Determination of the energetic coefficient of restitution of maize grain based on laboratory experiments and DEM simulations

Journal Pre-proof Determination of the energetic coefficient of restitution of maize grain based on laboratory experiments and DEM simulations

Lijun Wang, Zhaohui Zheng, Yongtao Yu, Tianhua Liu, Zhiheng Zhang PII:

S0032-5910(19)31129-5

DOI:

https://doi.org/10.1016/j.powtec.2019.12.024

Reference:

PTEC 15031

To appear in:

Powder Technology

Received date:

8 July 2019

Revised date:

18 November 2019

Accepted date:

12 December 2019

Please cite this article as: L. Wang, Z. Zheng, Y. Yu, et al., Determination of the energetic coefficient of restitution of maize grain based on laboratory experiments and DEM simulations, Powder Technology(2019), https://doi.org/10.1016/j.powtec.2019.12.024

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© 2019 Published by Elsevier.

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Determination of the energetic coefficient of restitution of maize grain based on laboratory experiments and DEM simulations Lijun Wang*, Zhaohui Zheng, Yongtao Yu, Tianhua Liu, Zhiheng Zhang (College of Engineering, Northeast Agricultural University, Harbin, Heilongjiang 150030, China)

*Corresponding author: Lijun Wang. Tel.: 86-451-55191897. E-mail address:

oo

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[email protected].

Abstract:

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The coefficient of restitution (COR) is one of the important mechanical properties of particle.

e-

The calculation model of the energetic COR of particle-particle collisions was derived. The

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method used to determine the energetic COR of irregular particles was proposed. Maize particle-particle collisions were simulated using the discrete element method (DEM). The kinetic

rn

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energy of the particles during the collis ion was investigated. The relationship between the input and calculated energetic COR was obtained via simulation. The energetic COR of maize grain was

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obtained and verified using laboratory experiments. The results indicate that it is accurate. The results of the simulation indicate that the energetic COR is more accurate than the kinematic COR for maize grain. The effects of the impact velocity and impact angle on the energetic COR were investigated. The results of the paper will be helpful for maize grain simulation and maize-processing machine design.

Keywords: Maize grain, Energetic COR, Particle-particle collision, DEM simulation, Laboratory experiment

1

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1.Introduction The DEM has been increasingly applied to study granular materials. In various industries such as pharmaceutical, food, agricultural production, and chemical processing, the DEM plays an important role [1-4]. To simulate the real process of the particle moving, it is necessary to accurately input their material and mechanical parameters. The COR is one of the important

f

mechanical parameters required to simulate the kinematics and kinetics of particles using DEM. It

oo

characterizes the energy loss during collis ion [5]. The COR has many definitions that are divided

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into three categories: kinematic, kinetic, and energetic definitions [6]. To date, the COR has

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mainly been obtained using the bulk calibration approach and experimental measurements.

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Some scholars obtained the mechanical parameters of particles using the bulk calibration approach. Hu [7] considered the angle of repose and bulk density of expanded graphite particles as

al

macroscopic responses and the particle density, sliding frictions, COR, and Poisson's ratio as

rn

microscopic variables. The optimization combinations of the microscopic variables were

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determined. Ghodki [8] obtained the best set of the coefficient of static friction, coefficient of rolling friction, and COR by comparing the angle of repose of soybeans in the simulations and experiments. Ma [9] used the coefficient of rolling friction, COR, JKR surface energy parameter, and coefficient of static friction as microscopic variables and the soil repose angle as the response value. The regression model of the soil repose angle was established and the regression model was optimized using the soil repose angles obtained via physical experiments. The optimal solution of the contact model parameters of clayey black soil particles was obtained. Based on these studies, it is customary to choose the appropriate combination of mechanical parameters that can describe the bulk response of the solids using the bulk calibration approach. Although this method can 2

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obtain the corresponding COR, the physical meaning of the COR might be difficult to understand. The kinematic COR is widely studied because the velocity of the particle before and after collision can be directly obtained using experimental measurements. Some scholars derived various calculation models of the COR based on the velocity to obtain the COR of the particle under different collision conditions. Jafarpur [10] simplified the kinematic COR to the function

f

derived based on the ball height before and after the impact and experimentally determined the

oo

COR of different sizes of balls using a drop test. Crüger [11] derived the calculation function of

pr

the COR based on the relative velocities of colliding partners and experimentally measured the

e-

COR of glass particles with two different diameters at varying relative velocities and liquid layer

Pr

thicknesses. Wang [12] ascertained the calculation function of the COR based on the grain velocities in the direction of the contact force of the particle-particle collision and experimentally

al

determined the COR of frozen maize partic le-particle collision. However, studies shown that the

rn

kinematic COR that is applied to oblique collis ion, with dry friction collision, and irregular

Jo u

particle collision can violate the law of conservation of energy [13-15]. The kinetic and energetic COR were primarily studied theoretically as it is challenging to obtain work done by the contact force and impulse during collision using experimental measurements. Stronge [13,16] found that for planar oblique collis ion with friction, the kinematic and kinetic definitions of the COR yield results that are energetically inconsistent at some initial velocities when sliding stops or changes direction before separation. If the initial sliding does not stop before separation or centric collision occurs between rough bodies, the kinematic, kinetic, and energetic definitions of the COR are equivalent. Djerassi [17] investigated collis ion with friction and found that the energetic COR yielded energetically consistent results. Pandrea [18] 3

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investigated collision without friction of two rigid bodies and proved the equality of the kinematic, kinetic, and energetic definitions of the COR. Ivanov [19] studied collisions with friction of two rigid bodies and found that the kinematic COR applied to an oblique impact can violate the law of conservation of energy. With the kinetic COR, the calculated energy dissipation is always positive and energetic COR is more realistic. Based on these studies, for collisions without friction or

f

centric collis ions with friction, the kinematic, kinetic, and energetic definitions of the COR are

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equivalent. For eccentric collision with friction, the energetic COR is superior to the kinematic

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and kinetic definitions of the COR.

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A method was proposed to determine the energetic COR of irregular particles. The calculation

Pr

model of the energetic COR of particle-particle collision was derived based on Stronge's impact mechanics model [20] and Newton-Euler dynamic equations [21]. The translational and rotational

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kinetic energy of the particles during collision were investigated. The energetic COR of maize

rn

grain at different impact veloc ities and angles was determined using laboratory experiments and

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simulations. The applicability of the kinematic and energetic COR for maize grain was compared. The effects of the impact velocity and impact angle on the energetic COR were investigated.

2.Theory The classic definition of the energetic COR was defined by Stronge [6]. The square of the energetic COR eE

2

is the negative of the ratio of the elastic strain energy released during

res titution to the internal energy of deformation abs orbed during compress ion [20]. The corresponding equations as follows:

eE   2

Wr Wz ( p f )  Wz ( pc )  Wc Wz ( pc ) 4

(1)

Journal Pre-proof Wz ( p)  

t ( p)

0

p

Fz vcz dt   vcz dp

(2)

0

Where Wr is the elastic strain energy released during restitution (J); Wc is the internal energy of deformation absorbed during compression (J); Wz ( p f ) is the work done by the normal contact force during collis ion (J); Wz ( pc ) is the work done by the normal contact force during compression (J); Fz is the normal contact force (N); v cz is the normal relative velocity between

f

the collision bodies at the contact point (m·s -1); p f is the normal component of impulse when the

oo

contact points finally separate (N · s). pc is the normal component of impulse when the

pr

deformation starts recovery (N·s).

e-

The translational and rotational motion of the particles can be described using

Pr

Newton-Euler dynamic equations. Therefore, the calculation model of the energetic COR of particle-particle collision is derived based on Stronge's impact mechanics model and

al

Newton-Euler dynamic equations.

rn

The particle-particle collis ion is shown in Fig.1. In global coordinate system XYZO , two

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particles collide at contact point. The contact point is defined as CG located at particle 1 and located at particle 2 for convenient analysis. The particles have centroids located at respectively. There is a position vector rG from GG to and CG ' , equal but opposite interaction forces FG and

CG ,

FG ' act

GG

and rG ' from GG ' to

CG '

and GG ' ,

CG ' .

At

CG

on the particles, respectively. The

particles have translational and angular velocities of VG , VG ' , G , and G ' , respectively. Information on particle-particle collis ion in the global coordinate system XYZO such the coordinates of the centroid and contact point, contact force including the normal contact force and tangential contact force, translational velocity, and angular velocity are obtained using the export data function available in EDEM2.6 software (DEM solution Ltd.,Edinburgh,UK). It is necessary 5

Journal Pre-proof to establish a local coordinate system xyzo whose origin is at CG for convenient analysis. The

z axis is parallel to the normal contact force

FGn . The collis ion contact plane is established. The

z axis is perpendicular to the collision contact plane and the origin of the local coordinate system is at the collis ion contact plane. The collision contact plane equation is solved based on the point normal form equation of a plane. A random line that passes through the origin of the local

f

coordinate system and is located at the collision contact plane is defined as the x axis. The y

oo

axis is obtained based on the mathematical idea of the vector product as the x , y , and z axes

pr

are perpendicular to each other. Information on the particle-particle collision is transferred to the

XYZO by coordinate

e-

local coordinate system xyzo from the global coordinate system

transformation. The calculation model of the energetic COR is derived based on the information

Pr

on the particle-particle collision in the local coordinate system xyzo .

al

The translational and rotational motion of the particle can be described using

Jo u

rn

Newton-Euler dynamic equations. The corresponding equations can be written as: dV  M  V  F dt

（3）

d    I  r  F dt

（4）

M

I

Where M is the mass of particle (kg); V is the translational velocity of particle (m·s -1 ); the angular velocity of particle (rad·s -1);

F

 is

is the interaction force ( N ); I is the moment of

intertia of particle (kg·m2 ); r is a position vector from centroid to the contact point (m). From Eqs. (3) and (4), we can obtain Eqs.(5) and (6), respectively.

dV  M -1dP   Vdt

d  I 1r  Fdt  I 1  Idt

（5） （6）

Where dP is the impulse (N·s). From Eqs. (5) and (6), the change values of the translational and angular velocities of particle 1 6

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and particle 2 in a unit of time can be obtained, respectively.

dVi  M -1dPi  i  Vi dt

(7)

dVi '  M '-1 dPi 'i 'Vi ' dt

(8)

1

1

di  I ij rj  Fk dt  I ij i  I iji dt

(9)

di '  I ij '1 rj 'Fk ' dt  I ij '1 i 'I ij 'i ' dt

(10)

(11)

oo

v V r

f

The velocity of the contact point of the particle can be calculated based on Eq. (11).

pr

From Eq.(11), we can obtain the velocities of the contact point of the particle 1 and particle 2,

e-

respectively.

(12)

vi '  Vi ' j 'rk '

(13)

Pr

vi  Vi   j  rk

al

The relative velocity between the particles at the contact point can be calculated based on Eq. (14).

vci  vi - vi '

rn

(14)

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The change value of the relative velocity between the particles at the contact point in a unit of time can be calculated based on Eq. (15).

dv ci  dvi - dvi '

（15）

Combining Eqs (7), (8), (9), (10), (12), (13), (14), and (15), dvci is expressed as:

dvci  mij dpi  Bi dt Where

1

mij  (M -1  M '-1 ) ij   ikm jln （I kl rm rn  I kl '1 rm ' rn '）

7

（16） （17）

Journal Pre-proof Bi   ijk（ -  jVk   j 'Vk '）

 

1

1

1

 

  ijk - rk I ji Ai  I jj A j  I jk Ak  rk ' I ji ' Ai ' I jj ' A j ' I jk ' Ak ' 1

1

1



（18）

Based on the index of B , the indices of A and A' can be determined. A and A' can be calculated based on Eqs. (19) and (20), respectively. Ai   ijk  j ( I kii  I kj j  I kk k )

（19）

Ai '   ijk j ' ( I ki ' i ' I kj '  j ' I kk ' k ' )

（20）

if i  j .

 ijk ,  ikm and  jln

the indices are in cyclic order,

if i  j

are the permutation tensor,

 ijk   ikm   jln  -1

 ijk   ikm   jln  1

if

if the indices are in anticyclic order, and

if the indices are in other orders.

e-

 ijk   ikm   jln  0

 ij  1

f

 ij  0

is the Kronecker delta defined as

pr

and

 ij

oo

Where i  j  k  m  l  n  x, y, z .

Pr

When the relative velocity between the particles at the contact point is generated in the collision contact plane, sliding motion occurs. The sliding velocity is the relative velocity between

al

the particles at the contact point in the collis ion contact plane. v cx and v cy are the sliding

 . The friction force hindering the sliding motion of the particles

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denoted as

rn

velocities in the x and y directions. The angle between v cx and the sliding velocity is will be generated.

The friction force is represented by Coulomb’s law and its direction is opposite the direction of the n sliding motion. The normal contact force FG

is defined as Fz in the local coordinate system

xyzo for convenient analys is. The x and y components of the friction force are defined as

Fsx and Fsy , respectively. Fsx and Fsy can be calculated based on Eqs. (21) and (22), respectively.

Fsx 

 v cx v cx  v cy 2

2

f s Fz

8

Fsx  - f s cos Fz

（21）

Journal Pre-proof  vcy

Fsy 

vcx  vcy 2

Fsy  - f s sin Fz

f s Fz

2

（22）

Where f s is the coefficient of static friction . From Eqs. (21) and (22), the x and y components of the impulse can be expressed as:

dpx  - f s cos dpz

（23）

dp y  - f s sin dp z

（24）

mxz   dpx   Bx   m yz  dp y    By  dt mzz   dpz   Bz 

oo

mxy m yy mzy

（25）

pr

dv cx  mxx dv   m  cy   yx  dv cz   mzx

f

Eq. (16) can be written as:

Combining Eqs. (23), (24), and (25), dv cz can be written as:

e-

dvcz  ( f s cos  mzx  f s sin mzy  mzz )dpz  Bz dt

（26）

dvcz  Cdpz  Bz dt

（27）

al

Pr

 f s cos  mzx  f s sin mzy  mzz is defined as C , Eq.(26) can be simplified as:

rn

v cz nonlinearly changes as pz changes because the angular velocity, translational velocity,

Jo u

position vector, and contact force constantly change during the collision. The calculation of the work done by the normal contact force is simplified based on the mathematical idea of the integral. The collision period is divided into n stages. The stages are defined as m ( 1  m  n ). The start and end times of the m stage are defined as

m  1 and m , respectively. We assume that the

information on the particle-particle collision in the m stage is the same as that of the time m . Therefore, v cz linearly changes as

pz changes in the m stage.

v cz can be obtained as a function of pz : v cz  Cm ( pz  pzm1 )  vczm-1  Bzmt Where t is the interval between

m  1 and m . 9

（28）

Journal Pre-proof The work Wm done by the normal contact force in the m stage can be calculated based on Eq. (29).

Wm  

p zm

p zm1

（29）

v cz ( pz )dpz

Combining Eqs. (28) and (29), Wm can be written as：

0.5v czm  0.5v czm1  0.5Bzm t 2  Bzm v czm1t Cm 2

Wm 

2

（30）

eE can be calculated based on Eq.(31). n

eE 

Wr  Wc

oo

f

The energetic COR

2

W

m c 1 c

m

（31）

Wm

pr

m 1

Pr

e-

Where c is the time that the deformation of the particle starts recovery.

rn

3.1. Contact model

al

3． Numerical simulation

The maize particle-particle collis ion was simulated using a three-dimensional DEM based on

Jo u

the soft-sphere model. In the DEM simulation, the translational and rotational motion of the particle was described by Newton-Euler dynamic equations. The cohesive force and liquid bridge between the particles were ignored as the moisture content of the maize grain was 18.6%. The literatures shown that the Hertz-Mindlin contact model can be applied to model the collision of the particle [22-24]. Thus, the no-slip Hertz-Mindlin contact model was chosen to calculate the contact force and torque of each maize partic le. The normal and tangential contact forces were calculated based on the Hertz’s theory and the Mindlin and Deresiewic z theory, respectively [25-26]. The EDEM2.6 was chosen to simulate the maize particle- particle collision. 5% of the Rayleigh time step was used as the fixed time step that is 9.19×10 -7 s to ensure the 10

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accuracy of the simulation results. The cell size was 2 Rmin and the Rmin was defined as the radius of the smallest sphere that is 1.9 mm. To increase the simulation efficiency, the data sampling interval was 1×10-6 s for the period between the time that the particles moved out of oblique plate and the time that the particles fell after the collis ion and that of the remaining

f

simulation time was set as 1×10-4 s.

oo

3.2. DEM model of particle and geometry

pr

In the DEM simulation, multi-sphere approach was often used to construct the model of

e-

irregular particle [27-29]. The number of overlapping sphere should be adopted as much as

Pr

possible to approximate the actual shape of an irregular particle. However, the excessive amount of the overlapping sphere would cause excessive computational expense and can't significantly

al

improve the accuracy of the simulation results [30]. Therefore, the reasonable number of the

rn

overlapping sphere should be adopted. The wedge maize grain was the object used because its

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quantity accounts for more than 90% of the total maize grain [31]. Thirty wedge maize grains were randomly selected to measure their dimension parameters including the heights, widths of the lower and upper shoulders, and thicknesses of the lower and upper shoulders (see Fig.2). The averages of the dimension parameters were calculated. A geometrical model of the wedge maize grain was attained using SolidWorks2016 (SP02, Dassault Systèmes Americas Corp., United States of America) and imported into EDEM2.6. The model was composed of 42 overlapping spheres to fit its real shape as shown in Fig. 3. The platform for testing the kinematic COR of the particle-particle collision is shown in Fig. 4(a), and the simple geometry of the DEM simulation is presented in Fig. 4(b). The simple 11

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geometry includes two oblique plates, two rotation shafts, and a trestle. a is the distance from the generation position of the particle to the lower end of the oblique plate. b is the width of the trestle. c is the 72 mm distance from the lower end of the oblique plate to the rotation shaft. d is the 12 mm thickness of the oblique plate. β is the angle between the oblique plate and the horizontal surface. The impact velocity of the particle can be adjusted by changing the values of a and b. The

f

impact angle of the particle can be adjusted by changing the value of β. The values of a, b, c, d,

oo

and β are the same as the laboratory experiment settings to ensure the accuracy of the simulation

e-

pr

results.

Pr

3.3. Material properties

The influence of the irregular shape in the simulation was considered.

Studies have shown

al

that the contact parameters such as the coefficient of rolling friction and the COR are affected by

rn

the shape of the particle [32,33]. The kinematic COR of the wedge maize grain at the different

Jo u

impact velocities and angles was experimentally measured and the process is the same as our previous research [12]. When the impact angle varies from 40° to 60° and the impact velocity changes from 0.9 m·s -1 to 1.7 m·s -1, the kinematic COR of the wedge maize grain is shown in Table 1. The coefficient of rolling friction of the wedge maize grain was referenced that we previously determined [34]. The coefficients of static friction between the maize grain and the oblique plate and maize grain were obtained using the inclined plane method, respectively. For the coefficient of static friction between the maize grain and the oblique plate, a maize grain is placed on a galvanized steel plate. For the coefficient of static friction between the maize grains, some maize grains are glued on the plate, and then a maize grain is placed on the maize plate. The 12

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inclination angle of plate is increased slowly. When the maize grain starts to slip, the inclination angle is measured. The tangent value of the inclination angle is the coefficient of static friction [35]. The wedge maize grain density was calculated based on its mass and its volume. The other physical parameters of the wedge maize grain and oblique plate were referenced from the literature [34]. The physical parameters of the wedge maize grain and galvanized steel plate are

f

reported in Table 2. The physical parameters of the wedge maize grain and galvanized steel plate

pr

oo

were input into EDEM2.6 software to simulate the collision between the wedge maize grains.

Pr

4.1. Behaviors of particle-particle collision

e-

4. Result & analysis

The microscopic properties and macroscopic behaviors of the particle-particle collision were

al

investigated based on the DEM simulation results to better understand the mechanism of the

rn

particle-particle collision. The behaviors of the particle-particle collision are described by setting

as an example.

Jo u

the collision between the particles at an impact angle of 60° and their impact velocity at 0.9 m·s -1

4.1.1. Microscopic properties of particle-particle collision The microscopic properties of the particle-particle collis ion such as the interaction force and kinetic energy of the particle during the collision were investigated. The collision between the particles occurred under the effect of gravity. When two particles collide, they exert the same forces on each other that are in the opposite direction [36]. So the interaction force acting on particle 1 and the kinetic energy of particle 1 were investigated. The forces acting on particle 1 are 13

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plotted in Fig.5. The magnitude of the gravity is 0.00303129 N. As the collis ion time increases, the magnitudes of the normal and tangential contact forces and friction force initially increase and then decrease. The maxima of the normal and tangential contact forces and friction force are 0.847827 N, 0.224173 N, and 0.3153916 N, respectively. The gravity is much smaller than the other forces. Thus, the gravity of the particle was ignored [36,37].

f

The interaction force acting on the contact point affects not only particle’s translational motion

oo

but also rotational motion [38]. Thus, the interaction force was transformed into a component

pr

force through the centroid of the particle and a rotational torque acting on the contact point of the

e-

particles as shown Fig.8(a) and Fig.9(a). The deformations and sliding motion of the particles

Pr

occur due to the interaction between the particles and change the interaction force. The interaction force includes the contact force and friction force. The contact force is affected by the particle

al

deformation [39]. The deformation process is reflected by the normal relative velocity between the

rn

particles at the contact point. The deformation period was divided into a compression period from

Jo u

0 to 0.000049 s wherein vcz  0 and the restitution period from 0.000049 to 0.00142 s wherein

vcz  0 based on the normal relative veloc ity (see Fig.6). The friction force is affected by the sliding motion [40]. The direction and speed of the sliding motion are reflected by the sliding velocity. The state of the sliding motion was divided based on the variations in the sliding velocity and the angle

 between it and its component in the x direction (see Fig.7). The state of the

sliding motion is divided into the deceleration slip in a positive direction whose duration is 0 to 0.000035 s wherein

  90 and the sliding velocity decreases, the acceleration slip in a reverse

direction whose duration is 0.000035 to 0.000085 s wherein

  90 and the sliding velocity

increases, and the deceleration slip in a reverse direction whose duration is 0.000085 to 0.000142 s 14

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wherein

  90 and the sliding velocity decreases. The analyses of the work done by the

interaction force and energy transformation during the collis ion were divided into four stages based on the deformation periods and the states of the sliding motion. In stage Ⅰ (0～0. 000035 s), the particles are in a compression period and the state of the sliding motion is deceleration slip in a positive direction. The particle deforms due to the

f

compression and the translational kinetic energy transforms into internal energy of deformation of

oo

the collis ion partner due to the work done by FR (see Fig.8(a)). The collision between the T

( see

pr

particles is eccentric and the particles have no angular velocity. The rotational torque

e-

Fig.9(a)) acting on the contact point is generated. The translational kinetic energy transforms into T

. Thus, the

Pr

the rotational kinetic energy of the particle due to the work done by rotational torque

rotational kinetic energy of the particle increases from 0 J to 2.35×10-6 J and its increase rate is

al

6.714×10-2 J ·s -1 (see Fig.9(b)). The translational kinetic energy of the particle decreases to

rn

1.147×10-4 J from 1.260×10-4 J and its decrease rate is 3.138×10-1 J ·s -1 (see Fig.8(b)).

Jo u

In stage Ⅱ (0.000035～0.000049 s), the particles are in a compression period and the state of the sliding motion is acceleration slip in a reverse direction. The direction of the FRf changes (see Fig.8(a)) as the direction of the sliding motion reverses. The transformation rate between the translational kinetic energy and internal energy of deformation of the collision partner reduces. The work done by FR deforms the particle.

T

increases with the particle deformation. The

transformation rate between the translational kinetic energy and rotational kinetic energy increases. The rotational kinetic energy increases from 2.35×10-6 J to 4.55×10-6 J due to the work done by

T

and its increase rate is 233% that of stage Ⅰ (see Fig.9(b)). The translational kinetic energy decreases to 1.114×10-4 J from 1.147×10-4 J and its decrease rate is only 75.11% that of stage Ⅰ 15

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(see Fig.8(b)). In stage Ⅲ(0.000049～0.000085 s), the particles are in a restitution period and the state of the sliding motion is acceleration slip in a reverse direction. The particle deformation starts to recover. The elastic part of the internal energy of deformation of the collision partner transforms into the translational kinetic energy during the restitution.

T

decreases with the recovery of the particle

f

deformation (see Fig.9(a)). The transformation rate between the translational kinetic energy and

T and

its increase rate is only 75.3% that of stage Ⅱ (see

pr

8.81×10-6 J due to the work done by

oo

rotational kinetic energy decreases. The rotational kinetic energy increases from 4.55×10-6 J to

e-

Fig.9(b)). The translational kinetic energy decreases to 1.08×10-4 J from 1.114×10-4 J and its

Pr

decrease rate is only 40.05% that of stage Ⅱ(see Fig.8(b)).

In stage Ⅳ (0.000085～0.00142 s), the particles are in a restitution period and the state of the

al

sliding motion is deceleration slip in a reverse direction. The energy transformation between the

rn

elastic strain energy and translational kinetic energy continues with the deformation recover. The

Jo u

rotational kinetic energy transforms into the translational kinetic energy when the particles have a large angular velocity [5]. The rotational kinetic energy decreases to 7.68×10-6 J from 8.81×10-6 J (see Fig.9(b)) and the translational kinetic energy increases from 1.08×10-4 J to 1.09×10-4 J (see Fig.8(b)). The mechanical energy transforms into heat due to the frictional slip. The increment of the translational kinetic energy is less than the decrement of the rotational kinetic energy.

4.1.2. Macroscopic behaviors of particle-particle collision Studies have shown that the macroscopic behaviors of particle collision are affected by the microscopic properties during the collisions [5,37]. The macroscopic behaviors of the 16

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particle-particle collis ion such as the translational and angular velocities of the particle were investigated. Fig.10（a）and (b) show the translational and angular veloc ities of the particles before and after the collision, respectively. The particles have only X and Z components of the translational veloc ity before the collision and have X, Y, and Z components of the translational velocity after the collis ion. The X component of the translational velocity decreases and the Z

f

component of the translational velocity slightly decreases. Overall, the translational velocities of

oo

the particles decrease after the collision. The particles have a small angular veloc ity before the

pr

collision. After the collision, the X component of the angular velocity increases slightly, the Y

e-

component of the angular velocity increases most obviously, and the Z component of the angular

Pr

velocity increases significantly. Overall, the angular velocities of the particles increase after the collision. The variation tendenc ies of the translational and angular velocities of the particles are

al

the same as that of the translational and rotational kinetic energies of the particles, respectively.

rn

The aforementioned analys is shows that the translational and angular velocities of the particle

Jo u

change after the collis ion as the kinetic energy is affected by the work done by the interaction force. This also proves that compared with the kinematic COR and kinetic COR, the energetic COR is better able to reflect the nature of the collision. The translational and angular velocities of the particle change after the collision. Therefore, the energetic COR of the maize grain obtained was verified by comparing the translational and angular velocities of the particle after the collision in the simulation and experiment.

4.2. Determination of the energetic COR The energetic COR of irregular particle was determined based on laboratory experiments and 17

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DEM simulations. The method of determination is as follows. The single-factor simulation was used to select the contact parameter of the particle [41,42]. The range of the energetic COR is important. We measured the kinematic COR of the maize grain (see Table 1) and referenced the kinematic COR of the maize grain determined by other scholars [33,43]. The energetic COR initially ranged from 0.75 to 1.25 times the kinematic COR obtained. The golden-section method

f

has the advantages of fine stability, fast convergence speed, and high precision compared with

oo

other single dimensional search method [44]. The initial range of the energetic COR was shrank to

pr

a relatively small range through the golden-section method and comparison of the input and

e-

calculated energetic COR in the silulation. The number of simulation experiments could be

Pr

decreased. Within the minimum range of the energetic COR, the relationship between the input and calculated energetic COR can be obtained through single-factor simulation. The more accurate

al

the parameters input, the more realistic the collis ion behaviors of the particle simulated by DEM

rn

[45]. When the energetic COR calculated is the same as the energetic COR input, the energetic

Jo u

COR can be obtained based on this relationship obtained. The energetic COR obtained was verified by comparing the translational velocities and the Y components of the angular velocities of the particles after the collision in the simulation and experiment.

4.2. 1. Selection of the energetic COR The 0.75 and 1.25 times the kinematic COR at the specific impact velocity and angle are denoted as A and B , respectively. The initial range of the corresponding energetic COR is denoted as A, B. In this range, two golden section points are selected, namely, points a and

b , as shown in Eqs. (32) and (33). 18

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Where   0.618 . a and

a  B   ( B  A)

(32)

b  A   ( B  A)

(33)

b were input into the software to simulate partic le-particle collision,

respectively. The corresponding energetic CORs that are calculated based on the simulation results are F a  and F b  , respectively. if

F (a)  a  F (b)  b

oo

f

Let a  A . Then, the range is [a, B] .if

(34)

A, b .

e-

Let b  B . Then, the range is

(35)

pr

F (a)  a  F (b)  b

Pr

New points are selected in the new range to continue the calculation and comparison. After several loops, the initial range eventually shrinks to a very small range. Within this range, the

al

relationship between the input and calculated energetic COR can be obtained through single-factor

rn

simulation. If the energetic COR calculated is the same as the energetic COR input, the energetic

Jo u

COR can be obtained based on this relationship obtained. To better understand the process of the selection of the energetic COR, the collis ion between the particles when their impact angle is 60° and impact velocity is 0.9 m·s -1 is used. The experimentally determined kinematic COR of the maize grain is 0.438. The initial range of the energetic COR is 0.3285-0.5475. Three iterations are used in the given initial range. The results obtained during the iteration process are shown in Table 3. The energetic COR of the maize grain ranges from 0.443 to 0.495. Within this range, the COR einput input into EDEM2.6 software is a factor, and the energetic COR ecalculated calculated based on the simulation result is a target. A single-factor simulation is conducted, The result is shown in Fig. 11. 19

Journal Pre-proof The data points in Fig.11 are fitted to determine the relationship between einput and the

ecalculated as shown in Eq.(36). ecalculated  11.41einput  12einput  3.598 2

（R2 =0.9805）

(36)

When ecalculated  einput , an energetic COR of 0.474 is obtained. Based on the aforementioned method, the energetic COR of the maize grain at the different

oo

f

impact velocities and angles was obtained as shown in Table 4.

pr

4.2.2. Physical verification experiment

e-

The energetic COR of the maize grain obtained was verified by comparing the translational

Pr

velocities and Y components of the angular velocities of the particles after the collision in the simulation and experiment. The particle-particle collisions at the different impact velocities and

al

angles were simulated based on the energetic COR shown in Table 4. The translational veloc ities

rn

and Y components of the angular velocities of the simulation partic les are obtained using the

Jo u

export data function available in EDEM2.6. Based on the platform for testing the kinematic COR of the particle-particle collision shown in Fig.4(a), the actual maize partic le-particle collis ion experiments were conducted under the same conditions used to simulate the particle-particle collis ion. The particle images were obtained using a high-speed digital video camera (Phantom v9.1, Vis ion Research, America) and processed via Phantom motion analysis software (8.0.606.0-CPhCon:606). At time t 0 , the coordinates of maize grains 1 and 2 were ( X10 , Y10 , Z10 ) and ( X 20 , Y20 , Z 20 ) , respectively. At time t n , the coordinates of maize grains 1 and 2 changed to ( X1n ,Y1n , Z1n ) and ( X 2n , Y2n , Z 2n ) , respectively. 20

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At time tn / 2 , the translational velocity of the particle is the same as the average translational velocity of the particle in the period from times t 0 to t n . The translational velocities of maize grains 1 and 2 at time tn / 2 can be calculated as shown in Eqs. (37) and (38), respectively.

X 1n  X 10 t n - t0

V1Y 

Y1n  Y10 t n - t0

V2 X 

X 2 n  X 20 t n - t0

V2Y 

V1Z 

Y2 n  Y20 tn - t0

Z1n  Z10 t n - t0

V2 Z 

Z 2 n  Z 20 t n - t0

（37）

（38）

f

V1 X 

oo

Due to restrictions in the measurement device and instrument, only the Y component of the

pr

angular velocity was measured. The positions of the maize grains are represented by the dotted lines. The angles between the dotted lines of maize grains 1 and 2 and the horizontal axis were

1n and  2 n respectively (see Fig.12 (b)) at time t n .

Pr

as

10 and  2 0 , respectively (see Fig.12 (a)) at time t0 , while the angles were defined

e-

defined as

al

At the time tn / 2 , the angular veloc ity of the particle is the same as the average angular

rn

velocity of the particle in the period from times t 0 to t n . The Y components of the angular

respectively.

Jo u

velocities of maize grains 1 and 2 at time tn / 2 can be calculated as shown in Eqs. (39) and (40),

1Y 

2Y 

1n  10 t n - t0

 2 n   20 t n - t0

（39）

（40）

The simulation and experiment were repeated 5 times. The translational velocity and Y component of the angular velocity were obtained by averaging all of the data from the experiments and simulations. The standard deviations of the translational veloc ity and Y component of the angular veloc ity were calculated to understand the dispersion of the data. The 21

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average values and standard deviations of the translational velocity and Y component of the angular velocity are shown in Tables 5, 6, 7, and 8, respectively. Fig.13 and 14 are based on Tables 5, 6, 7, and 8. The simulation result is close to that of the experiment. Hence, the determined energetic COR is accurate and the method used to determine the energetic COR of the maize grain is valid. However, there is still a difference between the

f

simulation and experiment. On the one hand, the collis ion positions of the maize grains are

oo

different in the experiment, while they are set exactly in the simulation. On the other hand,

pr

compared with the experiment, the simulation environment is ideal and the maize grain energy

Pr

e-

loss is smaller.

4.2.3. Comparison between the kinematic and energetic COR

al

The validity of the method used to determine the energetic COR of the maize grain was

rn

demonstrated by comparing the translational velocities and Y components of the angular velocities

Jo u

of the particles after the collision in the simulation and experiment. For the maize grain-grain collision, the applicability of the kinematic and energetic COR was compared based on the simulation.

The particle-particle collisions at the different impact angles and impact velocities were simulated based on the kinematic COR shown in Table 1 and the energetic COR shown in Table 4, respectively. The corresponding energetic COR was calculated based on the simulation results. The input and calculated CORs were compared. The simulation experiments of the particle-particle collis ions at the different impact angles and velocities were repeated 5 times. The average values and standard deviations of the energetic COR 22

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obtained were obtained as shown in Table 9. For convenient analysis, the kinematic and energetic COR that were input to simulate particle-particle collis ions are defined as eKinput and eEinput , respectively. The energetic COR eEcalculated

eKcalculated

was calculated based on eKinput and the energetic COR

was calculated based on eEinput .

As shown in Fig.15, the difference between eKinput and

eKcalculated

is large. The difference reaches

f

a maximum of 0.11798 when the impact angle is 40° and the impact velocity is 0.9 m·s -1. The

1.3 m·s -1 . The difference between eEinput and

is small. The difference reaches a maximum

pr

eEcalculated

oo

difference reaches a minimum of 0.0565 when the impact angle is 60° and the impact velocity is

e-

of 0.005 when the impact angle is 60° and the impact velocity is 0.9 m·s -1. The difference reaches

Pr

a minimum of 0.0004 when the impact angle is 60° and the impact velocity is 1.5 m·s -1 . The energetic COR is more accurate than the kinematic COR for maize grain. However, there is still a eEcalculated

because the calculation of the work done by the normal

al

difference between eEinput and

Jo u

rn

contact force during collision was simplified.

4.3. Effects of different factors on the energetic COR The determined energetic COR was verified by the laboratory experiment, which indicates that the determined energetic COR is accurate. The maize particle-particle collisions at the different impact velocities and angles were simulated based on the determined energetic COR of the maize grain shown in Table 4. The effects of the impact angles and velocities on the energetic COR were investigated based on the simulation results. Fig.16 is based on Table 4. Fig.16（a）and (b) show the effects of the impact angle and velocity on the energetic COR of the maize grain, respectively. When the impact velocity is fixed at 0.9 23

Journal Pre-proof m·s -1 , the energetic COR increases as the impact angle increases. The contact force decreases as the impact angle increases as shown in Fig.17(a). The energy loss decreases and the energetic COR increases because the particle deformation decreases as the contact force decreases. When the impact angle is fixed at 60°, the energetic COR decreases as the impact velocity increases. The contact force increases as the impact velocity increases as shown in Fig.17(b). The energy loss

f

increases and the energetic COR decreases because the particle deformation increases as the

pr

oo

contact force increases.

e-

5. Discussion

Pr

The calculation of the work done by the normal contact force during the collision was simplified based on the mathematical idea of the integral as the information on the particle-particle

al

collision constantly changes. The number of section points that divide the collision period should

rn

be adopted as much as possible to approximate the real value of the work. However, an excessive

Jo u

number of section points causes unreasonable computational expense. The interval between the two section points was 1×10-6 s. The energetic COR determined was verified by comparing the results in the simulation and experiment. The results of the simulation and experiment are close. Only the energetic COR of the wedge maize grain was investigated as its quantity accounts for more than 90% of the total maize grain [31]. Only the effects of the impact velocities and angles on the energetic COR were investigated. The effect of the collision position of the maize grain will be investigated in the future. The energetic COR was compared with the kinematic COR that we previously measured [12]. The energetic COR and kinematic COR have similar trends with the variations in the impact 24

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velocities and impact angles as the kinematic and energetic definitions of the COR can reflect the energy loss during collisions. The value of the kinematic COR compared with the energetic COR is smaller as the kinematic COR does not separate the energy dissipation due to deformation from that due to friction. The result that the energetic COR is more accurate than the kinematic COR for

f

maize grain is consistent with the findings of Stronge [13].

oo

6.Conclusions

pr

The calculation model of the energetic COR of particle-particle collis ion was derived based on

e-

Stronge's impact mechanics model and Newton-Euler dynamic equations. -1

Pr

The collision period of the maize particle-particle when their impact velocity was 0.9 m·s

and their impact angle was 60° was divided into four stages: the deceleration slip in a positive

al

direction and acceleration slip in a reverse direction during the compression period and the

rn

acceleration slip and deceleration slip in a reverse direction during the restitution period.

Jo u

The translational and rotational kinetic energies of the particle are affected by the work done by the interaction force. Thus, the translational and angular velocities of the particle change. The energetic COR derived based on the work done by the normal contact force reflects the nature of the collision. When the impact velocity and impact angle are 0.9 m ·s -1 and 60°, respectively, the energetic COR of the maize grain was 0.474. When the impact velocity is fixed at 0.9 m·s -1 and the impact angle increases from 40° to 60°, the energetic COR of the maize grain increases from 0.334 to -1

0.474. When the impact angle is fixed at 60° and the impact velocity increases from 0.9 m·s to 1.7 m·s -1 , the energetic COR of the maize grain decreases from 0.474 to 0.304. 25

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The simulation result is consistent with the result of the experiment, which demonstrates the validity of the method used to determine the energetic COR of irregular particles based on laboratory experiments and simulations. The energetic COR is more accurate than the kinematic COR for maize grain.

f

Acknowledgments

oo

This work was supported financially by the Chinese Natural Science Foundation (51475090),

pr

the Natural Science Foundation of Heilongjiang Province, China (E2017004), New Century

e-

Excellent Talents of General Universities of Heilongjiang Province, China (1254-NCET-003) and

Pr

the Science Backbone Project of the Northeast Agricultural University, China.

al

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[44] Z. Chu, Z.Y. Wu, L.P. Chen, J.W. Ding, Fast MARS based on golden section method, J. Syst. Simul. 24 (2018) 1561-1566.

https://doi.org/10.16182/j.cnki.joss.2012.08.001.

[45] J. Horabik, A. Sochan, M. Beczek, R. Mazur, M. Ryżak, P. Parafiniuk, R. Kobyłka, A. Bieganowski, Discrete element method simulations and experimental study of interactions in 3D granular bedding during low-velocity impact, Powder Technol. 340 (2018) 52–67.

Jo u

rn

al

Pr

e-

pr

oo

f

https://doi.org/10.1016/j.powtec.2018.09.004.

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Figure legends Fig.1. Particle-particle collision. Fig.2. Dimension parameters of wedge maize grain. H is the height, W is the width of the lower shoulder, w is the width of the upper shoulder, T is the thickness of the lower shoulder, and t is the thickness of the upper shoulder.

f

Fig.3. Simulation model of the wedge maize grain.

oo

Fig.4. (a) Platform for testing the kinematic COR of the particle-particle collision. (b) Simple

pr

geometry of the platform presented in the DEM simulation.

e-

n Fig.5. The interaction force FG acting on the particle 1, the normal contact force FG , the

Pr

g t f tangential contact force FG , the gravity FG , and the friction force FG .

Fig.6. Variations in the normal relative velocity v cz with the collision time.

 between the sliding velocity and the sliding velocity in the

al

Fig. 7. (a) Variation in the angle

rn

x direction with the collision time. (b) Variations in the sliding velocity with the collision time.

Jo u

Fig.8. (a) The component forces FR passing through the centroid of the particle of the interaction force, the component force of the normal contact force contact force

FRt ,

FRn ,

the component force of the tangential

and the component force of the friction force FRf . (b) Translational kinetic

energy Ek . Fig.9. (a) Rotational torque T acting on the contact point generated by the interaction force, the X component of the rotational torque TX , the Y component of the rotational torque TY , and the Z component of rotational torque TZ . (b) Rotational kinetic energy Erk .

33

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Fig.10. Translational and angular velocities of the particle before and after the collision: (a) translational velocity and (b) angular velocity. Fig.11. The effect of the input value of the COR einput on the energetic COR ecalculated calculated based on simulation result when the impact angle is 60° and the impact velocity is 0.9 m·s -1 .

f

Fig. 12. Images of the rotating maize grains ( front view): (a) t0 and (b) t n .

oo

Fig.13. Numerical and experimental comparison of the translational velocity and Y component of

pr

the angular velocity at the different impact angles when the impact velocity is 0.9 m·s -1 : (a) X

e-

component of the translational velocity V X , (b) Y component of the translational veloc ity VY , (c)

Pr

Z component of the translational velocity VZ , and (d) Y component of the angular velocity ωY . Fig.14. Numerical and experimental comparison of the translational velocity and Y component of

al

the angular velocity at the different impact velocities when the impact angle is 60°: (a) X

rn

component of the translational velocity V X , (b) Y component of the translational veloc ity VY , (c)

Jo u

Z component of the translational velocity VZ , and (d) Y component of the angular velocity ωY . Fig.15. The comparison between the input and calculated COR at the different impact angles and velocities : (a) the impact angle varying from 40° to 60° and the impact velocity is 0.9 m·s -1 and (b) the impact velocity changing from 0.9 m·s -1 to 1.7 m·s -1 and the impact angle is 60°. Fig.16. The effects of different factors on the energetic COR: (a) impact angle and (b) impact velocity. n

Fig.17. The effects of different factors on the normal contact force FG and tangential contact t

force FG ：(a) impact angle and (b) impact velocity.

34

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Jo u

Fig.2.

rn

al

Pr

e-

pr

oo

f

Fig.1.

35

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Fig.3.

(a)

Jo u

rn

al

Pr

e-

pr

oo

f

Fig.4.

(b)

36

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Fig.5.

1.0

FnG FtG FgG FfG

FG /(N)

0.8

0.6

oo

f

0.4

pr

0.2

0 0.2

0.4

0.6 0.8 t /(s)

1

1.2

1.4 1.5 ×10-4

al

Pr

e-

0

rn

Fig.6.

Jo u

v cz /(m·s-1)

0.5

0.0

compression

restitution

-0.5

0

0.2

0.4

0.6

0.8

t /(s)

37

1

1.2

1.4

1.5 ×10-4

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Fig.7.

θ /(°)

300

200

deceleration slip in a positive direction

100

deceleration slip in a reverse direction

acceleration slip in a reverse direction

(a)

v cx

0.10

vcy 0.05

f

v cx vcy /(m·s-1)

0

-0.05

0

0.4

0.2

0.6

0.8

1

1.2

(b)

1.4

1.5 ×10-4

rn

al

Pr

e-

pr

t /(s)

oo

0.00

Jo u

Fig.8.

FR /(N)

1 0.5

FnR FtR FfR

0

Ek /(10-4 J)

(a) 1.4

Ⅰ

Ⅱ

particle 1 particle 2

Ⅳ

Ⅲ

1.3 1.2 (b) 1.1 0

0.2

0.4

0.8

0.6

t /(s)

38

1

1.2

1.4

1.5 ×10-4

Journal Pre-proof

T /(10-3 N·m)

Fig.9.

2 （ a） 0 TX TY TZ

-2

Ⅱ

Ⅰ

particle 1 particle 2

Ⅳ

Ⅲ

f

10 5

0

0.4

0.2

0.6

0.8

1

e-

pr

t /(s)

oo

Erk /(10-6 J)

-4 15

0.9 0.8

0.5

particle 1

particle 2

rn

V /(m·s-1)

0.6

al

0.7

VX VY VZ

Pr

Fig.10.

0.4

Jo u

0.3 0.2 0.1 0.0

Pre-collision

Post-collision

(a)

39

1.2

（ b） 1.4

1.5 ×10-4

Journal Pre-proof 80

ωX 70

ωY ωZ

ω /(rad·s-1)

60

particle 1 50

particle 2

40 30 20 10 0

Pre-collision

Post-collision

e-

pr

oo

f

(b)

Pr

Fig.11. 0.52

al

0.48 0.46

rn

ecalculated

0.50

Jo u

0.44

0.45

0.46

0.47

einput

40

0.48

0.49

0.50

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rn

al

Pr

e-

(a)

pr

oo

f

Fig.12.

(b)

41

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Fig.13. 0.28

Simulated particle 1 Simulated particle 2

0.26

Experimental particle 1 Experimental particle 2

VX /(m·s-1)

0.24 0.22 0.20 0.18 0.16

40

45

50

oo

f

0.14 55

60

Pr

e-

(a)

pr

Impact angle /(°)

0.07

Experimental particle 1 Experimental particle 2

rn

0.04

Simulated particle 1 Simulated particle 2

0.03 0.02

Jo u

VY /(m·s-1)

0.05

al

0.06

0.01 0.00

-0.01

40

45

50

Impact angle /(°)

(b)

42

55

60

VZ /(m·s-1)

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0.85

Simulated particle 1 Simulated particle 2 Experimental particle 1

0.80

Experimental particle 2

0.75

0.65

50

45

55

pr

40

oo

f

0.70

60

e-

Impact angle /(°)

Pr

(c)

105

95

rn

Simulated particle 1 Simulated particle 2 Experimental particle 1 Experimental particle 2

85

Jo u

ωY /(rad·s-1)

90

al

100

80 75 70 65 60

40

45

50

Impact angle /(°)

(d)

43

55

60

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Fig.14.

0.40

Simulated particle 1 Simulated particle 2 Experimental particle 1

0.35

VX /(m·s-1)

Experimental particle 2 0.30

0.25

0.15 0.9

1.1

1.3

oo

f

0.20

1.5

1.7

Pr

e-

(a)

pr

Impact velocity /(m·s-1)

0.07

Simulated particle 1 Simulated particle 2

Experimental particle 1 Experimental particle 2

rn

0.06

al

0.08

0.04

Jo u

VY /(m·s-1)

0.05

0.03 0.02 0.01 0.00

-0.01 0.9

1.1

1.3

Impact velocity /(m·s-1)

(b)

44

1.5

1.7

Journal Pre-proof 1.6

Simulated particle 1 Simulated particle 2

1.5

Experimental particle 1

VZ /(m·s-1)

1.4

Experimental particle 2

1.3 1.2 1.1 1.0 0.9 0.8 0.9

1.7

1.5

1.3

1.1

f

Impact velocity /(m·s-1)

e-

pr

oo

(c)

Simulated particle 1

160

Pr

Simulated particle 2

Experimental particle 1 Experimental particle 2

120

al

ωY /(rad·s-1)

140

Jo u

80

rn

100

60

0.9

1.1

1.3

Impact velocity /(m·s-1)

(d)

45

1.5

1.7

Journal Pre-proof

Fig.15.

0.55

eEinput eKinput eEcalculated eKcalculated

0.50

e

0.45

0.40

0.35

45

50

55

oo

40

f

0.30 60

Pr

e-

(a)

pr

Impact angle /(°)

0.55

rn

0.40 0.35

0.25

Jo u

e

0.45

0.30

eEinput eKinput eEcalculated eKcalculated

al

0.50

0.9

1.1

1.3

1.5 -1

Impact velocity /(m·s )

(b)

46

1.7

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Fig.16.

0.48

eE

0.44 0.40 0.36 0.32 45

50

Impact angle (°)

60

pr

oo

(a)

0.48

e-

0.44

eE

55

f

40

0.40

Pr

0.36 0.32

1.1

1.3

Impact velocity (m·s-1)

(b)

Jo u

rn

al

0.9

47

1.5

1.7

Journal Pre-proof

Fig.17.

F /(N)

1.8 1.6

40°

45°

1.4

55°

60°

50°

1.2

FnG

1 0.8

FtG

0.6

0 0.2

0.4

0.6

1

0.8

1.5

1.4

×10-4

Pr

e-

(a)

1.2

pr

t /(s)

oo

0

f

0.4 0.2

rn

al

2

1.1m·s-1

1.3m·s-1

1.5m·s-1

1.7m·s-1

FnG

Jo u

F /(N)

1.5

0.9m·s-1

1

FtG

0.5 0

0

0.2

0.4

0.6

0.8

t /(s) (b)

48

1

1.2

1.4

1.6 ×10-4

Journal Pre-proof

Tables Table 1 Kinematic COR of the wedge maize grain. Table 2 Physical parameters and their values in simulation. Table 3 Results from different iterations. Table 4 The energetic COR of the maize grain at the different impact velocities and angles.

f

Table 5 Numerical and experimental comparison of the X component of the translational velocity

oo

V X at the different impact angles and velocities.

e-

VY at the different impact angles and velocities.

pr

Table 6 Numerical and experimental comparison of the Y component of the translational velocity

Pr

Table 7 Numerical and experimental comparison of the Z component of the translational velocity

VZ at the different impact angles and velocities.

al

Table 8 Numerical and experimental comparison of the Y component of the angular velocity ωY

rn

at the different impact angles and velocities.

velocities.

Jo u

Table 9 The comparison between the input and calculated COR at the different impact angles and

49

Journal Pre-proof Table 1 Kinematic COR

Impact velocity（m·s -1 ）

40

0.9

Average value 0.304

STDEV 0.032

45

0.9

0.353

0.026

50

0.9

0.401

0.017

55 60

0.9 0.9

0.418 0.438

0.021 0.039

60

1.1

0.404

0.032

60 60

1.3 1.5

0.365 0.308

0.025 0.030

60

1.7

0.257

0.026

Table 2 Parameters

Pr

Type

e-

pr

oo

f

Impact angle（°）

-3

Density, ρp (kg·m )

Galvanized steel

Poisson ratio, νp Shear modulus, Gp (Pa) Density, ρg (kg·m-3 )

al

Maize particle

Jo u

particle— particle

rn

Poisson ratio, νg Shear modulus, Gg (Pa)

particle—galvanized steel

Coefficient of restitution, epp Coefficient of static friction,

Value 1164 0.4 1.37×108 7850 0.28 8.10×1010 0.257～0.438 0.372

μ spp Coefficient of rolling friction, μ rpp

0.0607

Coefficient of restitution, epg Coefficient of static friction,

0.613 0.549

μ spg Coefficient of rolling friction,

0.0311

μ rpg

50

Journal Pre-proof Table 3

Iteration

Input value

Calculated value

The second iteration

0.412 0.463 0.464

0.5334 0.491 0.485

The third iteration

0.495 0.443

0.452 0.518

0.463

0.491

The first iteration

oo

f

Table 4

Impact angle（°）

Impact velocity（m·s -1 ）

40 45

0.9 0.9

50

0.9

55 60

0.9 0.9

60 60

1.1 1.3

0.419 0.397

60 60

1.5 1.7

0.326 0.304

pr e-

0.334 0.403 0.425

al

Pr

0.434 0.474

rn Jo u

Table 5

Energetic COR

Simulation

Experiment

Impact Impact

Particle1

Particle2

Particle1

Particle2

velocity angle（°）

（m· s-1 ）

Average

Average STDEV

value

Average STDEV

value

Average STDEV

value

STDEV value

40

0.9

0.2558

0.00335

0.25981

0.00324

0.25889

0.01195

0.26066

0.00769

45

0.9

0.22404

0.00773

0.22543

0.00726

0.22954

0.00831

0.2301

0.00659

50

0.9

0.19989

0.0043

0.20052

0.00575

0.20637

0.01248

0.20179

0.01209

55

0.9

0.17621

0.00548

0.17871

0.00468

0.17193

0.0103

0.17906

0.01391

60

0.9

0.15553

0.00715

0.15753

0.00271

0.15413

0.01115

0.15573

0.00762

60

1.1

0.17547

0.01007

0.1754

0.00802

0.16623

0.01217

0.16466

0.0082

60

1.3

0.21299

0.0086

0.21299

0.0086

0.19642

0.01357

0.19642

0.01357

60

1.5

0.27146

0.01088

0.26955

0.00852

0.27004

0.01212

0.26991

0.01055

60

1.7

0.34294

0.01354

0.34092

0.01205

0.33327

0.01037

0.34102

0.01976

51

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Table 6

Simulation Impact

Impact

angle

velocity

（°）

（m· s ）

Experiment

Particle1 -1

Particle2

Average

Particle1

Average

Average

STDEV value

Particle2

STDEV value

Average STDEV

value

STDEV value

0.9

0.01753

0.00722

0.01576

0.00974

0.01664

0.01127

0.01749

0.01155

45

0.9

0.00856

0.00575

0.01033

0.00515

0.02749

0.01378

0.00856

0.00575

50

0.9

0.0216

0.01972

0.02183

0.02036

0.01719

0.02267

0.01033

0.00515

55

0.9

0.00991

0.00682

0.01052

0.00604

0.02118

0.01416

0.02749

0.01378

60

0.9

0.02079

0.02105

0.02079

0.02105

0.01907

0.01438

0.02804

0.01623

60

1.1

0.02529

0.01398

0.02954

0.0179

0.01108

0.01207

0.01633

0.01001

60

1.3

0.03189

0.01664

0.03189

0.01664

0.03356

0.02005

0.03356

0.02005

60

1.5

0.02323

0.02006

0.02283

0.02071

0.01823

0.01123

0.01856

0.01089

60

1.7

0.02453

0.01998

0.02453

0.01994

0.01946

0.01537

0.01946

0.01538

e-

pr

oo

f

40

Pr

Table 7

Simulation Impact

angle

velocity

（°）

（m· s ）

Particle1 Average

Average

Average

STDEV value

Particle1

STDEV

value

Particle2 Average

STDEV value

STDEV value

rn

-1

Experiment

Particle2

al

Impact

0.0091

0.65338

0.00654

0.65544

0.01027

0.65206

0.00933

0.00922

0.71969

0.01138

0.73146

0.00752

0.72326

0.01352

0.00918

0.76699

0.01423

0.76472

0.00843

0.77862

0.00839

0.79271

0.00678

0.79333

0.00715

0.7935

0.01119

0.79905

0.01872

0.82758

0.00444

0.83021

0.00828

0.83837

0.01173

0.82683

0.00831

0.97085

0.02802

0.98813

0.03064

0.99928

0.04334

1.01661

0.03585

1.3

1.24811

0.0194

1.24955

0.02094

1.25502

0.01519

1.23149

0.01426

60

1.5

1.3629

0.02072

1.35916

0.0212

1.36544

0.0121

1.35753

0.01305

60

1.7

1.56109

0.00878

1.54339

0.0223

1.55757

0.01138

1.56366

0.00512

0.9

0.65463

45

0.9

0.72646

50

0.9

0.77426

55

0.9

60

0.9

60

1.1

60

Jo u

40

52

Journal Pre-proof Table 8

Simulation Impact

Impact

angle

velocity

（°）

（m· s-1 ）

Experiment

Particle1

Particle2

Average

Particle1

Average

Average

STDEV value

Particle2 Average

STDEV

STDEV

value

STDEV

value

value

0.9

98.19266

2.42444

99.4543

2.11521

95.606

3.31989

96.98

3.60305

45

0.9

92.33102

1.42073

93.09682

1.00399

89.26

2.61209

88.662

2.79682

50

0.9

81.7773

1.01063

80.69374

1.82674

78.312

0.80061

78.36

2.32013

55

0.9

75.8215

1.28341

76.03728

3.11258

74.62

1.2518

73.9

1.95448

60

0.9

67.08734

0.88628

68.30458

1.07859

65.9

2.05426

65.36

1.35388

60

1.1

75.5266

1.38429

76.89626

1.53528

69.26

1.78269

70.66

2.56671

60

1.3

88.98746

0.82203

88.1176

0.99218

85.26

2.28539

83.84

1.48425

60

1.5

126.9478

2.68354

124.6128

4.21623

121.82

2.96766

120.72

2.84377

60

1.7

153.475

3.15357

153.1688

2.75525

154.62

3.205

150.62

6.80419

Pr

e-

pr

oo

f

40

（°）

velocity -1

（m·s ）

45

0.9

0.334

Jo u

40

eEinput

rn

Impact Impact angle

al

Table 9

eEcalculated

eKcalculated

eKinput Average value

STDEV

Average value

STDEV

0.304

0.3306

0.00439

0.42198

0.00581

0.9

0.403

0.353

0.3986

0.00522

0.44006

0.00841

0.9

0.425

0.401

0.427

0.00406

0.45788

0.00579

0.9

0.434

0.418

0.4334

0.00462

0.4774

0.00488

0.9

0.474

0.438

0.469

0.00539

0.52266

0.00605

60

1.1

0.419

0.404

0.421

0.0051

0.50276

0.00558

60

1.3

0.397

0.365

0.3932

0.00789

0.4215

0.00634

60

1.5

0.326

0.308

0.3256

0.00365

0.3887

0.00559

60

1.7

0.304

0.257

0.3076

0.00503

0.3532

0.00466

50 55 60

53

Journal Pre-proof

Jo u

rn

al

Pr

e-

pr

oo

f

Graphical Abstract

54

Journal Pre-proof Highlights · The calculation model of the energetic coefficient of restitution was derived. · The kinetic energy of particle during collision was investigated. · The velocity of particle is affected by the interaction force and kinetic energy. · The energetic coefficient of restitution of maize grain was determined.

Jo u

rn

al

Pr

e-

pr

oo

f

· The collision period was classified based on the sliding velocity and deformation.

55