Flow behavior of non-spherical particle flowing in hopper

Abstract

Ellipsoidal particles flowing in the hopper were simulated by using the discrete element method (DEM), and described by the multi-element method. The contact detection algorithm and equations for ellipsoidal particle motion in hopper were developed. And the simulation results were confirmed by experiment. Additionally, the mass flow rate, pressure distribution and velocity distribution of two kinds of particles were examined. The results show that the mass flow rate of ellipsoidal particles is smaller than that of spherical particles. There is a maximum value of pressure drop at the top of the junction. Besides, the pressure drop decreases with the discharging time increasing. The velocity of spherical particle is larger than that of ellipsoidal.

Keywords

discrete element method / ellipsoidal particle / flow behavior / hopper

Introduction

The processing of granular is important in many engineering applications. These encompass operations such as storage, conveying, mixing and sizing from small scale pharmaceutical or food processing operations, where composition control may be critical, to large scale industry storage where wall stresses may be important [1]. As with solids, they can withstand deformation and form heaps; as with liquids, they can flow; as with gases, they exhibit compressibility [2]. Non-spherical granular particles, e.g. ellipsoidal, are frequently encountered in the chemical, food and pharmaceutical industries. The flow mechanisms of these particles in hoppers and silos are more complex than those of spherical particles [3,4].

The discrete element method (DEM) is widely used to model the behavior of granular materials [5-7], most of which focuses on the spherical particles. But particles in real life are largely non-spherical. Kohring et al. [8] simulated non-spherical particles flowing in hopper. Li et al. [9] simulated the disc-like particles using DEM. Grof et al. [10] simulated the needle shaped particles using DEM. Cleary and Sawley [11] researched the effect of particle shape on hopper discharge. Matuttis et al. [12] studied the contact detection of polygons. The method of potential particles introduced by Houlsby [13] can model any convex particle shape from circular to roughly polygonal in 2-D and from spherical to roughly polyhedral in 3-D. C.W. Boon [14] developed a new contact detection algorithm to simulate cube-shaped particles. Wu and Cocks [15] and Mack et al. [16] compared the results of DEM simulations with real experimental data in 2-D and 3-D, respectively. Mustoe and Miyata [17] employed super-quadrics to determine the effect of particle squareness on the dynamic angle of repose formed by cubic particles within a 2D horizontal rotating cylinder. The results showed that the dynamic angle of repose increases monotonically with particle squareness, having an upper limit of approximately 40°. Scott and Müller [18] also investigated the tangential velocity profile of non-spherical particles within a horizontal rotating cylinder using the DEM by gluing spheres together. Shardt et al. [19] described a method for the direct simulation of high-solids-volume-fraction suspensions of non-spherical rigid particles that are non-colloidal and slightly denser than the interstitial fluid.

However, the flow behaviors of ellipsoidal particles are not fully understood by researches mentioned above. The current work is aimed mainly to simulate the behavior of ellipsoidal particles flowing in hopper by developing a multi-element model, and validate the simulation by experiment. Besides, it also intends to investigate the mass flow rate, pressure distribution and velocity distribution of ellipsoidal particles.

Computational models

The contact detection algorithm and equations for ellipsoidal particles flowing in hopper can be found in Refs. [20-23].

Results and discussion

Comparison of DEM and experimental flow behavior

A three-dimensional hopper constructed by Plexiglas for convenient viewing was used in the experiment. The geometrical of the apparatus was depicted in Fig. 1. The hopper is 200 mm × 200 mm × 750 mm. The initial packing height is 700 mm. The wedge-shaped outlet has an angle of θ= 60°, and the square orifice of size is 50 mm × 50 mm.

The ellipsoidal particle used in the experiment is black soybean. To gain the flow behavior of non-spherical particles, tracer particles were used. Much attention was paid to making the property of tracer particle match with the test particle (shape, density, friction coefficient). For black soybean, the lima-bean was used as the tracer because it can keep the identical physical properties with the initial particles.

The black soybean, whose dimension is presented in Fig. 2 (the unit is mm), is described by three spherical elements. The simulation parameters used in this work are listed in Table 1. The process of the ellipsoidal particles discharging from the hopper was simulated by DEM, and the calculated flow pattern and discharge rate were compared with the experimental result to validate the applicability of the model.

Figure 3 shows the flow pattern of the ellipsoidal particles when discharging from the hopper. Although there are some difference between the DEM and experimental results, the modeling can express clearly the funnel flow in the discharging process, which is consistent with the experiment. However, the flow pattern is only a qualitative comparison. Therefore, the flow rate was studied in this paper to make a quantitative comparison between the DEM and the experiment. Figure 4 displays the comparison of the simulated mass fraction discharged and the experimental result, which indicates that the DEM predicts a flow rate slightly higher than the experimental value. The error is within 15%, which might have been caused by the deviation of simulation parameters from the actual value.

The comparisons above demonstrate that the simulated results are consistent with the experiment, indicating that the model is reasonable.

Effect of particle shape on discharge rate

The discharge rate, an important parameter when studying the flowing behavior of non-spherical particles in the hopper, was studied by many researchers [24-28]. To save computation time, the cross section of the hopper used in this work is 200 mm × 100 mm, with the other conditions invariable.

Figure 5 exhibits the comparison of the mass flow rate of spherical and ellipsoidal particles. It is seen that the mass flow rate increases sharply for two kinds of particles when the outlet is turned on instantly, because at this time the force supporting the materials in the hopper vanishes suddenly. The particles near the outlet are unloaded. This is the initial acceleration period which finishes in approximately 0.2 s. Along with the flowing wave spread over the whole hopper, a relatively stable period occurs. And a regular oscillation of almost a constant value occurs, which is caused by the dynamic arch that appears and vanishes periodically in the discharging process. The fluctuation value depends on the contact force in the shear direction. It is obviously seen that the mass flow rate of spherical particles is larger than that of ellipsoidal. And the discharging process of ellipsoidal particles finishes slower.

Figure 6 presents the comparison of the discharge rate of spherical and ellipsoidal particles. The discharging time of spherical particles is 5.5 s, but that of the ellipsoid particles is approximately 7 s. So the discharge rate of spherical particles is larger than that of ellipsoidal particles, which is corresponding to the result of Fig. 5. Besides, no particle remains in the hopper because the wedge-shaped outlet used is different from the flat-bottomed hopper.

Effect of particle shape on pressure drop

Pressure drop is an important parameter for evaluation of the flow behavior [29-36]. The wall normal pressure increases linearly with the depth for the fluid in the container. But the container is stacked in the particles, and the pressure changes are very different. Figure 7 shows the wall normal pressure distribution in the vertical when the ellipsoidal particle is static in the bed. It is to say the pressure is the stable state of all the particles after being stationary for some time when they are falling in the bed freely under the action of gravity. On the vertical axis, 0.0 m is the junction between the main part and the wedge-shaped section. It is the coordinate origin for researching the pressure changes of the main part and the wedge-shaped section. It is seen that the pressure drop is increasing with the depth at the top of the bed, and it reach the maximum value of 100 Pa, 0.15 m above the junction. Below this point, the pressure drop is decreasing with the depth. The reason for this might be that the arch formed by many particles stack together which relieves the pressure on the particles of the bed.

Figure 8 shows the normal pressure distribution in the horizontal. It can be seen that the pressure drop fluctuates around a constant value.

Figure 9 shows the pressure drop distribution in the vertical when the discharge time is 0 s, 0.2 s, 0.4 s, and 0.6 s respectively. It is seen that the pressure drop reaches the maximum value when t = 0 s. When t = 0.2 s the pressure drop decreases sharply. The reason for this is that an expansion wave is generated within the bed material and quickly spreads throughout the whole bed. A mass flow region is formed in the whole bed. As a result, the whole particles in the bed tend to flow downward. The particles accumulated become loose and the voidage increases. So the wall normal pressure drop decreases obviously. When t = 0.4 s, the pressure continues to decrease but only slightly. The materials in the bed keep losing. When t = 0.6 s, the pressure drops below 0.1 m, but it is much smaller than t = 0 s. At this time, the voidage of particles decreases, and the contact force increases, so the pressure drops correspondingly.

Figure 10 shows the wall normal pressure drop in the horizontal when the discharge time is 0 s, 0.2 s, 0.4 s, and 0.6 s, respectively. It is seen that the pressure drop reaches its maximum when the particle is static. The pressure drop in the horizontal decreases sharply when the particles discharge. And the pressure drop has no obvious change with the continual increasing in discharge time.

Effect of particle shape on velocity distribution

A variable probability density distribution (p_d) is defined to describe the velocity distribution. p_d represents the particles with a certain velocity in the percentage of the total particle number. The particle’s number with any velocity can be gained because each particle is tracked in the discrete element simulation.

Figure 11 shows the p_d of ellipsoidal and spherical particles. It is obviously seen that there is only one peak value for two kinds of particles. This illustrates that the flow regime is mass flow. The peak value on the vertical axis of spherical particles is larger than that of ellipsoidal particles. And the velocity distribution range on the abscissa is narrower. So, most spherical particles move downwards at the same speed.

Figure 12 shows the effect of particle shape on velocity. The horizontal velocity u and the vertical velocity v are dimensionless by

$$\begin{gathered} u^{*}={\displaystyle \frac{u}{{(g{W}_{\mathrm{0}})}^{1/\mathrm{2}}}}, \\ {v}^{*}={\displaystyle \frac{v}{{(g{W}_{\mathrm{0}})}^{1/\mathrm{2}}}}, \end{gathered}$$

where W₀ represents the outlet size.

The results show that the velocity is rapid at the center and slow at the side wall for two kinds of particles in the moving bed, which corresponds to the fluid flowing in the container. It is worth noticing that there are some similarities between the flowing of the particle and fluid. But the particle system relative to fluid shows more discontinuities, such as block emergence.

It also seen that the velocity of spherical particle is larger than that of ellipsoidal, which corresponds to the result in Fig. 10.

Conclusions

DEM modeling was developed to simulate ellipsoidal particles flowing in the hopper and the multi-element method was used to describe the ellipsoidal particle. The contact detection algorithm and equations of ellipsoidal particle motion in hopper were developed. And the simulation results were validated by experiment. The following conclusion can be reached based on the study.

1) The mass flow rate of ellipsoidal particles is smaller than that of spherical.

2) There is a maximum value of pressure drop at the top of the junction, and the pressure drop decreases with the discharging time increasing.

3) The velocity of spherical particles is larger than that of ellipsoidal.

Further work will focus on the flow behavior of mixed non-spherical particles.

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