Hajra et al. [8], it is evident that external effects like chute geometry play a significant role in segregation. As our ultimate goal is towards reducing segregation to obtain homogeneous end products, we want to investigate the reduction in segregation caused by periodic flow inversion i.e periodic perturbations in the granular flow direction by employing bends in the chute [7e9]. Specifically, we are interested in studying the extent of segregation in chute bend by varying bend angle. This is unlike Hajra et al.’s work [8] which focused on the frequency of bends in periodic flow inversions. Additionally, we will consider flows in the dilute, moderate and dense regimes, as previous researchers (e.g.
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Fan and Hill [10]) have observed that segregation is significantly affected by the mixture concentration. Since it is quite difficult to establish theoretical framework for complex geometries with different flow regimes, Discrete Element Method (DEM) simulations were performed to model the bulk movement of the binary mixture consisting of equal volume concentrations of 2 mm and 3 mm spherical particles. Section 2 contains a description of the chute geometry, while details regarding DEM simulations are described in Section 3. The numerical results are verified and validated using experimental results in Section 4.
Table 1 Parameters used for numerical simulation. Diameter of particles (d1 ; d2 )
2 mm, 3 mm
Particle density (r) Young's modulus (E) Particle-Particle restitution coefficient ðeÞ Particle-Particle friction coefficient (m) Poisson's Ratio (v) Simulation time step (△t)
2000 kg/m3 5 106 Pa 0.9 0.3 0.45 5 106 s
Fig. 1. Sketch of the chute used to study size segregation in periodic ow inversion. (a) Three-dimensional sketch of the chute; (b) Front face of the chute (in the z ¼ 0 plane) where the region AOCD is shaded blue; (c) Back face of the chute (in the z ¼ 0.05 plane) where the region EFGH is shaded yellow; (d) Side view of the chute showing the thickness. The coordinate system is shown with origin at point O. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 2. (a)e(e) Snapshots of smaller (light) and larger (dark) particles at t ¼ 0, 2, 5, 10 and 20 s for 4 ¼ 0.43, q ¼ 90 .
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The extent of segregation is quantified by plotting the local solids volume fraction profile and the results are presented in Section 5. 2. Chute geometry Fig. 1 illustrates the chute considered in this study. The chute consists of a bend with length h ¼ 0.02 m. Chute width w is 0.05 m and thickness t is 0.05 m in X and Z dimensions respectively. The chute consists of two inclined legs, each of length l ¼ 0.15 m. This value was chosen to be less than the critical length for segregation, 5.4 m, as described in Ref. [11]. Bend angle q is the angle between the inclined legs. Gravity acts along -Y direction. The boundaries are periodic in Y direction (vertical). There is a pair of walls along Z direction (front and back faces) which are highlighted in Fig. 1. The length of the leg is less than critical length, where the critical length is the minimum length required for the mixture flowing down the inclined plane to get segregated [11]. 60 , 90 , 120 , 150 and 170 bend angles were investigated for global solids volume fraction of 0.25, 0.43 and 0.54 (dilute to dense regime). The local solids volume fraction of the particles were extracted from the cuboidal domain AOCD EFGH having dimensions of 0.05 0.01 (m2) in XY plane and 0.05 m along Z dimension space.The simulation parameters are detailed in Table 1.
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there was a need to model such systems differently. One such technique, proposed by Cundall and Strack (1979) [15], is the Discrete Element Method (DEM). In recent years, DEM proved to be a promising tool for simulating multiple sphere particle dynamics as there is more freedom to specify particle-particle interactions, particle shape and material properties [10,16e19]. In this approach, particles are tracked in Lagrangian frame and interact in pair-wise manner. The discrete nature of the output of such simulations gives a better idea of the granular motion, mixing and segregation patterns of mixtures. Hence, DEM is employed in this study to describe the flow dynamics. The individual particle motion is governed by Newton's Laws and integrated using suitable time step [15]. Here, the contact force we use for particle-particle interactions is a non-linear soft sphere contact force model based on Hertzian and Mindlin contact theories [19,20], and the Coulomb law of friction. The time step used in the simulations is taken as 0.2 times the Rayleigh critical time step [21,22] and the overlap/diameter varies from 0.001 to 0.005. The soft particle approach is used here, where the particle stiffness in the simulations is artificially reduced to increase the numerical time step without significant effect on the kinematics [23]. In this paper, simulations are run using the open-source code LIGGGHTS [24] with Velocity Verlet scheme for time integration.
3. Discrete Element Method (DEM) simulations In dilute flows, the granular system is dominated by binary collisions, hence Kinetic Theory of Granular Flow (KTGF) which is derived from molecular gas dynamics predicts the particle dynamics well [12,13]. KTGF assumes chaotic motion and uses a statistical approach where the particle velocities are decomposed into time-averaged and fluctuating velocity components and the energy related with the random motions is represented by granular temperature corresponding to thermodynamic temperature. However, in moderate and dense flows, the particles usually remain in contact with several neighbors for longer periods of time and hence kinetic theory fails to predict the dynamics accurately [14]. Thus
Table 2 Total number of particles in each simulation case. Bend angle/Regime
60+
90+
120+
150+
170+
Dilute Moderate Dense
18374 31604 39689
24383 41939 52668
28994 49869 62627
31892 54854 68887
32770 56365 70784
Fig. 4. 4S profile along X direction at t ¼ 10 s, 4 ¼ 0.54. The three simulation cases differ only in the random number generator seed (prime number greater than 10000) which is responsible for the particle insertion.
Fig. 3. (a) Evolution of 4S profile; (b) Average vertical velocity variation with time for 4 ¼ 0.43, q ¼ 90 .
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The local truncation errors for position and velocity are Oð△t 4 Þ and Oð△t 2 Þ respectively and global truncation error for both position and velocity is Oð△t 2 Þ [25]. Simulations are run systematically for 4 ¼ 0.25, 0.43 and 0.54 for different bend angle and bend length. In all the results described here, the mixture consists of equal global solids volume fraction of 2 mm and 3 mm spherical particles.
In order to quantify the extent of segregation, local solids volume fraction (4) is extracted from the cuboidal volume AOCD EFGH as shown in Fig. 1. For this purpose, bins of size 0.01 0.01 0.0166 (m3) are created throughout the region ABCD. The reason for choosing this bin size is to have reliable packing fraction. Sufficient bins are required along X direction in order to quantify the segregation and also this is the direction in which the focus of the paper is. To ensure more particles for bin, the bin size along Z direction is increased to 0.0166 m. The number density of 2 mm and 3 mm particles is extracted from the bins, which is used for calculating the local solids volume fraction 4 (equations (1) and (2)).
4 ¼ 4S þ 4L 4S ¼
NS V S N VL ; 4L ¼ L VABCD VABCD
(1)
(2)
where V is volume, N is number of particles and subscripts S and L denote 2 mm and 3 mm particles respectively. In order to quantify the extent of mixing, intensity of segregation (S) of each species (i) is calculated using the following expression:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP u Nbin ci ci 2 t n¼1 n S¼ ðNbin 1Þ Fig. 5. Segregation flux in vertical chute (180 bend) for dense regime, 0e50 s.
(3)
where Nbin is the number of bins, cin ¼ 4i =4 is concentration of
Fig. 6. The experiment used to approximate the periodic flow.
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species i in bin n and ci is mean concentration of species i in the system. Since global solids volume fraction 4s ¼ 4l in all simulation cases, cs ¼ cl ¼ 0:5 and csn þ cln ¼ 1. S greater than 0.25 implies a poorly mixed system, while lower values of S imply a better mixed
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system [26]. In this study, we are more interested in the relative changes in S, rather than the absolute value of S. To better understand the dynamics that result in mixed and segregated states, normalized segregation flux (Sf ) given by equation (4) is also
Fig. 7. The top most section of the experimental setup, where W ¼ 0.07 m, L ¼ 0.15 m and h ¼ 0.02 m.
Fig. 8. (a) Snapshot at steady state (data extracted at 15000 fps) of 2 mm (light) and 3 mm (dark) particles; (b) comparison of local solid volume fraction at steady state for 4 ¼ 0.25, q ¼ 90 , h ¼ 0.02 m.
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plotted.
Sf ¼ 4i Dui ¼
4i ðui uÞ CvD
(4)
where CvD is steady-state spatially averaged vertical velocity. The sign of the flux indicates the relative segregation direction. Also, Lacey Mixing Index (as shown by equation (5)) is plotted for moderate and dense regimes.
M¼
s2o s2 s2o s2r
(5)
where s2 ¼ S2 is the variance of concentration in each bin, s2o ¼
iÞ ci ð1 ci Þ is the maximum value of the mixture variance, s2r ¼ ci ð1c n is the minimum variance in the bin, n is the possible number of particles in the bin.
First, the 2 mm sized particles are filled in the chute followed by the 3 mm sized particles. A virtual wall is present along the plane OCGF during the particle filling process. Once the filling is finished, the virtual wall is removed at time t ¼ 0. In all the simulations, particles are packed in the chute as shown in Fig. 2(a). Initially, smaller particles are inserted and allowed to settle under gravity, followed by larger particles. The number of particles inserted depends on the global solids volume fraction and the geometric specifications. The total number of particles in the entire system for all the simulation cases is summarized in Table 2. Fig. 2(a)e(e) show snapshots of the mixture for 4 ¼ 0.43, q ¼ 90 , h ¼ 0.02 m at t ¼ 0, 2, 5, 10 and 20 s respectively. Fig. 3(a) shows the temporal variation of 4S profile for q ¼ 90 , 4 ¼ 0.43 and h ¼ 0.02 m. At t ¼ 0 s, volume AOCD EFGH is completely filled with 2 mm sized particles and hence 4S is maximum. As time progresses, larger particles mix with small particles and 4S decreases in region AOCD EFGH. Eventually, the system reaches a mixed state and 4S
Fig. 9. (a)e(e) Snapshots of the dilute mixture (4 ¼ 0.25) at steady state for different bend angles; smaller (light) and larger (dark) particles.
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attains a steady profile. From Fig. 3(a), it is observed that there is no significant difference in 4S profile at t ¼ 10 s and t ¼ 20 s and hence we consider the system to have reached a steady state by t ¼ 10 s. We confirmed that the kinematics had also reached a steady state by considering the particle-averaged vertical velocities plotted in Fig. 3(b). There is an initial transience due to the sudden removal of the virtual wall at t ¼ 0 and it is observed that the averaged velocity reaches a steady state quite quickly. It is essential to ensure that the simulations are repeatable and the initial randomness in the particle positions does not influence
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the steady state results. The particle insertion depends on the random number generator seed which is a prime number greater than 10000. For this purpose, three test cases with different particle insertion seeds were simulated for 90 bend with 4 ¼ 0:54 and h ¼ 0.02 m. Fig. 4 shows 4S variation along X direction for the three cases. All three curves show similar qualitative trend with negligible differences in peaks. For the volume AOCD EFGH, the space averaged local solids volume fractions for the three cases are 0.298, 0.311 and 0.295 respectively with a standard deviation of 0.006. Hence, we believe that the steady state results are relatively
Fig. 10. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.25 of smaller (light) and larger (dark) particles.
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independent of the initial particle positions. 4. Verification and validation In this section we describe how our simulation results were validated. For the dense flows we consider the work of Fan and Hill [10]; for the dilute flows we use a small-scale physical experiment. 4.1. Verification of dense flow regime Fan and Hill [10] numerically studied the size segregation in vertical chute and reported in the dense regime, larger particles segregate to the region of higher shear rate gradients and smaller particles segregate to the region of lower shear rate gradients. We thus checked our results for 180 bend angle i.e. vertical chute, to see if the same phenomenon is observed. Fig. 5 shows the segregation flux profile of the particles. We note that even though the different chute dimensions and data extraction times led to a difference in magnitude, our simulation results exhibit the same segregation flux direction as that of Fan and Hill, and therefore the same segregation phenomenon. 4.2. Validation of dilute flow regime In order to validate the numerical results in dilute regime,
experiments are performed for 90 bend. Fig. 6 shows the schematic representation of the experimental setup. The complete setup (Fig. 6) is composed of 4 sections (8 legs). Acrylic is used for fabricating the four sections. Glass particles of 2 mm and 3 mm in diameter are used, where the density of material is 2500 kg=m3 . We interchange the legs to periodically invert the flow. Initially, the mixture is inserted in Section 1, closed by a stopper at the bottom (stopper 1). The insertion is done similar to that of the numerical simulations. Stopper 1 is then opened to allow the mixture to flow through Sections 2, 3 and 4. The mixture then settles in Section 4 as stopper 2 is in closed condition. The positions of Section 1 and Section 4 are then interchanged. At this point, stopper 2 is opened to allow the mixture to flow through the remaining three sections, where stopper 1 is in closed condition. Again Sections 4 and 1 are interchanged. This process is continued till a mixed state is reached. Fig. 7 shows Section 1 of the experimental setup. Since all four sections of the experimental setup are identical, only one section has been shown. The 2 mm and 3 mm particles are yellow and red, respectively. There is slight change of settled particle positions for dilute regime as most of the flow is a free surface flow. The local solids volume fraction of the particles close to the wall is measured in the bend region of Section 3 using high-speed video camera, when the mixture is in dynamic state. Since the data is measured in the dynamic state, the small change in the previous settled particle positions in dilute regime will not have any influence on results.
Fig. 11. (a) Variation of segregation intensity (S) with bend angle; (b)e(d) Profile of segregation flux (Sf), along chute width, for different bend angles for 4 ¼ 0.25.
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Initially, the particles in Section 1 are inserted such that the global solids volume fraction in Section 3 is close to 0.25 in dynamic state. We used ImageJ (an image processing software [27]) to determine the local solids volume fraction close to the chute wall. Fig. 8(a) shows the snapshot of the particles at steady state and Fig. 8(b) shows comparison of 4S profile obtained from experiments and numerical simulations. The brightness of the particles is used as threshold parameter to distinguish the particles at the wall. Since acrylic is used for fabricating the chute wall in experiments, the friction coefficient of the wall is slightly different from that used in the simulations. Accumulation of small particles close to the bend wall is observed, similar to that of the simulations. Hence, 4S is higher close to the bend wall. The experimental and numerical results have a decent agreement. This serves as a benchmark validation for dilute regime. From the results in verification and validation cases, we believe that the numerical results obtained in this study are credible in both dilute and dense regimes. 5. Results and discussion Granular flows are divided into three categories [28] based on the local inertial number given by I ¼ g_ dðr=PÞ0:5 where d is average particle diameter, P is pressure, r is particle density and g_ is shear rate.
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First, the dense regime, where the flow has considerable inertia, while the contact network still prevails and percolates through the particles. Second, the quasi-static regime where the inertia of the granular particles is negligible. Third, the inertial regime where inertial number is high enough and the flow is dilute and rapid. Our simulations span both the inertial and dense regimes, the results of which are described in Sections 5.1, 5.2 and 5.3 respectively. As described in Section 3, we observe both the rate of segregation as well as the change in kinematics to determine whether the flows have reached steady state. The kinematics reach a steady state more quickly than segregation intensity, so the time taken to reach steady state is determined based on segregation intensity (S). For our simulations, steady state for bend angles of 60 , 90 , 120 and 150 is reached in 20 s; for 170 , 300 s and for 180 , 800 s. 5.1. Dilute regime Fig. 9 shows snapshots of the dilute mixture (4 ¼ 0:25) at steady state for different bend angles. The flow is mostly a free surface flow as the regime is dilute (4 ¼ 0:25) and symmetric in both the legs. Fig. 10 (a)e10(e) show the plot of 4 along X direction. For 60 , 90 and 120 small particles accumulate near bend, hence 4S is higher near the bend wall AO (Fig. 1) and decreases with increasing X. Even though periodic inversion is employed, at the downstream of the bend, the mixture is expected to segregate in conventional manner
Fig. 12. (a)e(e) Snapshots of the moderate mixture (4 ¼ 0.43) at steady state for different bend angles; smaller (light) and larger (dark) particles.
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(with less segregation intensity) where smaller particles segregate to the bottom of the flow while large particles segregate to the top of the flow. However, for 60 , 90 and 120 bends, 4S shows a different trend altogether. This is primarily because of accumulation of small particles near the bend wall AO (Fig. 1). At 150 , accumulation no longer occurs and almost half of the bins close to wall AO remain empty. Hence, 4 is zero in the left half of the bins. At 170 , the mixture again occupies all bins. Fig. 11(a) shows the variation of segregation intensity with bend
angle. There is no variation in segregation intensity with change in bend angle except for 150 . At 150 , half of the bins are empty, therefore segregation intensity is larger. This is because segregation intensity (S) also considers how uniform the mixture is distributed in all bins. Fig. 11(b)e11(d) show the variation of segregation flux of small particles with bend angle. The direction of segregation flux (Sf ) for both particle species is similar, except that the peaks are of different magnitudes. Since the direction is of primary interest, Sf for only one particle species is plotted. With increasing bend angle,
Fig. 13. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.43 of smaller (light) and larger (dark) particles.
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the segregation flux magnitude is found to decrease and a change in segregation flux is observed at 170 bend. As shown in Section 5.3, flow inversion is effective only till 150 . For 170 bend, the flow phenomenon is similar to that of a vertical chute and the segregation flux direction is inline with Fan and Hill results [10]. Even though the magnitude of Sf decreases, there is no pronounced effect on S. This might be due to the high diffusive nature of the particles. 5.2. Moderate regime Fig. 12(a) -12(e) show snapshots of the mixture at moderate solids concentration (4 ¼ 0:43). The flows are again mostly free surface flows and symmetric in both legs. However, the flow is confined i.e. non-free surface flow near the bends for 60 , 90 and 120 . Fig. 13(a)e13(e) show the variation of 4 along X direction. For 60 bend, inverse grading occurs and smaller particles segregate towards the bottom of the flow while large particles segregate towards the top of the flow. Hence 4S peak is formed at high value of X/W and 4L peak is formed at low value of X/W. It is observed that small particles accumulate near the bend wall AB and thus another peak of 4S is observed near to the wall AO. With increasing bend angle, accumulation of small particles near the bend is observed to be less. For 90 , the peak due to small particles shifts slightly towards the intermediate values of X/W, while large particles
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segregate towards the top of the flow and form a peak in 4L near the bend wall AO. Khakhar et al. [29] and Wiederseiner et al. [30] reported S shaped 4S profile downstream of an inclined plane. Excluding the bins close to the wall because of accumulation, similar S shaped profile of 4S is observed for 60 bend. Fig. 14(a) shows the variation of intensity of segregation (S) with bend angle for 4 ¼ 0:43. Segregation intensity decreases with bend angle till q ¼ 150 and then increases, resulting a minimum at q ¼ 150 . Fig. 14(b) shows the Lacey mixing index for all the bend angles; the 150 case exhibits the highest mixed state. This behavior can be better explained with the help of segregation flux (Sf ), plotted in Fig. 14(c) and (d) when the system is about to reach a steady state. The magnitude of segregation flux reduces as the bend angle is increased until q ¼ 150 . At q ¼ 150 , the mixture shows a slight tendency of segregation reversal. At q ¼ 170 , a complete change in segregation direction is observed. Since the mixture is in the state of segregation reversal at 150 , a severe drop in segregation intensity is observed at q ¼ 150 . 5.3. Dense regime Fig. 15(a)e(e) show snapshots of the dense mixture (4 ¼ 0:54) at steady state for the five bend angles, with corresponding solid volume fraction profiles in Fig. 16(a)e(e). Unlike dilute and moderate regimes, the mixture is in contact with all the walls and the flow is generally not a free surface flow. A free surface with very
Fig. 14. (a) Variation of segregation intensity (S) with bend angle; (b) Variation of Lacey mixing index (M) with bend angle; (c)e(d) Profile of segregation flux (Sf), along chute width, for different bend angles for 4 ¼ 0.43.
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Fig. 15. (a)e(e) Snapshots of the dense mixture (4 ¼ 0.54) at steady state for different bend angles; smaller (light) and larger (dark) particles.
small surface area is formed in both legs for 90 and 120 , which can be considered almost negligible. At 150 , the flow is not symmetric and a free surface of larger area compared to that of 90 and 120 is formed in second leg. The flow no longer has a free surface for 170 and 180 . Kinetic sieving and squeeze expulsion are the dominant segregation mechanisms in dense regime due to the high packing fraction. This phenomenon is clearly observed for 60 bend (Fig. 16(a)) where the small particles are concentrated at the bottom of the flow and large particles at the top of the flow. However, segregation is not strong when flow inversion is employed. As the bend angle is increased, the mean shear rate increases. The small particles tend to move to the region of low shear rate gradient i.e. towards the center (intermediate X/W values). The peak of small particles is thus slightly shifted towards center (intermediate X/W values) as seen for 90 bend (Fig. 16(b)). The large particles tend to move to the regions of high shear rate. Since the shear zone thickness is larger at the top of the flow and smaller at the bottom of the flow, there is more tendency for the large particles to accumulate at the top of the flow. For 120 , the shear rate increases further. So, the small particles move further towards the center (intermediate X/W values) and large particles towards walls. This phenomenon continues with increasing bend angle. For 170 , no free surface is observed and the flow is almost equivalent to a vertical flow. In this
case, a clear segregation is observed, where large particles segregate towards the walls (region of high shear rate gradient) and small particles towards center (region of low shear rate gradient). The cross sectional view of the chute for 170 is shown in Fig. 18. Since the mixture is influenced by the four walls, larger particles segregate to the four shear zones close to the walls. Fan and Hill [10] have reported similar observations for vertical chute in dense regime. The shear zones that form near walls DJMH (bottom) and AKNE (top) are of primary interest as we are dealing with the change in bend angle. Based on the velocity gradients, we can estimate shear zone thickness [31]. Higher velocity gradients imply larger shear zone thickness and vice versa. For confined flows i.e. non-free surface flows in inclined chute, the shear zone thickness is not same on both sides of the flow. Shear zone thickness is found to be larger at the top of the flow and maximum for 60 bend. With increasing bend angle, the shear zone thickness decreases on both sides, but significantly on the top side of the flow. At 170 , there is a slight asymmetry in the shear zone thickness and finally at 180 bend angle, the shear zone thickness is found to be minimum with equal thickness on both sides and velocity profile is symmetric as shown in Fig. 17(a) and (b), where the vertical velocity (v) is nondimensionalized by the maximum vertical velocity (vm ). The flow exhibits a plug flow i.e. compact flow close to center (X/W ¼ 0.6)
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Fig. 16. (a)e(e) Local solids volume fraction profile along X direction (chute width) at steady state, 4 ¼ 0.54 of smaller (light) and larger (dark) particles.
while the mixture is less compact in the shear zones, where fluctuations in solid concentration occur. Due to the limitation in the number of bins, the fluctuations could not be visualized in this work. GDR MiDi [6] presented velocity profiles for dense granular chute flows with bend angle (inclined with the vertical) 0 , 33 and 59 . The normalized velocity profiles are in good qualitative agreement with the present results. Fig. 19(a) shows the variation of segregation intensity (S) with bend angle. S initially increases till q ¼ 120 and then decreases till q ¼ 150 , resulting in a lower segregation intensity. S again
increases for q > 150 . Fig. 19(b) shows the variation of Lacey mixing index with bend angle and the highly mixed state at 150 bend angle can be observed. This behavior can be explained by segregation flux plotted in Fig. 19(c) and (d) for large particles. The trend of segregation flux (Sf ) in dense regime is found to be different from that of dilute (inertial) regime. The magnitude of Sf increases from q ¼ 60 to q ¼ 120 and then decreases. Hence, S also initially increases and then decreases. The Sf profile at 90 and 150 overlap in the left half of the bins. Even though the magnitude of Sf is minimum at 170 , it is observed that flow inversion does not
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Fig. 17. (a) and (b) Normalized vertical velocity profile along X direction (chute width) for 4 ¼ 0.54 at bend angles of (a) 60 e150 , and (b) 170 and 180 .
experiment for free surface flows and tube length of 0.46 m. In a crude comparison, similar segregation intensity values close to 0.35 are observed in the dilute regime (free surface flow) in our simulations. Bhattacharya and McCarthy [9] studied the mixing mechanism in periodic chute flow and a mixed state with a Lacey mixing index close to 1 is obtained after a characteristic segregation time, which is the similar behavior observed in our study. Khakhar et al. [32] reported temporal variation of segregation intensity in half filled mixers (approximately moderate regime); the asymptotic segregation intensities in the range of 0.1e0.2, which is similar to the values reported in the present work. 6. Conclusion
Fig. 18. Cross-sectional view (along CD) of the chute for 170 bend angles at steady state (t ¼ 300 s) for 4 ¼ 0.54.
occur beyond 150 . Hence, segregation flux alone is not sufficient to comment on the state of the system. It has to be complemented by the evolution of segregation intensity as well. Fig. 20(a) and (b) show the evolution of S for 150 and 170 respectively. The time taken for the intensity of segregation to reach an asymptotic value is the characteristic segregation time (tcs ) [9]. For 150 , the characteristic segregation time is approximately 15 s. The order of magnitude of tcs in our simulations compare well with that of Bhattacharya and McCarthy's work. This comparison reiterates the credibility of our numerical results. And, it is obvious that for any q 150+, tcs would be less than 15 s. However, for 170 , tcs is observed to be 200 s. It is quite clear that the flow inversion is effective only up to 150 as the tcs is bounded to a small value. Therefore, tcs in a way dictates whether the flow inversion is effective or not. Fig. 21 shows the segregation intensity as a function of bend angle, for the three different flow regimes. Overall, the segregation intensity values are found to be maximum for dilute regime followed by moderate and dense regimes. Hajra et al. [8] reported segregation intensity values close to 0.3 during the rock and rotate
In this work we have shown that bend angle plays a vital role in controlling size segregation in periodic flow inversion. Segregation intensities were found to be maximum in dilute regime followed by moderate and dense regimes. This is due to the fact that the overall segregation flux magnitudes are maximum for dilute flows, followed by moderate flows and least for dense flows. In addition, when flow inversion is employed, the segregation intensities assume very low values for dense flows followed by moderate and dilute flows. Dilute flows were least affected by the change in bend angle. However, moderate and dense flows have shown significant differences in flow behavior and segregation with varying bend angle; in particular minimum segregation is observed for bend angles in the range of 120 to 150 . From the temporal evolution of segregation intensity, q greater than 150 is found to be ineffective for flow inversion. With increasing bend angle, streamwise velocity increases for all the regimes. In the inertial regime, the magnitude of segregation flux decreased with increasing bend angle. Also, a change in the direction of segregation flux is observed at 170 . Unlike inertial regime, a different phenomenon is observed in dense regime. Regardless of increase in streamwise velocity, segregation flux magnitude initially increased from 60 to 120 and then decreased for q > 120 . No change in the direction of segregation flux is observed. This emphasizes that segregation phenomenon is specific to the granular flow regime. It is well known that the size segregation occurs in the vertical chute flows under gravity. We propose that, instead of vertical chute, introducing 150 degree bend periodic flow inversion will help in reducing the segregation considerably. We recognize that the chute and mixture simulated here are very simple, and do not contain all the complexities of a realistic industrial system, such as a non-binary granular mixture with
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Fig. 19. (a) Variation of segregation intensity (S) with bend angle; (b)e(d) Variation of segregation flux (Sf) with bend angle for 4 ¼ 0.54.
Fig. 20. Evolution of segregation intensity (S) for (a) 150 and (b) 170 , 4 ¼ 0.54.
different composition or chutes with non-rectangular cross-sections. However we believe that our results are still generally applicable e since such granular flows will be largely similar, the trends we observed here should remain qualitatively the same. We would expect perhaps a difference in the segregation rate, that is, the time to reach steady state may differ. There is also the
possibility that the cross-sectional geometry of some chutes may lead to a minor shift in the optimal bend angle. For such systems, our results may perhaps be useful as a starting point from which to begin a more detailed investigation.
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Fig. 21. Segregation intensity as a function of bend angle, for the three different flow regimes.
Acknowledgement The physical experiments in this work were funded by the Singapore Ministry of Education Tier 1 grant, WBS no. R-265-000591-114. We would like to thank fellow research group members Dr. Ho-Minh Dao, Tong Jian and Ong Yin Xian for their helpful discussions. References [1] J.K. Prescott, J.W. Carson, Analyzing and overcoming industrial blending and segregation problems. IUTAM Symposium on Segregation in Granular Flows, Springer, 2000, pp. 89e101. [2] T. Shinbrot, Non-equilibrium patterns in granular mixing and segregation, Phys. Today (2000) 25e30. €ter, Focus on granular segregation, New J. Phys. 15 (3) [3] K.E. Daniels, M. Schro (2013), 035017. URL, http://stacks.iop.org/1367-2630/15/i¼3/a¼035017. [4] C.-C. Liao, S.-S. Hsiau, Transport properties and segregation phenomena in vibrating granular beds, KONA Powder Part. J. 33 (0) (2016) 109e126. [5] A. Levy, D.J. Mason, The effect of a bend on the particle cross-section concentration and segregation in pneumatic conveying systems, Powder Technol. 98 (2) (1998) 95e103. [6] G. MiDi, On dense granular flows, Eur. Phys. J. E 14 (4) (2004) 341e365. [7] D. Shi, A.A. Abatan, W.L. Vargas, J. McCarthy, Eliminating segregation in freesurface flows of particles, Phys. Rev. Lett. 99 (14) (2007) 148001.
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