Numerical analysis of the effect of the contraction rate of the curved hopper on flow characteristics of the silo discharge

Journal Pre-proof Numerical analysis of the effect of the contraction rate of the curved hopper on flow characteristics of the silo discharge

Hanru Liu, Fuguo Jia, Yawen Xiao, Yanlong Han, Gengrun Li, Anqi Li, Shigang Bai PII:

S0032-5910(19)30759-4

DOI:

https://doi.org/10.1016/j.powtec.2019.09.033

Reference:

PTEC 14707

To appear in:

Powder Technology

Received date:

5 July 2019

Revised date:

7 September 2019

Accepted date:

11 September 2019

Please cite this article as: H. Liu, F. Jia, Y. Xiao, et al., Numerical analysis of the effect of the contraction rate of the curved hopper on flow characteristics of the silo discharge, Powder Technology(2018), https://doi.org/10.1016/j.powtec.2019.09.033

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© 2018 Published by Elsevier.

Journal Pre-proof Numerical analysis of the effect of the contraction rate of the curved hopper on flow characteristics of the silo discharge Hanru Liu, Fuguo Jia*, Yawen Xiao, Yanlong Han, Gengrun Li, Anqi Li, Shigang Bai (College of Engineering, Northeast Agricultural University, Harbin, Heilongjiang 150030, China) *Corresponding author: Fuguo Jia. Tel. 86-451-55190620; Fax: 86-451-55191321. E-mail address: [email protected].

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Abstract: The jamming of particles during the silo discharge is a long-standing problem. The silo

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with a curved hopper can effectively prevent jamming. In this work, discrete element method was

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used to study the effect of contraction rate on the discharge flow characteristics in the silo with a

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curved hopper. Froude number was adopted to evaluate the flowability of the silo. It was found

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that a smaller contraction rate is propitious to granular flows. Particularly, the contraction rate was found to have a distinguishing point. Meanwhile, the effectiveness of the distinguishing point is

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experimentally confirmed. Moreover, the discharge rates of the curved hopper and the conical

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hopper under the same conditions were compared. Finally, the mathematical model between the

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contraction rate and Froude number was established. Understanding of the flow mechanism is useful to the design, scale-up and optimization of the silo with a curved hopper and similar structural devices.

Keywords: Curved hopper; Discrete element method; Contraction rate; Flow characteristics; Froude number

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1. Introduction Silos are extensively used in the storage of granular materials in mining, agriculture, chemical, food and other industries [1, 2, 3]. Despite the structural simplicity of silos, the granular system inside is complex. Gravity-drivenin discharge flow in silos demonstrates many complex flow phenomena which contain jamming flow [4, 5], transformation of the flow pattern [6, 7] and

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fluctuation of dynamic flow characteristics [8, 9], etc. However, these phenomena can cause some

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serious engineering problems, such as arching [10, 11], the inhomogeneity of granular flows [12,

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13] and so on. In fact, one of the main objectives of the discharge process is to obtain a constant

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flow without jamming [14]. Therefore, in order to ensure the reliable outflow of particles during

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the discharge process, scholars have proposed the silo with a curved hopper [15, 16], which can effectively prevent jamming. Nevertheless, the flow mechanism of the silo with a curved hopper is

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unclear. It is vital to have a thorough understanding of the flow mechanism which can be used to

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design, scale-up and optimize the silo with a curved hopper, and be benefit to improve the

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comprehension of the granular flows theory. In terms of the perception of the discharging process, prior experimental research and theoretical analyses of the curved hopper can typically be classified as macroscopic approaches observing the jamming of granular materials and microscopic approaches studying the change in wall resistance to granular materials during the discharging process [15, 16, 17, 18, 19, 20]. Lee [15], who is the first person to propound the curved hopper with a constant contraction rate, performed a force analysis on the granular materials at the hopper wall, and pointed out that the particles suffer less resistance when flowing in the curved hopper. In addition, Zhang et al. [19] also conducted a similar theoretical study and optimized the curved hopper to make the granular materials flow

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more easily in the hopper. Their research played an essential role in avoiding the formation of arches within the hopper and was fundamental to the successful design of curved hopper. Whereas, these research usually only consider the interaction between the granular materials and the wall. Actually, due to the solid-like and liquid-like characteristics of the granular system [21], the interaction between granular materials, which including collision, friction and sliding etc., has a

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significant influence on the formation of arches during the discharge process. For instance,

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Ahmadi et al. [22] indicates that arching is generated from interactions among contacting granular

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materials. Terzaghi [23] described the arching effect as the load transfer between the granular

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materials. Obviously, the interaction between granular materials is worthy of our attention. On the

not be in line with the actual situation.

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other hand, if only considering the interaction between granular materials and the wall , this will

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In order to verify the actual use of the curved hopper, Wang et al. [20] proved that the curved

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hopper has better flowability and indicated that no jamming occurred during the experiment.

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However, most of these studies seek knowledge to optimize the contraction rate by relating the observed process outcomes with input parameters. These studies have been difficult to clarify the relationship between the contraction rate and granular flows due to they have lacked the understanding of the flow mechanism of the curved hopper. As a matter of fact, because of the granular system has complex time and spatial variations [24], understanding the flow mechanism is the critical step to have an insightful grasp of granular motion. Unfortunately, traditional experimental method is difficult to explore complex motion information of granular materials simply. Compared with traditional experimental methods, numerical simulations have been proven to be

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an effective tool for obtaining microscopic information of granules in the studying of granular systems [25, 26, 27]. Discrete element method (DEM) proposed by Cundall and Strack [28] can provide more detailed information on the granular flow on microscope level, such as the velocity, stresses and torque [29, 30] etc. In particular, advances have been made in the research into granular motion during the processes of granular flows [31, 32, 33]. These research indirectly

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proved that DEM is an available method to understand the flow behavior of particles. Moreover,

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to the best of the authors' knowledge, there is currently no studies of numerical simulation in the

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literature that would understand the flow characteristics of granular materials in the curved hopper.

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Based on the above analyses, in this study, the DEM simulation approach is used to investigate the

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discharging flow of rice particles in silos with curved hoppers, which aim to clarify the effect of the contraction rate on the granular flow. For this purpose, the flow characteristics of rice particles

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will be analyzed in terms of velocity, porosity and discharge rate fluctuations during

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discharging. Then, we found that the contraction rate has a distinguishing point. Meanwhile, the

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effectiveness of the distinguishing point is experimentally verified. Moreover, the discharge rates of the curved hopper and the conical hopper under the same conditions were compared. The result shows that the performance advantage of the curved hopper is greater as the height of the hopper increases. Finally, the Froude number was used to quantitatively evaluate the flowability of the particles, and a prediction model for the flowability of the silo with the curved hopper was proposed. The results obtained enabled understanding of the flow mechanism of the silo with the curved hopper and provide guidelines to the design. 2. Theory 2.1. The contraction rate In order to clarify the flow mechanism of the silo with the curved hopper, the design concept of a

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curved silo must be known. The advantage of a curved silo over a conical silo is that it has a constant contraction rate. First introducing the definition of the contraction rate, it defined as the ratio of area reduction of a certain section of the hopper per unit distance of descent with respect to the original area. The formula for calculating the contraction rate is as follows(see Fig. 1):

C

dA 2πxdx 2  2    ln  x  Ady πx 2dy y  d 

(1)

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Where C is the contraction rate of the hopper section; A is the cross-sectional area; dA is the differential of the cross-sectional area; dy is the small variables in the direction of height.

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It can be known that different vertical-sections have different contraction rates. For a hyperbolic

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section, the equation of the section curve is as follows (C0 is a constant):  C0  y 2 

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d  x  e 2

(2)

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To introduce Eq. (2) into Eq. (1), the contraction rate C is listed as follow:

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 C0  2  2  2  2 d  2 y   C  ln  x   ln e  C0  y d  y d 2  

(3)

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For a conical section, the equation of the section curve is as follows (k is a constant):

1 x  (y  b) k

(4)

To introduce Eq. (4) into Eq. (1), the contraction rate C is listed as follow:

C

2  2  2  2 y  b  ln  x   ln   y  d  y  dk 

(5)

It can be seen that the contraction rates of the curved hopper and the conical hopper are constant and logarithmic functions, respectively. This indicates that the contraction rate of the conical hopper gradually increases as the height of the hopper decreases, where as the contraction rate of the curved hopper is constant. Actually, the contraction rate has an important influence on the formation of the arch during the silo discharge process [15]. The particles overcome the resistance

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and flow downward during the discharge process, which are constantly rearranged to accommodate changes in cross-sectional contraction rate during this time. Here we analyze the individual particle as the research object. Fig. 2 shows the force acted on the individual particle in different hoppers. The formula for the normal force is as follows:

Fn  G cos

(6)

Where Fn is the normal force; G is the gravity;  represents the angle between the

oo

f

tangential force ( Fr ) and the horizontal direction. It is clear that the  gradually increase with

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the decreases of the height in the curved hopper. While in the conical hopper, the  is constant.

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In other words, considering only the motion of the individual particle, it can be seen from Eq (6)

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that the Fn of the particle in the conical hopper is constant. However, the Fn of the particles decreases as the height of the hopper declines in the curved hopper. Therefore, the Fn in the

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lower part of the curved hopper is smaller than that of the conical hopper when the G of the

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particles is the same. As we all know, a large Fn will lead to greater horizontal compression

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between the particles. The particles will stop moving and form arches when the horizontal compression between the particles is too large. Therefore, smaller normal forces will reduce the probability of jamming. On the other hand, we will find that the Fn is larger in the upper part of the curved hopper, but arches are generally formed near the outlet, the large cross-sectional area of the upper part do not cause arching. On this basis, the hopper section is designed to have a constant contraction rate to avoid jamming. 2.2 The Froude number Research suggest that it is possible to use the Froude number Fr as a measure of flowability as their relationships were also established [34]. In this paper, to quantitatively characterize the flowability of particles in the silo with the curved hopper, the Froude number is used. And if its

Journal Pre-proof value is small, the flowability of the discharge is low. The advantage of introducing the Froude number is that it can eliminate the effect of opening size on flowability, and considering only the effect of the contraction rate on flowability. The Froude number is defined as the ratio of the inertia to gravitational forces.

Fr 

v gL

(7)

analyses of the above formula, Lehmann [35] derived a non-squared Froude number for axially symmetric containers as

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M ρs  g  DA2.5

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Fr 

(8)

 s is the density of the particles, g is the acceleration of

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where M is the mean discharge rate,

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gravity, DA is the diameter of the discharge port. In this study, Eq (8) is used to calculate Fr.

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3. Materials and Methods

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3.1. Simulated system description 3.1.1. Mechanical contact model

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To simulate the discharge of rice particles in a curved silo, a three-dimensional DEM was used

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based on the soft-sphere model. Dry rice particles were selected to establish the model which is free from cohesive force and liquid bridge. In the model, the translational and rotational motion of each particle in a system is described by Newton's law of motion, given by

n dv i m i  m g    F  F d  F  F d  i dt i n n t t  j  1

(9)

n dw i  i T  T i dt t r j 1

(10)

I





Where vi and wi are the translation and angular velocities respectively of particle i with d mass mi and moment of inertia Ii , ni is the number of particle j in collision with particle i. Fn

is the normal damping force and Fn is the normal contact force.Similarly, Ftd is the tangential damping force and Ft is the tangential contact force. The torque, in turn, includes two terms,

Journal Pre-proof arising from the tangential force, Tt , and the rolling friction, Tr . In DEM simulations, the no-slip Hertz-Mindlin contact model was chosen to calculate the forces and torques of each rice particle in the present study. The equations used to calculate the normal total force, the tangential total force, the tangential torque and the rolling friction torque can be found elsewhere [36, 37], and are not repeated here for brevity. 3.1.2. DEM model of particle and geometry In the simulation, a rice particle is usually simplified as the axis-symmetric ellipsoid particle

oo

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model [38]. As shown in Fig. 3, the model of a rice particle is designed as an ellipsoid with a 7 mm long axis and a 2.8 mm short axis. The properties of rice particles are the same as those used

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in previous work [8].

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The schematic of the silo with the curved hopper is presented in Fig. 4. The silo is essentially two

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parts that a cylinder section and a curved hopper, and the vertical-section of a curved hopper as shown in Fig. 5. In simulations, the physical properties of the rice particle and the silo are reported

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in Table 1, which are the same as those used in the literature [39]. Note that the coordinate origin

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of the z-axis.

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is placed at the bottom of the hopper and the direction of particle gravity is the positive direction

In a conical hopper, the height of the hopper changes with the change of the half angle when the diameter of the silo and the outlet size are constant, this means that each half angle corresponds to a different hopper height. The value of the half angle is usually taken every 5 degrees [6], the conical hopper height at different half angles was calculated with constant silo diameter and outlet size. In this paper, we bring the parameters obtained in the conical hopper into Eq (6) to obtain different contraction rates, such as silo diameter, outlet size, and hopper height. In other words, the side wall of the conical hopper is replaced by a curved surface without changing the other structure in a conical silo, such as silo diameter, hopper height and outlet size. The value of

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contraction rates C are shown in Table 2. The simulations started with random generation of 15000 rice particles at a distance of 50 mm from the top of the silo. And then, at 0s, rice particles were allowed to fall onto the bottom of the silo under gravity. When all the particles were located in their stable position, the outlet at the bottom opened and all the rice particles flowed out under gravity. Note that the particles begins to

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of each particle,was recorded within the 0.01 s time interval.

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flow out at 3.3s. During the discharge, the information, such as position, velocity and orientation

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In the discharging process, to observe and analyze the movement information of particles in a silo

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conveniently, a silo slice (i.e., the width of the slice is in the range of −5 mm < X < 5 mm) was

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studied. All movement information used in this work was obtained within the silo slice. 3.2. Experimental setup and procedure

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The components of the experimental setup are shown in Fig. 5. Table 2 shows the specifications of

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the seven special curved silos that were used in the experiment. The silos used for experiment and

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simulation are exactly same. The detailed structure of the silos is shown in Fig. 5(a). In conducting the experiment, the outlet was blocked and filled with 15,000 grains of rice. Then, the outlet is opened and the stopwatch is used to record the time required for discharge. The measurements are repeated three times per replicate. The goal of the experiments is to indirectly verify the effectiveness of the distinguishing point of the contraction rate by comparing the discharge rate. 4. Results and discussion 4.1. Model validation A curved silo model was used firstly to examine the accuracy of the simulation. The consistency of the experiment and simulation was examined by observing the change of the top particle. The

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schematic of the experimental equipment is shown in Fig. 5. Note that only one curved silo which with a contraction rate of 0.092 is selected here for verification. Moreover, three different colored particles are filled from the silo axis to the silo wall, respectively(see Fig. 6). Fig. 7 shows the snapshots obtained from both experiment and simulation at equivalent time during the discharge. It can be observed that the area of the annular colored ribbon in the

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discharging experiment is basically consistent with the simulation. Additionally, the annular

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ribbon is gradually gathered toward the axis of the silo as the discharge continues. The accuracy of

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DEM is confirmed by comparing the results of simulation with those experiment. 4.2. Macroscopic flow patterns

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In order to observe the flow of the particles during discharge, the previously mentioned silo slice

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was selected. Fig. 8 shows the snapshots obtained at two moments, one is the 3.3s of the simulation and the other is the 5s of the simulation. Moreover, data is recorded when the discharge

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tend to reach steady state, here we select the data during 4-5s in which is a stable condition. Four

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layers of 10mm thick layers of dyed granules are selected (see Fig. 8). Moreover, start recording

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when the discharge reaches a steady state, the subsequent data is obtained within 4s-5s. Four layers of 10mm thick layers of dyed granules are selected (see Fig. 8). The particles form the “dead zones” near the junction of the hopper and the cylinder during the discharge process, the position of the “dead zones” is similar to the stagnant zones position of the conical silo [40]. It can be seen that the red particles staying at the bottom gradually increase and the stagnation zones is gradually expanding with the increase of the contraction rate. Besides, the difference in flow between the upper layer of particles(purple particles) at different contraction rates is most pronounced. The purple particles layer is relatively flat when the contraction rate is small, but as the contraction rate increases, the purple particles layer gradually appears v-shaped. This is

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because the flat particle layer is in the mass flow and all particles move down at a relatively equal velocity. The particle velocity at the center is obviously greater than the particle velocity near the wall in the v-shaped particle layer, this is consistent with the funnel flow characteristics. Actually, there is a transition from mass flow to funnel flow during the discharge flow for silos [6]. Therefore, it can be seen from Fig. 8 that the transition height is lower when the contraction rate is

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small, as the contraction rate increases, the height of the transition also increases. Because mass

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flow presents the advantages of stable material flow, no agglomeration and no segregation, the

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particles have better flowability in a relatively small contraction rate of the curved hopper. 4.3. Flow characteristics

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Based on the above analyses, it is known that the change of the contraction rate has a certain

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influence on the flowability. However, observing from macroscopic flow patterns still has limitations. Therefore, in this research, we tried to explain the difference in flow characteristics

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discharge fluctuation.

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based on the statistical distribution of microscale variables, such as particle velocity, porosity and

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4.3.1. The effect of the contraction rate on particle velocity. To observe and analyze the flow characteristics of particles in curved silos, the silos are divided into two parts: the hopper section and the cylinder section. The height of the hopper is different due to different contraction rates, so all hoppers are divided into three layers. Furthermore, in order to ensure that the observation window contains at least one vertically falling rice grain, a 7mm high observation window is selected for each layer (see Fig. 9(a)). Moreover, in order to observe the flow pattern transition in the cylinder section, the axial range of 0 mm ≤ Z ≤ 70 mm within the silo slice is uniformly divided into a series of cuboid units. The cuboid units are at the center of the silo and length, width and height are 20mm, 10mm, 10mm, respectively (see Fig.

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9(b)). We now introduce the well-known velocity scaling, to attempt to confirm the difference in flow characteristics at different contraction rates. In this study, we define the average vertical velocity of all particles in the observation window as the velocity scale V. Therefore, we rescale our velocities such that

Ve V

(11)

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f

Vn 

Where Ve is the (unscaled) vertical velocity component of each particle in the observation

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window, and Vn is the normalised (scaled) components.

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Fig. 10 illustraties the probability density distribution of Vn with different contraction rates in the

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hopper. At the bottom of the hopper (see Fig. 10(a)), it can be observed that the peak is gradually decreasing as the contraction rate increases and the width of the peak gradually widens. Note that

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higher peaks and narrower widths indicate more consistent motion of the particles. Therefore, Fig.

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10(a) indicates that the consistency of particles motion decreases as the contraction rate increases.

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Due to the particles in the hopper are in the funnel flow, the velocity difference between the central particles and side ones is greatest. The worse consistency of the motion of particles indicates a large difference in velocity between the center and the wall. Thus, the difference between the velocity of the particles at the center and the velocity of the particles at the wall increases with the increase in the contraction rate. Interestingly, when the contraction rate increases to 0.110, the probability density distribution of Vn appeares two peaks. The reason is that the number of particles in the “dead zones” increases as the contraction rate increases during the discharge process. The left peak corresponds to the particles in the “dead zones”, while the right peak is related to the particles in the central region. The “dead zones” particles gradually become

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more and more as the hopper height and the contraction rate increases as shown in Fig. 10(b) and Fig. 10(c), which is consistent with what we have observed visually (Fig. 8). Thus, this means that the flowability of the particles decreases as the contraction rate increases, and the possibility of jamming the silo increases. Interestingly, when the contraction rate increases to 0.110, we can see that the left peak suddenly rise in Fig 10(c), this indicates that the number of “dead zones”

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particles is much higher than before. Moreover, the peak value is gradually reduced before the

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contraction rate is less than 0.110, where as when the contraction rate is greater than 0.110, the

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peak hardly changes.

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In order to investigate the effect of contraction rate on flow pattern transform, the cylinder section

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will be studied. Fig. 11 shows the vertical mean velocity of particles in the center as a function of the cylinder height at different contraction rates. This shows that velocity changing with the

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contraction rates can be divided into two stages, one stage is that the vertical mean velocities vary

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significantly, showing the velocity of the particles at the center increases as the height of the

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cylinder decreases, which conforms to the characteristics of funnel flow. Another stage is that the difference in vertical mean velocity of the particles is small, the velocity hardly changes with the decrease in height, which conforms to the characteristics of mass flow. The transition region between these two stages is called the transition region of the flow pattern [6]. Fig. 11 shows that the particle velocity decreases as the height of the cylinder increases, the particle velocity no longer changes significantly when a certain height is reached. This indicates that the particles are already in the mass flow. Besides, the height of the transition of the flow pattern gradually increases as the contraction rate increases. The characteristics of the flow pattern transition during discharge are consistent with the phenomenon that is observed as shown Fig. 8. However, when

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the contraction rate increases to 0.110, the two fitted lines almost coincide, which indicates that the height of the flow pattern transition is almost no longer changing. This has the same contraction rate as the observed mutation point in the hopper section. 4.3.2. The effect of the contraction rate on the spatial distribution of porosity Porosity has been widely used to describe the flow structure of particles in silos. To clarify the flow structure of the particles and explore the effect of contraction rate on particles flow, the

is calculated as follows: Vu  N  Vd Vu

pr

P

oo

f

previously designed observation windows are selected (see Fig. 9(b)), and the porosity in each unit

(12)

e-

where P, N are the porosity and average number of particles in each unit respectively when the simulation was carried out between 4s and 5s . Vu is the volume of each unit. Vd is the volume of

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each particle. Note that if the center of the particle is within the unit then the particle is considered to be inside the unit, vice versa.

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The vertical distribution of porosity as a function of the silo height in the central region is shown

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in Fig. 12. It can be seen that the porosity increases with the decrease in the silo height, this is

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because the particles in the lower part of the silo are in the funnel flow. The larger porosity leads to weaker contact between particles which indicates that the resistance of particles is small. Conversely, particles near the wall are subject to greater resistance under the action of the wall. The particles experience a difference in velocity between the wall and the center ,which gradually increases with different resistance, and then lead to the formation of a funnel flow. In addition, there is a transition zone (the yellow zone) for changes in porosity at all contraction rates, which indicates that the flow pattern begins to transform. Furthermore, it can be seen that the height of the transition zone increases as the contraction rate increases. However, when the contraction rate increases to 0.110, the height of the transition zone is almost no longer changing. This indicates that the transition height, which the flow pattern transforms from the mass flow to the funnel flow, no longer changes significantly after the contraction rate increases to 0.110, which is consistent with the previous analyses.

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4.3.3. The effect of the contraction rate on the discharge rate fluctuations The discharge rate of the particles at each moment during the discharge process is not constant, one of the reasons is that dynamic arches are generated during the discharge process, and there is the possibility of jamming [41]. When the discharge rate fluctuates greatly, the flowability of the particles is weak during the discharge process, and there is a high probability of jamming. Actually, there are two forms that are often used to describe the discharge rate. One is the mean discharge rate (W), which covers the entire discharge process, and other is the instantaneous discharge rate

f

(Wt), which represents that at any particular instant. Fig. 13(a) shows the mean discharge rate at

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different contraction rates. As shown Fig. 13(a), it is clearly that the mean discharge rate decreases as the contraction rate increases which indicates that the smaller contraction rate is

pr

beneficial to granular flows. Interestingly, the mean discharge rate almost never changes when the

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contraction rate increases to 0.110. Besides, it is well known that the discharge rate is not

Pr

constant during the discharge process, select one of the contraction rates (C=0.092) to display here (see Fig. 13(b) ). The blue line in Fig. 13(b) shows the change in Wt over the simulation period of C=0.092, and the red line represents the mean discharge rate W. Note that the

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different Wt values are calculated using an averaging time of Δt=0.1 s. It can be seen that

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the Wt oscillates around the mean W, the reason for the fluctuation is the formation and collapse

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of the arch during the discharge process [41]. Unfortunately, the magnitude of the oscillation cannot be known from the figure. Therefore, the standard deviation of discharge rate, which reflects the degree of fluctuation, is used. For example, the larger standard deviation means a more violent fluctuation. In this work, the first difference method is used to obtain the standard deviation, here the mean velocity and standard deviation are obtained by: σ

t

u  u' 2

j 1

t 1



(13)

Where u and u' are the discharge rate at time step j and the average discharge rate from 4 s to 5 s, respectively; t is the total time step; σ is the standard deviation of the discharge rate. The variation of the standard deviation of the discharge rate as a function of the contraction rate is shown in Fig. 14. It is clearly observed that when the contraction rate is less than 0.110, the

Journal Pre-proof standard deviation of the fluctuation floats at a lower value. However, when the contraction rate increases to 0.110, the value of the fluctuation deviation suddenly increases. Furthermore, the standard deviation of the fluctuation is still at a higher value as the contraction rate continues to increase. This may be due to the increased number of particles in the stagnation zone, resulting in the more dramatic formation and collapse of dynamic arches. When the contraction rate increases to 0.110, the number of particles in the stagnant zone will no longer change significantly. Indeed, the stagnant zones are helpful for the formation of arches of particles that find the way of stable, at least for a short time, slowing down the flow at the outlet [42]. The change in the stagnation zones

oo

f

is consistent with the previously observed phenomenon.

Based on all the above researches, it is known that the flowability of particles in the curved hopper

pr

is affected by the contraction rate. Moreover, the analyses of the velocity, porosity and fluctuation

e-

amplitude show that the flowability of the particles no longer changes significantly when the contraction rate is increased to 0.110. So here we define a distinguishing point, the concept of

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distinguishing point is used to distinguish the influence of contraction rate on liquidity. When the contraction rate is less than the distinguishing point, there is a significant influence on the

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flowability. When the contraction rate is greater than the distinguishing point, the flowability of

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the particles no longer changes significantly.

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4.4. Quantitative indicators of flow characteristics By observing the flow characteristics of the particles during the discharge process, the flowability of the particles is significantly different as the contraction rate of the curved hopper changes. Furthermore, in order to quantitatively evaluate the flow characteristics of the particles, the Froude number Fr can be used as a dimensionless characteristic. The power function relationship between the Froude number Fr and the contraction rate (see Fig. 15). Fig. 15(a) shows that values of the Froude number when C=0.042 to 0.132. It indicates the Froude number Fr decreases as the contraction rate increases. The value of Fr can represent the flowability of the particles, a larger Fr represents a better flow of the particles. Actually, the relationship between the Froude number and the contraction rate can be explained by previous conclusions. For example, the number of particles in the stagnant zone increases and the height of the flow pattern transition rises.

Journal Pre-proof Furthermore, it can be seen here that the curve gradually becomes flat with the contraction rate increases. Nevertheless, the trend which is gradually flattening is not obvious. In order to verify this trend, three models which with contraction rates of 0.160, 0.198, and 0.254 were selected for simulation. Note that these contraction rates corresponds to hopper half angles of 60°, 65°, 70° , respectively. The reason for choosing these values is to take a value every 5° as described previously. One the other hand, it is only to verify the flattening tendency after the contraction rate is increased, so only the portion where the contraction rate continues to increase is taken. Fig. 15(b) shows that values of the Froude number when C=0.042 to 0.254. It can be seen that the trend of

oo

f

the curve does gradually tend to level. In other words, after the contraction rate increases to a certain extent, which will no longer have a significant effect on improving flowability. However,

pr

the relationship of Fr - C is a constantly changing smooth curve, it is difficult to infer from the

e-

curve of the Fr-C that the contraction rate is increased to what value does not have a significant influence on the granular flows. But what is certain is that there is indeed a distinguishing point

Pr

that can distinguish the effect of the contraction rate on flowability. Indeed, a distinguishing point was found for the contraction rate by analyzing the particle velocity, porosity and discharge

rn

0.110.

al

fluctuation, and it is known that the distinguishing point of the contraction rate is approximately

To predict the flowability of the silo with the curved hopper under different contraction rates, a

Jo u

correlation is formulated based on the present simulation results. It can be seen that the Froude number consistent with the power function distribution, as given below:

y  axb  c

(14)

Here a, b and c are three coefficients. Based on the present simulations, they are 0.02413, -0.7351 and 0.0567. The R2 value is 0.9967 for the present model, which represents the accuracy of the proposed correlation. This predictive model can be used to guide the design of curved silos. 4.5. Comparison of performance between curved hopper and conical hopper In order to clarify that the advantage of curved hopper compared to the conical hopper under the same conditions, such as diameter of hopper outlet, height of hopper and diameter of the cylinder, five sets of comparative experiments were conducted. In the experiment, the outlet size and the

Journal Pre-proof

diameter of the cylinder are the same as previously selected. Fig. 16(a) shows that the mean discharge rate of the curved hopper and the conical hopper at different hopper heights, it is found that the discharge rate of the curved hopper and the conical hopper is increasing as the height of the hopper increases. This indicates that the particles have better flowability as the half angle increases and the contraction rate decreases. In addition, Fig. 16(b) shows that the difference in the

f

mean discharge rate of the hopper under the same conditions. It can be clearly seen that the

oo

difference in discharge rate gradually increases with the increase of the height of the hopper. This

pr

indicates that the performance advantage of the curved hopper is greater as the height of the

e-

hopper increases when the other conditions are the same.

Pr

4.6. Discussions

In above simulations, the distinguishing point of the contraction rate is presented. A definite fact is

al

that the contraction rate which is greater than the distinguishing point does not have a significant

rn

effect on the flow characteristics of the particles. In actual discharge, the mean discharge rate and

Jo u

flow characteristics have similar changes [43]. Therefore, it can be thought that there is no significant difference in the discharge rate when the contraction rate is greater than the distinguishing point. So, although it is difficult to experimentally verify the distinguishing point by measuring the flow characteristics of particles like simulations, but the related experiments that reflect the critical of the contraction rate are carried out. The purpose of present experiments is to indirectly verify the range of the distinguishing point of the contraction rate by investigating whether the mean discharge rate no longer changes significantly. The results of the mean discharge rate by using different contraction rates are listed in Table 3. Note that W is experimentally measured data and W s is the data obtained by simulation. As

Journal Pre-proof

expected, the mean discharge rate decreases as the contraction rate increases in the experiment, which is in agreement with the results in the simulation. Nevertheless, there is a slight difference between the simulation and experimental results. This may be due to the rice particle model used in the simulation is assumed to be mono-size, cohesionless and without liquid bridge, which are different from the properties of real rice particle. Moreover, it can be clearly observed that the W

f

shows significant difference (p < .05) at different contraction rates. Furthermore, the W decreases

oo

as the contraction rate increases. These results might due to the effects of the number of particles

pr

in the dead zone and the height of the flow pattern transition. More importantly, the W shows no

e-

significant difference when contraction rates are greater than 0.110. This means that the change in

Pr

contraction rate no longer has a significant effect on the W when the contraction rate is greater than 0.110. Overall, the results of the tests in this paper indicated that there is a distinguishing

al

point for the effect of the contraction rate on granular flows, the distinguishing point of

rn

contraction rate in our study is around 0.110, and this experiment verifie the location of the

5. Conclusions

Jo u

distinguishing point from a quantitative perspective.

DEM simulations were used to investigate the flow characteristics during discharge of rice grains in a curved silo. The flow characteristics is analyzed qualitatively and quantitatively. Based on the presented results the following conclusions can be drawn: • According to theoretical analysis, the resistance of the particles in the curved hopper near the outlet is smaller than that of the conical hopper. In addition, it has been found through experiments that the curve hopper has a faster discharge rate than the conical hopper under the same outlet size, hopper height and silo diameter. The performance advantage of the curved hopper is greater as the height of the hopper increases when the other conditions are the same. • During the discharge process in the silo with the curved hopper, there are significant changes in the flow characteristics as the contraction rate changes. By analyzing the particle

Journal Pre-proof velocity, porosity and discharge fluctuations in the DEM simulations, it is known that there is a distinguishing point of the contraction rate. When the contraction rate is greater than the distinguishing point, the flowability of the particles no longer changes significantly. In addition, the location of the distinguishing point is proved quantitatively by experiment. • The flowability was quantitatively evaluated by the Froude number. In addition, the flowability of curved silos with different contraction rates can be estimated using the method of the power function fitting. On the whole, the flowability of the silo with the curved hopper

f

becomes better as the contraction rate decreases.

oo

Acknowledgements

pr

This work was financially supported by the Chinese Natural Science Foundation (11802057),

for

Scientific

and

Technological

Innovation

Talents

of

Harbin,

China

Pr

Project

e-

Natural Science Foundation of Heilongjiang Province of China (LC2018010), Special Fund

(2017RAQXJ073)and the “Young Talents” Project of Northeast Agricultural University, China

al

(18QC21).

rn

Reference

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[14] R.O.Uñac, A.M.Vidales, O.A.Benegas, I.Ippolito, Experimental study of discharge rate

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[25] C. Gonzálezmontellano, J.M. Fuentes, E. Ayugatéllez, F. Ayuga, Determination of the

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[29] X.Y. Meng, F.G. Jia, H.L. Qiu, Y.L. Han, Y. Zeng, Y.W. Xiao, P.Y. Chen, DEM study of white rice separation in an indented cylinder separator, Powder Technol. 348 (2019) 1-12. [30] P.Y. Chen, F.G. Jia, H.R. Liu, Y.L. H, Y. Zeng, X.Y. Meng, Y.W. Xiao, G.R. Li, Effects of feeding direction on the hulling of paddy grain in a rubber roll huller, Biosystems Engineering. 183 (2019) 196-208. [31] M. Umer, M.S. Siraj, DEM studies of polydisperse wet granular flows, Powder Technol. 328 (2018) 309-317. [32] Y.Y. Zhou, Z.M. Shi, Q.Z. Zhang, W.R. Liu, M. Peng, C.Z. Wu, 3D DEM investigation on the morphology and structure of landslide dams formed by dry granular flows, Engineering Geology. 258 (2019) 105151.

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[36] S.D. Liu, Z.Y. Zhou, R.P. Zou, D. Pinson, A.B. Yu, Flow characteristics and discharge rate of ellipsoidal particles in a flat bottom hopper, Powder Technol. 253 (2014) 70-79.

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[37] Z.Y. Zhou, D. Pinson, R.P. Zou, A.B. Yu, Discrete particle 400 simulation of gas fluidization

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[41] R.O. Unac, A.M. Vidales, O.A. Benegas, I. Ippolito, Experimental study of discharge rate fluctuations in a silo with different geometries, Powder Technol. 225 (2012) 214-220. [42] Y. Li, N. Gui, X. Yang, J. Tu, S. Jiang, Effect of wall structure on pebble stagnation behavior in pebble bed reactor, Ann. Nucl. Energy. 80 (2015) 195-20. [43] S.D.Liu, Z.Y.Zhou, R.P.Zou, D.Pinson, A.B.Yu, Flow characteristics and discharge rate of ellipsoidal particles in a flat bottom hopper, Powder Technol. 253 (2014) 70-79.

Journal Pre-proof Table Captions: Table 1 Physical parameters and their values in simulations. Table 2 List of simulation tests. Table 3 The statistical analyses of the experimental results.

Jo u

rn

al

Pr

e-

pr

oo

f

Figure Captions: Fig. 1 The vertical-section of a curved hopper. Fig. 2 The force analyses diagram of the particles in different hoppers which have the same outlet size, height and diameter: (a) A curved hopper; (b) A conical hopper. Fig. 3 Model of a rice particle in the present DEM simulation: (a) 2D model of a rice particle; (b) 3D model of a rice particle. Fig. 4 Geometry of a curved silo used in the present DEM simulation: (a) 3D model of a curved silo; (b) 2D model of a curved silo. Fig. 5 Experimental setup of silos. Fig. 6 Three different colored particles are filled the silo with the curved hopper (C=0.092). Fig. 7 Comparison of flow patterns between experiments and simulations at different heights: (a) h= 150mm; (b) h= 125mm; (c) h= 100mm; (d) h= 80mm; (e) h=70mm. Fig .8 The flow patterns at two different moments in the simulation: (a) t=3.3s; (b) t=5s. Fig .9 Observation windows of the silo slice: (a) The hopper section; (b) The cylinder section. Fig. 10 Probability density distribution of Vn in the hopper: (a) Lower layer; (b) Middle layer; (c) Upper layer. Fig. 11 The vertical mean velocity in the center with cylinder height. Fig. 12 The porosity in the center of the silo with different contraction rates: (a) C= 0.043; (b) C= 0.053; (c) C= 0.065; (d) C= 0.078; (e) C= 0.092; (f) C= 0.110; (g) C= 0.132. Fig. 13 The discharge rates : (a) Mean discharge rates provided by different contraction rates; (b) Instantaneous and mean discharge rates provided by C=0.092. Fig. 14 Standard deviation of the discharge rate fluctuations to the mean discharge rate at different contraction rates. Fig. 15 The Froude number with different contraction rates: (a) C=0.043 to 0.132; (b) C=0.043 to 0.254. Fig. 16 The mean discharge rate of different hoppers : (a) The mean discharge rate of the curved hopper and the conical hopper at different hopper heights; (b) The difference in the mean discharge rate of the hopper under the same conditions.

Journal Pre-proof Table 1 Physical parameters and their values in simulations. Parameters

Value

Rice particle

Density, ρR (kg/m3) Poisson ratio, νR Shear modulus, GR (Pa) Density, ρS (kg/m3) Poisson ratio, νS Shear modulus, GS (Pa) Hopper half angle, α (°) Outlet size, D0 (mm) Silo diameter, D (mm) Restitution coefficient, eRR Coefficient of static friction, μs,RR Coefficient of rolling friction, μr,RR Restitution coefficient, eRS Coefficient of static friction, μs,RS Coefficient of rolling friction, μr,RS Time step, Δt (s)

1350 0.25 3.75 × 108 7800 0.29 7.5 × 1010 30–60 22–38 80–110 0.6 0.3 0.01 0.5 0.5 0.02 1.62 × 10−6

Silo

pr

ePr al rn

Simulation

Jo u

Particle-silo

oo

Particle-particle

f

Type

Journal Pre-proof Table 2 List of simulation tests. Silo diameter(mm)

Outlet size(mm)

contraction rate C

Hopper half angle(°)

1 2 3 4 5 6 7

80 80 80 80 80 80 80

20 20 20 20 20 20 20

0.043 0.053 0.065 0.078 0.092 0.110 0.132

25 30 35 40 45 50 55

Jo u

rn

al

Pr

e-

pr

oo

f

Test

Journal Pre-proof Table 3 The statistical analyses of the experimental results. C

W* (g/s)

Ws (g/s)

a

0.065 0.078 0.092 0.110 0.132

61.82 ±1.005 55.27b±0.553 50.36c±0.514 43.36d±0.143 41.61d±0.891

57.29 52.59 47.07 41.58 39.25

Jo u

rn

al

Pr

e-

pr

oo

f

“*”denotes passing of the p < .05 significance test; means of the W with the same letter are not significantly different in different discharge rates.

Journal Pre-proof

Highlights •The silo with a curved hopper was simulated using discrete element method. •Effects of the contraction rate on flow characteristics was analyzed. •The distinguishing point of the contraction rate was determined.

Jo u

rn

al

Pr

e-

pr

oo

f

•Close, almost power relation between Froude number and contraction rate.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16