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i
공학석사 학위논문
Analysis on Jamming Probability of Ball-type SMR Secondary Shutdown System Using
Discrete Element Method (DEM)
개별요소 해석기법을 통한 소형모듈원전의 Ball-type 이차정지계통 막힘 확률 해석
2018 년 8 월
서울대학교 대학원 에너지시스템공학부
박 수 산
ii
Analysis on Jamming Probability of Ball-type SMR Secondary Shutdown System Using
Discrete Element Method (DEM)
개별요소 해석기법을 통한 소형모듈원전의
Ball-type 이차정지계통 막힘 확률 해석
지도교수 김 응 수
이 논문을 공학석사 학위논문으로 제출함
2018 년 7 월
서울대학교 대학원 에너지시스템공학부
박 수 산
박수산의 석사 학위논문을 인준함
2018 년 7 월
위 원 장 조 형 규 ( 인 )
부위원장 김 응 수 (인)
위 원 정 경 재 ( 인 )
i
Abstract
Analysis on Jamming Probability of Ball-type SMR Secondary Shutdown System Using
Discrete Element Method (DEM)
Su-San Park Department of Energy System Engineering The Graduate School Seoul National University
Over the past few years, there have been many attempts and studies on eliminating soluble boric acid in small modular reactors (SMRs). Soluble boron- free concepts can eliminate boric-acid-induced corrosion and simplify the large components related to Chemical and Volume Control System (CVCS). For this reason, some alternative concepts for replacing soluble boron control system have been proposed as the Secondary Shutdown System. Of the concepts, injecting solid neutron absorbers through independent guide tubes has been widely proposed in various studies. In this Ball-type Secondary Shutdown System, jamming can occur during injection of the absorbers, which can lead to a severe accident. However, the studies on jamming of Ball-type Secondary Shutdown
ii
System were insufficient. In particular, jamming is a complex phenomenon influenced by various physical parameters, and it is difficult to repeat experimentally. Therefore, the purpose of this study was determined as follows.
First, performing the physical modeling of Discrete Element Method (DEM) to be used for jamming analysis, and validating this model through experiments.
Second, performing the quantitative analysis of the jamming probability through the validated DEM model.
Discrete Element Method (DEM) is the most commonly used numerical model for describing the mechanical behavior of the granular material. Various force and torque models have been proposed for DEM analysis. Among them, Hertz-Mindlin contact force model is the most commonly used DEM force model. Therefore, it is used in this study to simulate the behavior of the neutron absorbers in the hopper. The jamming probability obtained from experiment and DEM simulation was compared under the same conditions to verify the validity of DEM for jamming analysis. The jamming probability was 3.21 % higher in the simulation than the experiment. Through this, it was confirmed that the Hertz-Mindlin model implemented the jamming phenomena well within the experimental range.
The validated DEM model was used to investigate the effect of various physical parameters (such as particle diameter, hopper outlet diameter, hopper angle, friction coefficient, geometry size, particle density, and the number of particles) on jamming probability. In all simulations, it was confirmed that the jamming probability decreases as the ratio of particle diameter to hopper outlet diameter increases. Especially, the number of particles was found as one of the most important variables in jamming.
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Finally, a jamming analysis method that can be applied when the number of particles is not enough was proposed. An analytic solution to calculate the change of jamming probability according to the number of particles was derived.
The variation of the particle passing probability according to the number of particles was confirmed through this solution. As a result, it was confirmed that as the number of particles in the hopper increases, the particle passing probability decreases. This study is significant in that it could be a basis for a jamming prediction model in Ball-type SSS.
Keywords
Discrete Element Method (DEM), Granular Material, Jamming, Ball-type Secondary Shutdown System, Soluble boron free reactor, Numerical Simulation, Probability and Statistics
Student Number: 2016-21293
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List of Contents
Abstract ... i
List of Contents ... iv
List of Tables ... vi
List of Figures ... vii
Chapter 1. Introduction ... 1
1.1 Background ... 1
1.2 Motivation ... 3
1.3 Objectives ... 4
Chapter 2. Discrete Element Method ... 8
2.1 Granular material ... 8
2.2 Basic concept of DEM ... 9
2.3 DEM contact force model ... 11
2.3.1 Previous studies for contact force models ... 11
2.3.2 Hertz-Mindlin contact force model ... 12
Chapter 3. Validation of DEM Analysis on Jamming ... 18
3.1 Experiment ... 18
3.1.1 Experiment setup ... 18
3.1.2 Experiment condition and results ... 19
3.2 DEM analysis model ... 19
3.2.1 DEM analysis setup and condition ... 19
3.2.2 DEM analysis process and results ... 23
3.3 Validation of DEM analysis model ... 24
Chapter 4. DEM Analysis on Jamming Probability ... 36
4.1 Effect of physical parameters on jamming ... 36
4.1.1 Effect of D/d ... 37
v
4.1.2 Effect of hopper angle ... 38
4.1.3 Effect of friction coefficient ... 39
4.1.4 Effect of geometry size ... 39
4.1.5 Effect of ball density... 40
4.1.6 Effect of number of particles ... 41
4.2 Summary of analysis result ... 41
Chapter 5. Effect of Number of Balls on Jamming Probability ... 52
5.1 Basic idea ... 52
5.2 Existing jamming analysis method ... 54
5.3 New jamming analysis method ... 56
Chapter 6. Conclusion and Recommendations ... 66
6.1 Conclusion... 66
6.2 Recommendations ... 67
References ... 68
Appendix A. ... 74
Appendix B. ... 76
국문 초록 ... 78
감사의 글 ... 80
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List of Tables
Table 1.1 The main pictures of jamming studies in various field ... 5
Table 2.1 DEM force and torque model... 15
Table 2.2 Summary of Hertz-Mindlin contact force model ... 15
Table 3.1 Comparison of properties of SUS 304 and Tungsten ... 25
Table 3.2 Experimental conditions ... 26
Table 3.3 Material properties entered in DEM simulation ... 27
Table 3.4 DEM simulation conditions ... 28
Table 3.5 DEM simulation process ... 28
Table 4.1 Summary of simulation results ... 28
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List of Figures
Figure 1.1 “Ball-3X” safety system of B-Reactor ... 6
Figure 1.2 Typical jamming event in 2-dimensional hopper ... 7
Figure 1.3 Typical jamming event in 3-dimensional hopper ... 7
Figure 2.1 Brazil-nut effect ... 16
Figure 2.2 Janssen effect ... 16
Figure 2.3 Forces acting on particle i from contacting particle ... 17
Figure 2.4 Application of Kelvin-Voight viscoelastic model... 17
Figure 3.1 Conceptual diagram of experimental setup ... 29
Figure 3.2 Jamming experiment results ... 30
Figure 3.3 Conceptual diagram of device ... 31
Figure 3.4 Friction coefficient measuring device ... 31
Figure 3.5 DEM analysis process ... 32
Figure 3.6 Jamming simulation results ... 33
Figure 3.7 Typical jamming event in experiment ... 34
Figure 3.8 Typical jamming event in DEM Simulation ... 34
Figure 3.9 Comparison of experiment and simulation results ... 35
Figure 4.1 Jamming probability as a function of D/d ... 44
Figure 4.2 Overall hopper shape according to angle change ... 45
Figure 4.3 Jamming probability according to hopper angle ... 46
Figure 4.4 Jamming probabiltiy according to friction coefficient ... 47
Figure 4.5 Jamming probabiltiy according to geometry size ... 48
Figure 4.6 Ball passing velocity over time (D=12 mm) ... 49
Figure 4.7 Ball passing velocity over time (D=24 mm) ... 49
Figure 4.8 Jamming probabiltiy according to particle density ... 50
Figure 5.1 Disc jamming probability ... 61
Figure 5.2 Comparison of simulation results and expected values ... 62
Figure 5.3 Comparison of experimental results and expected values ... 62
Figure 5.4 Avalanche size distribution according to s ... 63
viii
Figure 5.5 Avalanche size distribution according to s ... 63
Figure 5.6 Avalanche size distribution according to s in new method... 64
Figure 5.7 Particle passing probability according to the number of particles ... 65
Figure A.1 Typical dilute and dense flow in 3-dimensional hopper ... 74
Figure A.2 Avalanche size distribution according to s ... 75
Figure A.3 Avalanche size distribution according to s in new method ... 75
1
Chapter 1 Introduction
1.1 Background
Over the past few years, there have been many attempts and studies on eliminating soluble boric acid in small modular reactors (SMRs) (Justin Mart et al., 2014; Mohd-Syukri Yahya, 2017). Soluble boron-free concepts can eliminate boric-acid-induced corrosion and simplify the large components related to Chemical and Volume Control System (CVCS). According to the US-NRC guide for Reactivity Control Systems, two completely independent and diverse reactivity control systems are required. In current PWR reactors shutting down the reactor is accomplished in two methods. The first method is the insertion of safety and control rods employing gravitation or other passive methods. The second method is the dissolution of the neutron absorbing boric acid into the primary water. However, for the soluble boron-free SMR, the second method cannot be used. Therefore, some alternative concepts for replacing soluble boron control system should be proposed as the Secondary Shutdown System (SSS) of soluble boron-free SMR.
Of the concepts, injecting solid neutron absorbers through independent guide tubes has been widely proposed in various studies (S. Vanmaercke et al., 2012;
2
R.K. Paschall and A.S. Jackola, 1976; E.R. Specht et al., 1976), and the concept was actually employed in the nuclear power plant such as Hanford N-Reactor and B-Reactor under the name "Ball-3X Safety System"(R. K. Wahlen, 1989) as shown in Fig. 1.1. Moreover, for the liquid-metal and gas-cooled GEN IV reactors (LFR, SFR, and GFR), the reactivity control method through boric acid cannot be used because there are no liquid absorbers that can be dissolved in sufficient quantity in the liquid metal or gas. Even if such an absorber would exist, cleaning the liquid metal after a SCRAM would be expensive. For this reason, the solid neutron absorber injection concept also has been widely proposed and designed as the SSS in GEN IV reactors (Simon Vanmaercke et al., 2012).
In this Ball-type Secondary Shutdown System (Ball-type SSS), neutron absorber balls can clog or jam while trying to flow through a narrow guide tube like Fig. 1.2, 1.3. If the absorbers are jammed, it makes failure to control reactivity, and it can lead to an unwanted increase in fission rate and reactor power. Power increase may damage the reactor core, and in severe cases, even lead to the disruption of the reactor. Therefore, it is essential to manage the jamming phenomenon in Ball-type SSS.
Jamming of discrete objects is a general phenomenon in various fields, and there have been many attempts to interpret it. The objects could be animals (A.
Garcimartin et al., 2015), pedestrians (D. Helbing et al., 2002; D.R. Parisi et al., 2005), insects (S. A. Soria et al., 2012), red blood cells (I. Pivkin et al., 2009), bacteria (E. Altshuler et al., 2013), cars (P. de Gennes, 1999) or grains (A. Ashour et al., 2015) as shown in Table 1.1. Although there are many theoretical, experimental studies in jamming of granular systems, little is known about how the transition from flowing to jamming occurs. With the advance in experimental
3
techniques and computer, studies in laboratory experiments (B. L. Brown and J. C.
Richards, 1960) and computer simulations (D. C. Hong and J. A. McLennan, 1992) showed that jamming is due to an arch formation at the hopper exit. Nevertheless, there is no quantitative description of the arch that leads to jamming. Therefore, the quantitative analysis of jamming in a hopper became the subject of this study.
1.2 Motivation
Although research on jamming is essential in Ball-type Secondary Shutdown System, it has not been done sufficiently. The study of jamming in ball-type SSS is challenging because of three reasons below. First, jamming is a complex phenomenon influenced by various physical parameters. If variables are not strictly controlled, different conclusions are drawn for the same phenomenon. For example, a certain study concludes that there is no variation of jamming probability for a two-dimensional disk within an angle range of less than 60˚ (Kiwing To, 2001), but another study has concluded that jamming decreases steadily as the angle increases (Chih-Yuan Chang et al., 2016).
Therefore, it is necessary to design so that the effect of one factor can be confirmed without other influences. Second, jamming is a statistical phenomenon. The results associated with jamming are expressed as probabilities, so it is necessary to iterate experiments or simulations many times.
However, jamming is difficult to repeat experimentally, and a long time is required for interpretation. Some studies solved this problem by interpreting jamming phenomenon through DEM, which is commonly used in the granular material analysis (M. Tsukahara et al., 2008; P. Parafiniuk et al., 2013). Finally,
4
it is difficult to apply the existing jamming analysis method directly to Ball-type SSS environment. For example, existing jamming analysis method assumes that there are enough particles in the hopper (Iker Zuriguel et al., 2003), but there are not enough absorbers in Ball-type SSS. Therefore, a new method to analyze jamming correctly in Ball-type SSS should be derived.
Based on this motivation, this study attempts to propose an appropriate approach for jamming analysis and interpret the jamming phenomena in Ball-type SSS. Details on this approach can be found in the following chapters.
1.3 Objectives
As mentioned earlier, there are several parameters affect hopper jamming, and many of them need further research. The goal of this work is to confirm the effects of these parameters. To achieve this goal, physical modeling for jamming analysis should be prioritized. Furthermore, an appropriate analytical model for jamming behavior is required to estimate the effect of some important parameters. In this respect, the objectives of this study can be summarized as follows:
Physical modeling for jamming analysis
Validating this model through experiments
Quantifying jamming probability with regard to various physical variables based on validated DEM simulation
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Table 1.1 The main pictures of jamming studies in various field
Objects Pictures Objects Pictures
Animals
A. Garcimartin (2015)
Red blood cells
I. Pivkin (2009)
Pedestrians
D. Helbing (2002)
Bacteria
E. Altshuler (2013)
Pedestrians
D. R. Parisi (2005)
Cars
P. de Gennes (1999)
Insects
S. A. Soria (2012)
Grains
A. Ashour (2016)
6
(a) Neutron absorber balls using in B-Reactor
(b) Upper shape of B-Reactor
Figure 1.1 “Ball-3X” safety system of B-Reactor, The liquid boron was replaced with 29 "ball hoppers" (one at the top of each VSR channel) that contained 3/8- inch to 7/16-inch nickel-plated carbon steel balls. These balls, which also acted to shut down the pile through neutron absorption, could funnel down into the VSR channels in the event of an emergency or a test. The balls could then be removed by a vacuum system. In January 1952, B Reactor became the first to be fitted with the new "Ball-3X" system.
7
Figure 1.2 Typical jamming event in 2-dimensional hopper (Shubha Tewari, 2013)
Figure 1.3 Typical jamming event in 3-dimensional hopper
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Chapter 2
Discrete Element Method
2.1 Granular material
A conglomeration of a large number of discrete solid particles in space is called granular material. Granular material, such as sand piles, grains, and pills are common in daily life. For any of these materials, the particle size is on a comparable scale to the total system size. This similarity of scales leads to the heterogeneous phenomena not observed in other states. A typical example is the
“Brazil-nut effect.” It is a common phenomenon that can be seen in the surroundings. When various grains are mixed, it seems that the grain mixes well on the surface, but when it shakes, big grains (typically Brazilian peanuts) float on to the surface like Fig. 2.1. It seems to be trivial and unremarkable, but it is one of the challenges that has not got clear answers about why this phenomenon has happened. Another unusual phenomenon of granular material is “Janssen effect.” H. A. Janssen (1895) discovered that in a vertical cylinder the pressure measured at the bottom does not depend upon the height of the granular material filling, i.e. that it does not follow the Stevin law which is valid for Newtonian fluids at rest. Fig. 2.2 is one of the experimental investigations of the Janssen effect. Because of these peculiar characteristics, it is not possible to apply
9
existing fluid or solid interpretation methods to granular material.
In principle, each position and velocity of granular particles can be calculated if the external force and the interaction between them are known. If the granular system consists of dry macroscopic particles, their interaction can be approximated by hard core repulsion, and a continuum theory can describe the dynamical behavior of the granular system under certain conditions (C.
Bizon et al., 1999). However, unlike gases and liquids in which collision between two particles is elastic, in granular systems the collision between two macroscopic particles is dissipative. Therefore, some of the kinetic energy is lost at each collision, and the particles will become immobile unless power is supplied. Such complicated inter-particle interactions give granular systems their highly nonlinear dynamical behavior (F.Radjai, 2002). Because of this, properties of granular material are described the using statistical arguments ultimately.
2.2 Basic concept of DEM
Discrete Element Method (DEM) is the most commonly used numerical model for describing the mechanical behavior of granular material flow. Since DEM was firstly proposed by Cundall (1979), it has been used in various industries. DEM is constructed by applying Newton's second law to the system containing the moving particles. At each time step, obtaining all the forces and moments acting on each particle and calculate the displacement to get the new position of each particle. The general Discrete Element Method modeling sequence is as follows:
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(1) Generation of particles + Definition of boundary and initial condition (2) Detection of contacts (between particles and between particles and
boundary)
(3) Calculation of Forces (F) and Moments (M) (4) Calculation of Acceleration and Rotation (5) Calculation of New Positions
(6) Repeat 2 to 5 until the stop criterion is achieved
The process of calculating the forces and moments acting on the particles (sequence (3)) is the core of the DEM analysis. The governing equations for the translational and rotational motion of the particle with mass and moment of inertia can be written as
, i ( n t) ,
i total i ij ij i g
j
m d
dtv
F F F F (2.1)
, i ( , t r)
i total i con i ij ij
j
I d
dtw
τ R F τ (2.2)
where vi and wi are the velocity and angular velocity of the particle i, respectively, Fij and τij are the contact force and torque acting on particle i by particle j or walls, Rcon is the vector from the center of mass to contact point, and τcon is the torque due to friction.
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2.3 DEM Contact force model
2.3.1 Previous studies for DEM contact model
In general, the contact between two particles is not at a single point but on a finite area, because the contact of two rigid bodies is allowed to overlap slightly in the DEM. Then, the contact force over this area can be decomposed into a component in the contact plane (or tangential plane) and one normal to the plane.
Fig. 2.3 schematically shows the conventional forces and torques involved in a DEM simulation. DEM generally uses a simplified force model to determine the forces and torques due to the contact between particles, and there are many approaches have been proposed for this purpose.
The DEM force model is classified into the Hertz theory-based model (Johnson et al. 1971) and the linear spring model (Cundall, 1979; Mishra et al., 2003). Generally, the linear spring model is the simplest model. “Linear spring–
dashpot model” proposed by Cundall and Strack (1979) is the most common linear spring model, where the spring accounts for the elastic deformation and the dashpot is used for the viscous dissipation. More theoretically model, Hertz–
Mindlin and Deresiewicz model, has also been developed. Hertz (1882) proposed a theory to describe the elastic contact between two spheres in the normal direction and he proved that the relationship between the normal force and normal displacement was nonlinear. Mindlin and Deresiewicz (1953) proposed a tangential force model in a similar way. This Hertz-Mindlin and Deresiewicz theory-based model is the most commonly used model in DEM because of its accuracy. Therefore, it is used in this study as well. Table 2.1
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shows the equations for some commonly used force models for spherical particles, including the linear spring–dashpot model and the simplified Hertz–
Mindlin and Deresiewicz model.
2.3.2 Hertz-Mindlin contact force model
When a collision occurs between particle-particle or particle-wall, the particle receives force from the collision point. During the collision, the particles received to translational acceleration in the direction of the force vector.
At the same time, they also received to angular acceleration, which is the driving force of the rotational motion. The forces at the contact point act on the same size and opposite direction for each particle. In this way, restoring force occurs after collisional deformation in the vicinity of the contact point, so that the solid particles have a characteristic of viscoelastic material during a collision.
This characteristic of the solid collision is represented by the Kelvin-Voight viscoelastic model in the Hertz-Mindlin contact force model. In the Kelvin- Voight viscoelastic model, the viscoelasticity of the collision is represented by a nonlinear spring-dashpot. The collision force between particles is separated into normal and tangential components as shown in Fig. 2.3, and a separate spring- dashpot model can be applied to each component like Fig. 2.4 (A. J. Caserta et al., 2016). Therefore, the forces acting on the particles are as follows:
total contact non contact g
F F F F (2.3)
n t
F F Fg
, , , ,
(Fn s Fn d) (Ft s Ft d) Fg
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where Fn s, is the normal contact force, Fn d, is the normal damping force, Ft s, is the tangential contact force, Ft d, is the normal contact force, and Fg is the gravitational force.
In this case, the normal force acting on the i-th particle, Fn is expressed as below:
3
2ˆ ˆ
n n
ij knn nn ijv n
F (2.4) where kn is normal stiffness of nonlinear spring, n is normal component of particle displacement at collision, nˆ is normal unit vector, contact force, n is normal damping constant, and vij is the normal component of the relative velocity between particles.
Similar to the normal force, the tangential force, Ft is expressed by the tangential stiffnesskt, the tangential component of particle displacement t, the tangential unit vector ˆt, the tangential damping constant t, and the tangential component of the relative velocity between particles.
3
2ˆ ˆ
t t
ij ktt t t ijv t
F (2.5) In the tangential direction force, the effect of friction must be considered. If
t
Fij is greater than the maximum static friction force Fijn (: static friction coefficient), slip in the tangential direction occurs at the contact point. Finally, the tangential force Fijt is as follows:
3
2 ˆ
min ,
t t n
ij ktt t ijv ij t
F F (2.6) The stiffness coefficientskn,kt and the damping coefficients n,t are the
14
material property. Misra and Cheung (1999) show the stiffness coefficients as a function of Poisson’s ratio, Young’s modulus, and shear modulus as follows:
4
n 3 ij ij
k E R (2.7)
t 8 ij ij n
k G R (2.8)
1
1 1
ij
i j
R R R
(2.9)
2 1
2 1
1 i j
ij
i j
E E E
(2.10) 2 1
2 i j
ij
i j
G G G
(2.11)
The i,j,Ei,Ej and Gi,Gj are Poisson’s ratio, Young’s modulus, and shear modulus of two colliding particles, and Ri,Rj are their radius. Also, the damping coefficients n,t is determined by restitution coefficient e, damping ratio , and the equivalent mass mij (M. Obermayr et al., 2013).
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n 3 m kij n
(2.12)
10
t 3 m kij t
(2.13)
1
1 1
ij
i j
m m m
(2.14)
2 2
ln( ) ln
e e
(2.15) These correlations were summarized in Table 2.2.
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Table 2.1 DEM force and torque models
Table 2.2 Summary of Hertz-Mindlin contact force model
Force model Normal force Tangential force
Linear spring- dashpot model
( )
n
ij knn cn ij
F n v n n
4 ,
n 3 ij ij
k E R cn2 m kij n
( )
t
ij kttct ij
F v n n
8 ,
t ij ij n
k G R ct2 m kij t
Hertz-Mindlin contact force model
3/2 ( )
n
ij knn n ij
F n v n n
10 ,
n 3 m kij n
2 2
ln( ) ln
e e
min ( )
t n
ij kt t t ij s ij
F v t F t
10 ,
t 3 m kij t
2 2
ln( ) ln
e e
Torque model Rolling friction torque Torque from tangential force
, ,
r
ij rRcon i n s i
τ F w ,
t
ij con i t
τ R F
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Figure 2.1 Brazil-nut effect (Rosato et al., 1987)
Figure 2.2 Janssen effect (Shawn Chester et al., 2009)
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Figure 2.3 Schematic illustration of the forces acting on particle i from contacting particle
Figure 2.4 Application of Kelvin-Voight viscoelastic model to Hertz-Mindlin Contact model (A. J. Caserta et al., 2016))
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Chapter 3
Validation of DEM Analysis on Jamming
3.1 Experiment
3.1.1 Experiment setup
Before interpreting the jamming with DEM, an experiment was designed to validate the DEM analysis model. The conceptual diagrams of the experimental setup are presented in Fig. 3.1. This experiment imitates the situation where a large amount of neutron absorber balls are injected into the guide tube.
A 3-dimensional hopper and guide tube are fabricated with an acrylic base plate for visualization. 3-D Hopper consists of a storage part for storing balls and a connection part connected to a guide tube. Considering the space in the reactor, the diameter of the hopper was designed to be 50 mm, and the diameter of the guide tube (it is same with the diameter of hopper opening) was designed to be D = 12 mm. The connection part of the hopper is 3 mm thick acrylic geometry having an angle of 60˚ with respect to the ground. SUS 304 balls are used to simulate B4C-doped tungsten neutron absorbers, and Table 3.1 shows the physical properties of tungsten and SUS304 in order to consider the effect of
19
materials. In addition, an electrically operated sliding gate was installed between the storage part and connection part of the hopper.
3.1.2 Experiment condition and results
The experiment proceeded in the way of putting monodisperse SUS 304 balls of d =2.5, 3, 3.175, 3.5, 3.572, 4, 4.762, 5 mm in diameter, and the number of balls was changed to N = 500, 1000, 2000. For each condition, the number of jamming events NJ was counted, and the jamming probability J (which is defined as J
t
N
N ) was obtained, where Nt is the number of the entire trial for each condition. These experimental conditions are summarized in Table 3.2.
The experiment was repeated 30 times, and the jamming probability in each condition was derived. These experimental results can be seen in Fig. 3.2. As a result, the jamming probability increases as the number of particles in the hopper increases, and as the D/d decreases. When D / d is large enough, the jamming probability converges to 0, and when D / d is small enough, the jamming probability converges to 1.
3.2 DEM analysis model
3.2.1 DEM analysis setup and condition
The experiment results are compared with those of DEM simulation
20
performed under the same conditions. In this study, the simulation was implemented through EDEM, a proven DEM analysis program according to the following process:
(1) Setting particles and geometry material property (2) Determining the simulation condition
(3) Setting the time step
(4) Proceeding DEM simulation
A. Particles and geometry material property
First, the properties of Acrylic (particle) and SUS 304 (geometry) materials were investigated and entered as follows:
① Acrylic Property
Poisson Ratio(ν): 0.37
Solid Density(ρ): 1180 kg/m3
Shear Modulus(G): 1.7 × 109 Pa
② SUS 304 Property
Poisson Ratio(ν): 0.3
Solid Density(ρ): 8000 kg/m3
Shear Modulus(G): 8.6 × 1010 Pa 8.6 × 106 Pa
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In the DEM analysis method, the interaction properties between materials are as important as the material property. However, the interaction properties related to contact between materials have not been sufficiently studied. In particular, no previous studies were found in which the static friction coefficient between specific materials had been accurately measured. In this study, the static friction coefficient between materials was directly measured by the friction force measuring device as shown in Fig. 3.3, and Fig. 3.4. Therefore, the interaction properties which affect the particle behavior were summarized as follow:
③ SUS 304 – Acrylic Interaction Property
Coefficient of Restitution: 0.7
Coefficient of Static Friction: 0.158
Coefficient of Rolling Friction: 0.04
④ SUS 304-SUS304 Interaction Property
Coefficient of Restitution: 0.8
Coefficient of Static Friction: 0.176
Coefficient of Rolling Friction: 0.01
The material properties applied to the DEM simulation are shown in Table 3.3.
B. Simulation condition
Table. 3.4 show simulation conditions. The hopper shape and condition of the simulation were set as follows including the experiment conditions.
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① Hopper angle: 40˚, 60˚, 80˚
② Diameter of the hopper opening (D): 12 mm
③ Diameter of the ball (d): 2 mm ~ 6 mm
④ Number of balls (N0): 500, 1000, 2000
C. Time step
The process of setting the time step is vital in DEM simulations. If the time step is too large, the particles will pass through each other before interacting with other particles or geometry. On the contrary, if the time stop is too small, the simulation will take a long time to run, and the simulation cannot be performed enough. According to Catherine O'Sullivan (2004), the time step of the DEM simulation is proposed to be 10 ~ 30% of the Rayleigh time step. The Rayleigh time step is determined by the physical properties of the particle as follows:
/
(0.1631 0.8766)
R
r G
t
(3.1)
where, r is the diameter of the smallest particle, is the particle density, G is the shear modulus of the particle, is the Poisson’s ratio of the particles.
The time step was suggested to be about 10-7 sec when deriving Rayleigh time step through real SUS 304 material property. In this case, too long analysis time was required. According to Eqs 3.1, the Rayleigh time step is inversely proportional to the shear modulus of the particles. In previous studies, it has been found that changing the shear modulus does not significantly affect the behavior
23
of the particles unless the particles are deformed under high pressure (R. Wood, 2001). Therefore, the problem of the time step being too short was solved by changing the shear modulus value from 8.6 × 1010 Pa to 8.6 × 106 Pa. In this case, the time step was determined about 10-5 sec.
3.2.2 DEM analysis process and result
DEM simulation was carried out under the same conditions of the experiment.
In each condition, the simulation proceeded in the following order. Between 0 and 1 second, balls are created randomly on top of the hopper using a random number generator. After that, balls are stacked without any operation. This process is done until the all of the ball's motion was stopped through the equilibrium of force. In 1.5 second, the sliding gate, which was blocking the balls, opens, and particles begin to be injected into the guide tube. The sliding gate is opened in 0.1 seconds like the experimental device. The simulation is finished when the injection of the balls is completed. This process can be summarized as follows:
(1) Generating particles at random position (2) Stacking particles in storage
(3) Opening the sliding gate
(4) Injecting particles into the guide tube
Details of the DEM analysis process can be found in Fig 3.5 and Table 3.5.
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The simulation was performed 30 times in the same manner as the experiment. The jamming probability for each condition was derived and expressed as a function of D/d. These simulation results can be seen in Fig. 3.6.
3.3 Validation of DEM analysis model
In this section, the validity of the DEM analysis was evaluated by comparing the DEM analysis results and the experiment results. Fig. 3.7 and Fig. 3.8 is typical jamming events form experiments and DEM simulations. As a result, it could be seen that the jamming occurs through the mechanism of forming arches in both cases. Fig. 3.9 shows the comparison of experiment and simulation results performed under the same conditions. Overall, the simulation obtained by DEM overestimated the actual jamming probability by 3.21%, which was a good match within the error range. Besides, it was confirmed that the difference between simulation and experiment results tends to increase as the D/d increases. The point at which the probability difference between the experiment and the simulation was the maximum was d=3.5 mm diameter, N0=2000 conditions. In this case, the simulation result was 21.933% higher than the actual jamming probability. From the above results, it was confirmed that the jamming phenomenon could be physically reproduced, and quantitatively interpret the jamming probability through the DEM analysis.
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Table 3.1 Comparison of properties of SUS 304 and Tungsten (W)
Properties Tungsten SUS 304
Density 19300 (kg/m3) 8000 (kg/m3)
Heat capacity 132 (J/kg K) 500 (J/kg K)
Thermal conductivity 173 (W/m K) 16.2 (W/m K)
Thermal expansion 4.3×10-6 (m/m K) 17.2×10-6 (m/m K)
Shear modulus 161 GPa 86 GPa
Young’s modulus 411 GPa 190 GPa
Poisson ratio 0.28 0.3
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Table 3.2 Experimental conditions
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Table 3.3 Material properties entered in DEM simulation
(a) SUS 304 property
(b) Acrylic property
(c) SUS 304 – Acrylic interaction property
(d) SUS 304 – SUS 304 interaction property
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Table 3.4 DEM simulation conditions
Table 3.5 DEM simulation process
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Figure 3.1 Conceptual diagram of experimental setup
① Hopper
② Storage part of hopper
③ Connection part of hopper
④ Guide tube
⑤ Sliding gate
① ②
③
④
⑤
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2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
(N=500) experiment data (N=1000) experiment data (N=2000) experiment data
Jammi ng probab il it y
Figure 3.2 Jamming experiment results
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Figure 3.3 Conceptual diagram of device
Figure 3.4 Friction coefficient measuring device
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(a) Generating particles (0 ~ 1 sec) (b) Stacking particles (1 ~ 1.5 sec)
(c) Opening the sliding gate (1.5 ~ 1.6 sec) (d) Injecting particles (1~1.5 sec)
Figure 3.5 DEM analysis process
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2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
(N=500) simulation fitting curve (N=1000) simulation fitting curve (N=2000) simulation fitting curve (N=500) simulation data (N=1000) simulation data (N=2000) simulation data
Jammi ng probab il it y
Figure 3.6 Jamming simulation results
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Figure 3.7 Typical jamming event in experiment
Figure 3.8 Typical jamming event in DEM Simulation
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2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
(N=500) simulation fitting curve (N=1000) simulation fitting curve (N=2000) simulation fitting curve (N=500) experiment data (N=1000) experiment data (N=2000) experiment data
Jammi ng probab il it y
Figure 3.9 Comparison of experiment and simulation results
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Chapter 4
DEM Analysis on Jamming Probability
4.1 Effect of physical parameters on jamming
Various parameters impact the jamming probability from a hopper, and lots of studies have investigated this issue. In this section, the DEM model, which was validated in the previous section, was used to analyze various parameters.
After that, its effect on the jamming probability in the hopper was evaluated.
Before simulations, parameters that are expected to have a major impact on jamming were selected through previous studies. The parameters considered in this study are as follows:
① Ratio of particle diameter to hopper outlet diameter (D/d)
② Hopper angle (Ф)
③ Friction coefficient between wall and particles (μ)
④ Geometry size (D)
⑤ Density (ρ)
⑥ Number of balls in the hopper (N0)
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The analysis results for each physical parameters were as below. Unless otherwise noted, each parameter defaulted to the following values reflecting actual physical quantities:
① Hopper outlet diameter (D) : 12 mm
② Ball diameter (d) : 2~6 mm
③ Friction coefficient between wall and particles (μ) : 0.158
④ Density (ρ) : 8000 kg/m3
⑤ Number of balls in the hopper (N0) : 1000
4.1.1 Effect of D/d
Jamming probabilities have been represented by the function of the hopper and ball diameter in various studies for jamming analysis (Angel Garcimartin et al., 2010; Kiwing To et al., 2001). In this study, jamming probability was arranged as a function of D/d like previous studies. As a result, it could be seen that the jamming probability tends to decrease as the hopper outlet diameter D increases and the ball diameter d decreases as shown in Fig. 4.1. If jamming occurs in the actual Ball-type SSS, it will lead to a severe accident. Therefore, D and d must be determined in the region where the jamming probability converges to zero. In particular, it was confirmed that balls were injected into the guide tube without jamming in the region of D/d> 4.5 regardless of the other
38
conditions. In this study, a minimum D/d value with the jamming probability of less than 1% was found, and it was named as critical D/d. In this case, the effect of each parameter on jamming was confirmed by measuring the change of critical D/d value.
4.1.2 Effect of hopper angle (Ф)
The influence of the hopper angle (Φ) on the granular flow has been confirmed by various studies. A previous study using both experimental and numerical methods for the analyzing stresses of static granular media inside hopper show that the hopper angle has a strong influence on the spatial distribution of maximum shear stress distribution inside the hoppers (Saeed Albaraki et al. 2013). In this study, the variation of jamming probability was observed by changing the hopper angle to 40˚, 60˚, and 80˚ as shown in Fig. 4.2.
As a result, the jamming probability tended to decrease as the hopper angle increases as shown in Fig. 4.3. As the angle changed from 40˚ to 60˚ and 80˚ the critical D/d decreased to 3.967, 3.818, and 3.011. This phenomenon occurs because as the hopper angle increases, the frictional force acting on the particles decreases. Reduced frictional forces make arch formation difficult in the hopper.
From this results, it could be expected that Ball-type SSS operate with higher reliability when the hopper angle is larger. However, as the hopper angle increase, the hopper volume increased as shown in Fig. 4.2. Therefore, the hopper angle should be determined to the maximum within the space allowed in an actual Ball-type SSS.
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4.1.3 Effect of friction coefficient (μ)
The effect of friction coefficient (μ) between the wall and the particles on jamming was confirmed. As a result, the jamming probability increased as the coefficient of friction increased to 0.1, 0.2, and 0.4 as shown in Fig. 4.4. In this case, critical D/d increased to 3.701, 3.947, and 4.227. This phenomenon occurs because as the friction between the particle and the wall decreases, the particles form fewer arches, and it is more difficult to balance forces within the hopper.
From this results, it could be expected that Ball-type SSS operate with higher reliability when the surface treatment is performed to minimize the friction between the ball and the wall.
4.1.4 Effect of geometry size (D)
For the same D/d, the effect of the whole geometry size on jamming was confirmed. As a result, the jamming probability decreased when the geometry size doubles as shown in Fig. 4.5. In this case, critical D/d decreased 3.818 to 3.734. This was because of the average particle passing speed at the hopper outlet increased as shown in Fig. 4.6, 4.7. Beverloo correlation can explain the increase in passing velocity at the hopper exit when the scale increased.
According to Beverloo (1961), the mass flow rate of the ball passing through the 3-dimensional hopper satisfies the relation of Eq.4.1.
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( )2.5
Q g Wkd (4.1)
When both the size of the hopper and ball are doubled, the mass flow rate becomes 25/2 times. However, the passing area becomes 22 times larger, and velocities of the balls through the hopper exit is 21/2 times faster. Jamming occurs when a particle forms an arch through interaction with surrounding particles. If the velocity of the ball passes fast, the probability of exiting before interacting with the surrounding particles increases and the jamming probability decreases. According to this result, the actual Ball-type SSS should be designed to the largest size as long as space is allowed.
4.1.5 Effect of ball density (ρ)
The effect of particle density on jamming was confirmed. As a result, the simulation performed with increasing particle density to 4000 kg/m3, 8000 kg/m3, and 16000 kg/m3 showed that the same jamming probability is obtained within the error range as shown in Fig. 4.8. In this case, critical D/d changed to 3.737, 3.818, and 3.808. This phenomenon occurs because both the frictional force that maintains jamming and the gravitational force that breaks jamming are proportional to the particle density. From this result, it can be confirmed that the material of the neutron absorber can be selected without density restriction in an actual Ball-type SSS.
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4.1.6 Effect of number of particles (N0)
Finally, the effect of the number of particles on jamming was confirmed. As a result, the jamming probability increased as the number of particles increased to 500, 1000, and 2000 as shown in Fig. 4.9. In this case, critical D/d increased to 3.571, 3.818, and 3.860. This phenomenon occurs because as the number of balls increases, the number of attempts to pass through the hopper exit increases, and the jamming probability increases. As in previous studies, if the probability of one ball passing is constant, the probability of all balls passing must increase exponentially with regard to the number of balls. However, the results in Figure 4.9 did not satisfy this trend. The interpretation of this was explained in the next chapter.
4.2 Summary of analysis result
In this chapter, the effect of various physical parameters on jamming was confirmed by comparing the value of critical D/d. The critical D/d value was compared with the standard conditions in the simulations related to each parameter as shown in Table. 4.1. As the hopper angle changed from 40˚ to 80˚, the critical D/d changed by a maximum of 0.807 from the standard condition.
As the friction coefficient changed from 0.1 to 0.4, the critical D/d changed by a maximum of 0.409 from the standard condition. As the geometry size changed from 6 mm to 12 mm, the critical D/d changed by a maximum of 0.084 from the
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standard condition. As the ball density changed from 4000 kg/m3 to 16000 kg/m3, without any tendency, the critical D/d changed by a maximum of 0.081 from the standard condition. As the number of particles changed from 500 to 2000, the critical D/d changed by a maximum of 0.247 from the standard condition. Through the above process, hopper angle, friction coefficient, and the number of particles were found as the important variables in jamming.
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Table 4.1 Summary of simulation results
Property Measurement Critical D/d* Difference of Critical D/d
Hopper angle (Φ)
40˚ 3.967 +0.149
60˚ 3.818 0
80˚ 3.011 -0.807
Friction coefficient (μ)
0.1 3.701 -0.117
0.158 3.818 0
0.2 3.947 +0.129
0.4 4.227 +0.409
Geometry size (D) 6 mm 3.818 0
12 mm 3.734 -0.084
Ball density (ρ)
4000 kg/m3 3.737 -0.081
8000 kg/m3 3.818 0
16000 kg/m3 3.808 -0.01
Number of particles (N0)
500 3.571 -0.247
1000 3.818 0
2000 3.860 +0.042
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2.5 3.0 3.5 4.0 4.5
0.0 0.2 0.4 0.6 0.8
1.0 (N0=1000) fitting curve
(N0=1000) simulation data
D/d
Jammi ng probab il it y
Figure 4.1 Jamming probability as a function of D/d
45
Figure 4.2 Overall hopper shape according to angle change
46
2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
N=1000 (o) N=1000 (o) N=1000 (o)
Jammi ng probab il it y
Figure 4.3 Jamming probability according to hopper angle
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2.0 2.5 3.0 3.5 4.0 4.5
0.0 0.2 0.4 0.6 0.8
1.0 () fitting curve
() fitting curve () fitting curve () simulation data () simulation data () simulation data
D/d
Jammi ng probab il it y
Figure 4.4 Jamming probability according to friction coefficient
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2.0 2.5 3.0 3.5 4.0 4.5
0.0 0.2 0.4 0.6 0.8 1.0
D/d
N=1000 (D = 12 mm) N=1000 (D = 24 mm)
Jammi ng probab il it y
Figure 4.5 Jamming probability according to geometry size
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Figure 4.6 Ball passing velocity over time (D=12 mm) 1
Figure 4.7 Ball passing velocity over time (D=24 mm) 2
1 When D = 12 mm and d = 3.3 mm, the average passing velocity of the ball is 0.5225 (m/sec).
2 When D = 24 mm and d = 6.6 mm, the average passing velocity of the ball is 0.74755 (m/sec).
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2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
(1.6 e+04) fitting curve (8 e+03) fitting curve (4 e+03) fitting curve (1.6 e+04) simulation data (8 e+03) simulation data (4 e+03) simulation data
Jammi ng probab il it y
Figure 4.8 Jamming probability according to particle density
51
2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0
D/d
(N=500) simulation fitting curve (N=1000) simulation fitting curve (N=2000) simulation fitting curve (N=500) simulation data (N=1000) simulation data (N=
