Study on Residence Time Distribution of Particles in a Quasi-Two-Dimensional Batch Discharge Silo Using the Multi-Simulation Averaging Method

Abstract

As the primary carrier for storing and transporting particles, the silo is widely used in the production process. The RTD is a promising method for studying the silo discharge process and has not been studied enough. This paper presents a study on the residence time distribution (RTD) and flow pattern of particles in a two-dimensional flat-bottom batch discharge silo under gravity using experiments and the discrete element method (DEM). Meanwhile, a multi-simulation averaging method is proposed to eliminate local fluctuations in the residence time. The results are as follows. The mean flow rate is 16.85 g·s${}^{-1}$ in simulations, which is only 2.7% larger than the experimental value. In the central area of the silo, the residence time contour lines take on elliptical shapes and the trajectories of particles are straight lines. The particles are distributed along the elliptical residence time contour lines all the time during the discharge process until they flow out of the silo. The particles near the side wall of the silo swiftly flow with a constant acceleration to the central line of the silo along the upper horizontal surface, which has become avalanche slopes, and then flow down the outlet together with the particles in the radial flow region. In this study, an elliptical distribution law during the silo discharge process was funded for the first time. An improved radial flow model was proposed with a higher accuracy and clearer physical meaning, which will be helpful in silo design and scaling up in industrial applications.

1. Introduction

Granular materials are widely used in industrial and agricultural processes, including food, pharmaceutical and chemical industries [1,2,3,4], where the gravity flow discharge of particles from silos is a major area of interest for its close relation to production efficiency [5]. Previous research is mainly focused on flow patterns [6,7], stress fields [8,9], velocity profiles [10], mass flow rate [11] and so on [12]. Most of them are image-based methods and would be affected by the transparency of particles [13,14]. Recently, the residence time distribution (RTD) of particles has attracted increasing attention in 3D silo investigation [15,16,17,18]. The residence time (RT) is the length of time that it takes for a particle to flow out of the silo after the outlet is opened and the spatial distribution of the residence time of all particles is defined as RTD [15].

Existing studies are mostly focused on the steady-state discharge process for its simplicity [19,20,21]. For instance, by integrating the velocity profile along the radial direction, a commonly used model was proposed by Cleaver and Nedderman [21] to investigate radial flows. As a matter of fact, in practical applications, the batch discharge mode is widely adopted, but few theoretical studies have analyzed the batch discharge silos and many researchers only conduct qualitative analysis. This is largely due to the transient nature and instability of the batch discharge process, which adds to the complexity and difficulty of research. Able [22] studied the batch discharge process and divided the residence time into two parts, i.e., the stagnant time and the flow time, based on the study of Nedderman [23]. The experimental results showed that this model could work well in the early flow stage with high accuracy but failed in the late stage [12]. For this reason, some researchers choose to ignore the short initial phase and study a specific area by assuming the particles in the steady state at different times [12,24].

After simplifying the batch discharge process, the temporal or spatial averaging methods can be used to investigate the flow characteristics [25]. However, considering that the batch discharge is a transient process, the time-based methods are not suitable. The averaging method of going from the discrete to the continuum is a good choice [26,27]. Goldhirsch et al. [28] put forward an explicit expression of the stress field by using a general coarse-graining formulation. This method has an advantage of high accuracy, but it is only applicable to a limited number of variables, like the velocity field and the stress field. Such a shortcoming can be addressed by the traditional binning method [29], but there is also a drawback with the traditional binning method; that is, the insufficient resolution. To this end, the multi-simulation averaging method is proposed, which can eliminate local fluctuations with a high spatial resolution.

Discrete element simulation (DEM) is widely used in granular matter investigation [1,30,31,32,33], which outperforms experimental methods in rapidness, cost-effectiveness and informativeness [18,34,35,36]. Li et al. [18] analyzed the flow pattern of particles using the whole-field residence time distribution. But the PTV (particle tracking velocimetry) method is only applicable to 2D cases. Du et al. [24] compared the discharge rate under different stirring velocities and dimensionless depths of granular materials for 3D cases. The DEM method has been proved to be a powerful tool [37,38]. Pascot et al. [37] found that the discharge rate would increase under higher vibration intensity using DEM, and this was confirmed by experiments two years later [11]. Xu et al. [35] established a super-ellipsoid DEM model to investigate the multi-level discharge behaviors. Golshan et al. [15] investigated the RTD of elliptical particles in the wedge-shaped and flat-bottom silos with the numerical method and validated the effectiveness of the multi-sphere method.

As a promising method for studying the silo discharge process, RTD has not been studied enough. The aim of this study is to explore the relationship between RTD and the velocity characteristics of particles in a flat-bottom batch discharge silo using DEM. In addition, the multi-simulation averaging method is proposed to eliminate the spatial randomness in the flow time of particles. The rest of this paper is organized as follows. Section 2 presents the experimental setup and the particle velocity calculation method. Section 3 introduces the DEM model and the multi-simulation averaging method in detail. Section 4 discusses the relationship of RTD and the flow pattern of particles in the batch discharge silo. First, the outlet velocity distributions and flow rates of the silo were measured in experiments and compared with the simulation results. Second, the RTD and flow patterns of particles in three different outlet sizes were studied. Finally, an improved radial flow model was proposed to describe the silo discharge process.

2. Materials and Methods

2.1. Experimental Setup

Figure 1 portrays the geometry of a quasi-2D flat-bottom silo. The silo is a cuboid made of transparent Plexiglas with anti-static coating to minimize the electrostatic forces [37]. The inner width (L) and the height (H) of the silo are 200 mm and 500 mm, respectively. At a larger inner length, the inner width W of the silo cannot remain a constant value due to the material flexibility. The inner width ($W=2.6$ mm) is slightly larger than the diameter of beads ($d_p=2\pm 0.12$ mm) so the silo is filled with beads for only one layer [39]. The diameter of beads $d_p$ is calculated by an average of 1000 randomly selected beads with the help of a micrometer. The silo is filled with glass beads for a height of approximately 480 mm with a horizontal upper surface. The beads that flow out of the silo are weighed by an electronic scale with a 10 mg resolution. The outlet of the silo D is adjustable within a range from 0 mm to 30 mm.

Figure 2 shows the experimental setup, which comprises a flat-bottom silo with an adjustable outlet, an electronic scale, the lighting equipment and the image acquisition system.

The lighting equipment is a high-power LED (light-emitting diode) panel that can shorten the exposure time (about 100 $\mu$s). The image acquisition system mainly includes a high-speed camera (X150, Revealer, Hefei, China) with a $2560\times 1920$ p resolution at 2000 FPS (frames per second) and a personal computer. The camera memory card capacity must be large enough (about 200 GB) to record a whole discharge process. The computer is connected with the camera for parameter setting and image processing.

A total of eight experiments are conducted with the outlet size of 20 mm, including one experiment for snapshotting the discharge process and analyzing the flow pattern and the particle velocity distribution at the outlet and seven experiments for measuring the flow rates.

2.2. Particle Tracking Velocimetry

The PTV is employed to investigate the kinematic characteristics of particles in experiments, which involves three steps: particle identification, particle matching and velocity calculation.

The diameters and centers of particles are identified using the Matlab function (the function name is imfindcircles and its reference can be see in the official document: https://www.mathworks.com/help/images/ref/imfindcircles.html, accessed on 11 November 2022) which is based on the circular Hough transform (CHT) algorithm. To obtain a high identification accuracy, the parameters of the algorithm are tuned manually with an arbitrarily selected frame image and, in turn, used to identify all the particles in the remaining frame images. This is because all the frame images show a high degree of similarity.

Particle matching is finding out the same particle in two successive frame images. If the center of a particle in the second frame image is within the radius of a particle in the first frame image, they are considered one particle. Therefore, the inter-frame time shall be short enough so that the displacement of a particle in two successive frame images is not larger than the particle radius [40]. The allowable maximum particle velocity shall be 2 m·s${}^{-1}$ according to the frame rate of 2000 FPS and the particle radius of 1 mm, which is larger than the actual maximum particle velocity of about 0.7 m·s${}^{-1}$ in this experiment, so accurate matching can be guaranteed. However, for the reasons that errors may occur during the particle radius identification or some particles may run out of the frame image, the matching of some particles may fail or a particle may match more than one particle, but the possibility of this happening is small and this does not make much difference to the results. Such particles are discarded from the velocity analysis.

After the matching of particles, their velocities can be derived from their displacements divided by the inter-frame time, which are expressed as the number of pixels per second and therefore need to be converted into the physical distance per second through calibration of pixels.

3. Simulation

3.1. DEM

In this paper, the discrete element method (DEM) is used to investigate the micro-mechanical properties of particles. The material is assumed to be isotropic. The effects of temperature, humidity and air are not taken into consideration. A soft-sphere DEM with the Hertz–Mindlin contact model is used to update data of the mechanical and dynamic behaviors of particles.

As in Equations (1) and (2), the translational velocity $v_i$ and rotational velocity ${\omega }_i$ of the particle i are governed by Newton’s second law of motion [1,41]:

$$\begin{array}{ccc}\hfill m_i\frac{\mathrm{d}{v}_i}{\mathrm{d}t}& =& \sum _k(F_{ik}^n+F_{ik}^t)+m_{i}g\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill I_i\frac{\mathrm{d}{\omega }_i}{\mathrm{d}t}& =& \sum _k(M_{ik}^t+M_{ik}^r)\hfill \end{array}$$

(2)

where $m_{i}g$ and k are the gravitational force and the contact particle number of the particle i; $F_{ik}^n$, $F_{ik}^t$, $M_{ik}^t$, $M_{ik}^r$ denote the normal contact force, the tangential contact force, the tangential torque and the rolling resistance torque, which are given by Equations (3)–(6):

$$\begin{array}{ccc}\hfill F^n& =& \frac{4}{3}{E}^{*}\sqrt{R^{*}}{\delta }_n^{\frac{3}{2}}-2\sqrt{\frac{5}{6}}\beta \sqrt{2E^{*}\sqrt{R^{*}{\delta }_n}{m}^{*}}\dot{{\delta }_n}\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill F^t& =& -S_t{\delta }_t-2\sqrt{\frac{5}{6}}\beta \sqrt{S_t{m}^{*}}\dot{{\delta }_t}\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill M^t& =& min(F^t,{\mu }_s{F}^N)\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill M^r& =& -{\mu }_{r}R{F}^n{\omega }^{rel}\hfill \end{array}$$

(6)

where ${\delta }_n$, ${\delta }_t$, ${\mu }_s$ and ${\mu }_r$ are the normal overlap, the tangential overlap, the coefficient of static friction and the coefficient of rolling friction. $E^{*}$, $G^{*}$, $R^{*}$ and $m^{*}$ are the equivalent Young’s modulus, the equivalent shear modulus, the equivalent radius and the equivalent mass, which are defined as in Equations (7)–(10):

$$\begin{array}{ccc}\hfill \frac{1}{E^{*}}& =& \frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \frac{1}{G^{*}}& =& \frac{2-{\nu }_1}{G_1}+\frac{2-{\nu }_2}{G_2}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \frac{1}{R^{*}}& =& \frac{1}{R_1}+\frac{1}{R_2}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \frac{1}{m^{*}}& =& \frac{1}{m_1}+\frac{1}{m_2}\hfill \end{array}$$

(10)

with $E_1$, $v_1$, $R_1$, $m_1$ and $E_2$, $v_2$, $R_2$, $m_2$ denoting the Young’s modulus, the Poisson’s ratio, the particle radius and the particle mass of two particles. The remaining parameters are defined as in Equations (11) and (12):

$$\begin{array}{ccc}\hfill \beta & =& \frac{lne}{{\left(\ln e\right)}^2+{\pi }^2}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill S_t& =& 8G^{*}\sqrt{R^{*}{\delta }_n}\hfill \end{array}$$

(12)

The critical time step for the simulations is given by Equation (13) [42]:

$$\Delta t_c=\frac{\pi R}{0.8766+0.163\nu }\sqrt{\frac{\rho }{G}}$$

(13)

where $\nu$ is the Poisson’s ration of the particle. In this work, $\Delta t=\frac{1}{4}\Delta t_c$ is used to maintain numerical stability [43].

The model size of the simulation and experiment is set to be the same. The simulation parameters are presented in

Table 1. Parameters used in the simulations.

Parameter	Value
N (particle number)	28,000
d (particle diameter)	$2\pm 0.12$ mm
${\rho }_p$ (particle density)	2674 kg·m${}^{-3}$
${\rho }_w$ (silo density)	1200 kg·m${}^{-3}$
$\nu$ (Poisson’s ratio)	0.23
E (Young’s modulus)	5 × 10${}^6$ Pa
${\mu }_{pp}$ (particle–particle static friction coefficient)	0.5
${\mu }_{pw}$ (particle–wall static friction coefficient)	0.2
$e_{pp}$ (particle–particle restitution coefficient)	0.95
$e_{pw}$ (particle–wall restitution coefficient)	0.85

, which come from the experimental setup and similar cases [37] with some corrections.

A single simulation runs for about 2 h on a workstation with Intel Xeon 16-core CPU E5-2620 V4 and 32 GB RAM.

3.2. Multi-Simulation Averaging Method

Due to the discontinuity of the particle system, the particle properties in the silo have randomness. The fluctuations can be eliminated by averaging the particle parameters. Better results can be obtained if more particles can be used for data processing. In this paper, the multi-simulation averaging method is proposed to attain the goal without changing the silo size.

The implementation of the multi-simulation averaging method is as follows. For each simulation, the particles are randomly generated at different places of the inlet at first and then they freely fall into the silo until the silo is filled. This can make sure that the initial spatial distribution of particles in the silo has no duplicates in different simulations and it is possible to study the transient discharge process through the multi-simulation averaging method. A single simulation consists of 3 s for particle generation, 2 s for setting and 25 s for particle gravity flows. After each simulation is completed, the residence time of particles is obtained. Subsequently, the silo is divided into small bins, and the total number and the residence time of particles in the $(i,j)$ bin of all simulations can be obtained, as shown in Figure 3. Finally, the results are averaged over all the simulations.

The averaged residence time in the $(i,j)$ bin $T_{i,j}$ can be calculated by Equation (14):

$$T_{i,j}=\frac{{\sum }_{k=1}^M{T}_{i,j}^k}{M}$$

(14)

where M is the total particle number in the $(i,j)$ bin of N simulations. As shown in Equation (14), there are two possible approaches used to further homogenize the RT of the particles in the silo: increasing the simulation number or decreasing the bin size. The former method can increase the value of M and reduce the parameter fluctuation of a single particle. The latter method is to increase the total number of bins while keeping the size of the silo unchanged.

In this study, the bin width is set to 1 mm to ensure the spatial resolution. Three different outlet sizes of 20 mm, 25 mm and 30 mm are used and ten simulations are performed with each outlet size. Testing with a smaller bin width or a greater number of simulations was also conducted, but it resulted in only a minor improvement.

4. Results and Discussion

4.1. Flow Pattern

To characterize the flow pattern of particles, the particles are marked with red and blue colors at intervals in simulations while transparent and opaque glass beads are used in experiments for comparison purposes. As shown in Figure 4, the discharge processes in a randomly selected simulation and the experiment with the outlet size of 20 mm have a high degree of similarity at different times. After the discharge process begins, the particles in the central region of the silo flow toward the central line of the silo and down the outlet at the same time. A V-shaped hollow is gradually formed on the upper horizontal surface of particles. The particles near the side walls keep stationary until they merge with the particles of the upper surface. It can be seen that the flow mode belongs to the funnel flow [44]. More precisely, this is a mixed flow [45].

As can be seen from the five image pairs, the particle height and the upper surface shape show good consistency between the simulation and experiment. The angles of repose at t = 22 s are ${42}^{°}$ and ${40}^{°}$ in the simulation and experiment, respectively. It is worth noting that the discharge processes in the simulation and experiment also show local transient similarity. In the red dashed circles A and B of Figure 4, the particles of the upper surface pile up perpendicularly along the inner wall of the silo for roughly the same height and later collapse quickly. Therefore, it can be inferred that the numerical model can simulate the dynamic discharge process with good accuracy.

4.2. Outlet Velocity

The particle velocity near the outlet is shown in Figure 5. Figure 5a,b are snapshots of the particles flowing through the outlet in a randomly selected simulation and the experiment with the outlet size of 20 mm, respectively. In the experiment, the centers and radiuses of particles are identified using the PTV method at first and then the velocities of particles are calculated with their displacements from two successive frame images and the inter-frame time. As can be seen from Figure 5c,d, the velocity distribution of particles flowing through the outlet in the simulation and experiment take on arch shapes and the maximum particle velocity is about 0.5 m·s${}^{-1}$ at $x=0$. The velocity distributions of the particles flowing through the outlet in the simulation and experiment show high consistency.

Figure 6 shows the results of flow rates calculated from seven experiments and seven simulations with an outlet size of 20 mm. The mean flow rate is 16.85 g·s${}^{-1}$ in simulations, which is only 2.7% larger than the experimental value of 16.4 g·s${}^{-1}$. This deviation value is considered sufficient for the study, although it has the potential to be further reduced by fine-tuning the simulation parameters over a longer period of time. The results in simulations have better consistency, for the variance of the mean flow rate (0.0056) in simulations is smaller than the experimental result (0.0572). The fourth and fifth discharge data in the experiment exhibit larger error bars, which may be attributed to the initial state of the particles. This issue could be resolved by enhancing the filling method.

Above all, the DEM model is proved to be accurate in predicting the discharge process of the silo. The slight error may result from the outlet size difference.

4.3. Residence Time Distribution

The RTD isolines of particles in the silo are plotted with the averaged results of simulations with different outlet sizes and are presented in Figure 7. The smoothed RTD isolines indicate that the multi-simulation averaging method has effectively reduced the temporal variability. It can be seen that the silo can be divided into three distinct regions in spite of different outlet sizes: the central radial flow region where the RTD isolines take on elliptical shapes, the near-sidewall regions where the residence time is influenced by the particle–wall frictions and the stagnant regions where the angles of repose are formed between the sloping upper surface and the bottom of the silo. Specifically speaking, with an outlet size of 20 mm, the RTD isolines in the central radial flow region show a high similarity and the vertexes of the elliptical RTD isolines basically lie in the central line of the silo. This can also be found with outlet sizes of 25 mm and 30 mm. Furthermore, for RT = 1 s, the height of the RTD isolines increases linearly with an increase in the outlet size. This is also the case for RT = 3 s and 5 s. This finding makes it possible to predicate the discharge process with a small outlet by simulations with a larger outlet to save time. However, the irregular RTD isolines in the two near-sidewall regions will pose challenges to modeling the discharge flows of particles.

To further investigate the boundaries between the central radial flow region and the near-sidewall regions, the RTD of particles at different normalized heights is plotted with the averaged results of simulations with different outlet sizes and is shown in Figure 8. In the central radial flow region, the residence time of particles at different normalized heights have almost the same curves in spite of different outlet sizes. The larger the height, the longer the residence time of the particles. The residence time of particles right above the outlet is the shortest. In the regions near the side walls in Figure 8a,b, the residence time of particles increases along with the increase in height, which differs from the findings of a previous study [15]. However, in the regions near the side walls in Figure 8c, with the increase in height, the residence time of particles decreases at first and then increases. Unlike the particles in the central radial flow region, the residence time of particles in the regions near the side walls is largely impacted by the stagnant time.

To explain the phenomenon above, the trajectories of particles at four different heights in a randomly selected simulation with an outlet size of 20 mm are investigated and shown in Figure 9. Only half-planes of particle trajectories are plotted due to bilateral symmetry. Particles of equal height are uniformly selected from the transverse direction. It can be seen that there are three types of particle trajectories, namely straight lines, curved lines and piecewise lines. In the central region of the silo, the particles flow from their initial positions toward the outlet along the radial direction. In the regions near the side walls, the particles flow downward at first, then rush to the central line at an angle equal to the angle of repose and finally flow out of the silo along the central line. In the transitional region between the central radial flow region and the regions near the side walls, the particle trajectories are slightly curved lines. The locations of the three types of particle trajectories are consistent with the regions of the silo divided by the RTD isolines in Figure 7.

In order to investigate the relationship between the flow pattern and the residence time of particles, the particles with RT = 10 s are colored red in the discharge process of a randomly selected simulation with the outlet size of 20 mm and are shown in Figure 10. The particles are distributed along the RTD isolines all the time. Furthermore, the red particles are relatively more densely distributed at the top of the elliptical RTD isolines. This is consistent with the hypothesis of a point sink at the origin proposed by Nedderman [23].

4.4. Radial Flow Model

The residence time of particles at position x = 0 with different outlet sizes is plotted in Figure 11. It can be seen that the particle motion is continuous in the silo, which conflicts with the free-fall arch hypothesis [46].

The radial flow can be expressed by Equation (15):

$$V_z=\frac{\mathrm{d}z}{\mathrm{d}t}$$

(15)

where $V_z$, z are the vertical velocity and the particle height, respectively. By integrating the vertical velocity in polar coordinates, the residence time $RT$ can be obtained as in Equation (16):

$$RT\left(z\right)={\int }_0^T\mathrm{d}t={\int }_z^0\frac{\mathrm{d}z}{V_z}=p_1{z}^2+p_{2}z$$

(16)

Equation (16) suggests that the residence time $RT$ and the particle height z maintain a two-degree polynomial relationship. As in Equation (17), the three lines overlap after introducing the outlet size D as the scaling parameter.

$$RT=p_1{\left(\frac{z}{D}\right)}^2+p_2\frac{z}{D}$$

(17)

Therefore, the vertical velocity can be expressed by Equation (18):

$$V_z=-\frac{1}{2p_1\frac{z}{D}+p_2}$$

(18)

Equation (18) differs from Equation (19) in a previous study [16]:

$$V_z=-\frac{f\left(\theta \right)}{z}$$

(19)

The equation used in this study is more consistent with physical principles.

5. Conclusions

A study on the flow pattern and residence time distribution of particles in a flat-bottom batch discharge silo is conducted by using the DEM model. In addition, the multi-simulation averaging method is proposed to eliminate local fluctuations in the residence time of particles. The results of experiments and simulations in terms of the discharge process and the flow rates are in good agreement, so the combination of the DEM model and the multi-simulation averaging method is proved to be an effective tool for analyzing the RTD of particles. The velocity field of the batch discharge silo can be obtained from RTD with high precision. It is found that the batch discharge process can be smoothed by the proposed multi-simulation averaging method, which may provide an analytical framework for other similar transient studies. In spite of different outlet sizes, the RTD isolines show similar shapes and distributions. The silo can be divided into three distinct regions and the trajectories of particles differ from one region to another. From the central region to the regions near the side walls, the trajectories of particles change from straight lines to curved lines and finally become piecewise lines. The residence time of particles in the central region of the silo obey the elliptical distribution law; that is, the particles are distributed along the elliptical RTD isolines all the time during the discharge process until they flow out of the silo.

The elliptical distribution law found in this study supports the point sink hypothesis proposed by previous researchers. The proposed radial flow model provides a more physical form for the silo discharge process. The findings can be used for silo design and scaling up in industrial applications in future study. In addition, the relationship of the particle velocity and the residence time distribution in the central radial flow region can be used to establish the rheological law of particles later. An issue that remains unsolved is how to describe the transitional region between the radial flow region and the avalanche flow region. Further work needs to be carried out to figure out whether the silos of different geometries present similar residence time distribution patterns.