# A DEM-Based Modeling Method and Simulation Parameter Selection for Cyperus esculentus Seeds

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurement and Analysis of the Physical Properties of Cyperus esculentus

## 3. Measurement and Analysis of the Mechanical Properties of Cyperus esculentus

^{7}Pa, 5.55 × 10

^{7}Pa, and 1.46 × 10

^{8}Pa, respectively.

_{0}is the release velocity of No. 1 seed, m/s; v

_{1}is the velocity of No. 2 seed (collided seed) after impact, m/s; v

_{2}is the velocity of No. 1 seed after impact, m/s; H is the release height of No. 1 seed, mm; h

_{1}is the rebound height of No. 2 seed (collided seed), mm; and h

_{2}is the rebound height of No. 1 seed, mm.

_{x}is the static friction coefficient; and θ is the angle between the inclined apparatus and horizontal plane, deg.

## 4. Modeling Method of Cyperus esculentus Seeds

#### 4.1. Particle Modeling

_{1}) with a radius of T/2 was filled at the center (point O

_{1}) of the ellipsoid, as shown in Figure 11a. Second, on the xoz plane, the width W of the ellipsoid was divided into trisections, and two line segments, $\overline{AB}$ and $\overline{CD}$, which were parallel to axis x and intersecting with the ellipsoid outline, were made through each equal diversion point. Then, the parallel line segments $\overline{AB}$ and $\overline{CD}$ were divided into trisections, as shown in Figure 11b. The equal diversion points (point O

_{2}, point O

_{3}, point O

_{4}, point O

_{5}, and point O

_{6}) were the centers of the new filing spheres. Thus, another five spheres that were tangential to the outline of the ellipsoid were filled, defined as sphere O

_{2}, sphere O

_{3}, sphere O

_{4}, sphere O

_{5}, and sphere O

_{6}, as shown in Figure 11c. Finally, the last sphere O

_{7}was filled at the position of the hilum. As a result, seven total spheres were filled. Based on the filing principle, the 7-sphere, 9-sphere, and 11-sphere models of Jinong 1 were successively modeled, as shown in Figure 14a–c.

_{1}, point O

_{2}, point O

_{3}, point O

_{4}, point O

_{5}, and point O

_{6}) were the centers of the filing spheres. Thus, six spheres that were tangential to the outline of the ellipsoid were filled, defined as sphere O

_{1}, sphere O

_{2}, sphere O

_{3}, sphere O

_{4}, sphere O

_{5}, and sphere O

_{6}, as shown in Figure 12b. In addition, on the xoy plane, the width W of the ellipsoid was divided into seven parts, and the equal diversion points (point O

_{7}and point O

_{8}) were only considered as the centers of the new filing spheres. Thus, another two spheres that were tangential to the outline of the ellipsoid were filled, defined as sphere O

_{7}and sphere O

_{8}, as shown in Figure 12c. Finally, the last sphere O

_{9}was filled at the position of the hilum. As a result, nine spheres in total were filled. Based on the filing principle, the 9-sphere, 11-sphere, and 13-sphere models of Jinong 2 were successively modeled, as shown in Figure 14d–f.

_{1}, point O

_{2}, point O

_{3}, and point O

_{4}) were the centers of the new filing spheres. Thus, four spheres that were tangential to the outline of the ellipsoid were filled, defined as sphere O

_{1}, sphere O

_{2}, sphere O

_{3}, and sphere O

_{4}, as shown in Figure 13b.

_{5}and point O

_{6}) were only considered as the centers of the new filing spheres. Thus, another two spheres that were tangential to the outline of the ellipsoid were filled, defined as sphere O

_{5}and sphere O

_{6}, as shown in Figure 13c. Finally, the last sphere O

_{7}was filled at the position of the hilum. As a result, seven total spheres were filled. Based on the filing principle, 7-sphere, 9-sphere, and 11-sphere models of Jinong 3 were successively modeled, as shown in Figure 14g–i.

#### 4.2. Contact Force Model of Multi-Sphere Particles

_{z}, E

_{y}, μ

_{z}, and μ

_{y}being the Young’s moduli and Poisson ratios of particle z and y, respectively; ${R}^{\ast}$ was the equivalent radius, and ${R}^{\ast}={\left(1/{R}_{za}+1/{R}_{yb}\right)}^{-1}$ with R

_{za}and R

_{yb}being the radii of partial sphere a and b, respectively; δ

_{zap}was the normal overlap; $\beta =\mathrm{ln}e/\sqrt{{\mathrm{ln}}^{2}e*+{\pi}^{2}}$ with e* being the coefficient of restitution; ${S}_{n}=2{E}^{\ast}\sqrt{{R}^{\ast}{\delta}_{zap}}$; ${m}^{\ast}$ was the equivalent mass, and ${m}^{\ast}={\left(1/{m}_{za}+1/{m}_{yb}\right)}^{-1}$ with m

_{za}and m

_{yb}being the masses of partial sphere a and b, respectively; ${\widehat{n}}_{zap}$ was the normal unit contact vector which pointed from the contact point p to the center of partial sphere a, and ${\widehat{n}}_{zap}=\left({x}_{za}-{x}_{zap}\right)/\left|{x}_{za}-{x}_{zap}\right|$ with x

_{za}being the position vector of the center of partial sphere a, and x

_{zap}being the position vector of the contact point p; ${S}_{t}=8{G}^{\ast}\sqrt{{R}^{\ast}{\delta}_{zap}}$, G* was the equivalent shear modulus; ξ

_{zap}was the total tangential displacement of partial sphere a; v

^{n}

_{zap}and v

^{t}

_{zap}were the relative normal and tangential velocities of partial sphere a at the contact point p; μ

_{s}was the coefficient of static friction.

_{z}was the position vector of the center of particle z; μ

_{r}was the coefficient of rolling friction; ${\hat{\mathsf{\omega}}}_{z}$ was the unit angular velocity of multi-sphere particle z, as shown in Figure 15 [11].

## 5. Determination of Simulation Parameters

#### 5.1. The Direct Shear Test

#### 5.2. Plackett–Burman Test and Path of Steepest Ascent Method

^{−7}s. The simulation time was 12 s. The Cyperus esculentus seed particles were generated by a normal distribution in terms of volume, and the sample mass generated was the same as the test mass for each case. At 2 s in the simulation, the plate was subjected to a normal force of 200 kPa in the +z direction and was stabilized for 0.5 s; at 2.5 s in the simulation, the down box started to move at a speed of 0.002 m/s in the +x direction until the end of the simulation, and the screenshots of the simulation for different times are shown in Figure 17a–d.

## 6. Analysis and Validation

#### 6.1. Piling Tests

#### 6.2. Bulk Density Tests

#### 6.3. Simulation Analysis

^{−7}s.

## 7. Conclusions

- (1)
- The sizes of the Cyperus esculentus seed particles all had a normal distribution, and a certain functional relationship was identified between the primary dimension and other secondary dimensions. The width of the seed was the primary dimension, and the other secondary dimensions (length and thickness) were calculated based on their relationships with the primary dimension. On this basis, an approach for modeling Cyperus esculentus seed particles based on the MS method was proposed. The 7-sphere, 9-sphere, and 11-sphere models were constructed for the seeds of Jinong 1 and Jinong 3, and the 9-sphere, 11-sphere, and 13-sphere models were constructed for the seeds of Jinong 2;
- (2)
- The mechanical properties of the Cyperus esculentus seeds were tested and analyzed. The elastic modulus of the seed, the restitution coefficient between seed–seed, the restitution coefficient between the seed and the contact material, and the static friction coefficient were all obtained through experiments. Thus, the value range of the simulation parameters was determined, and then, significance analysis of the simulation parameters was carried out by using the PB test design method through the direct shear test in the simulation. It was found that the static friction coefficient between seed–seed had the most significant effect on the results. On this basis, the value of the simulation parameters was further confirmed through the path of steepest ascent method;
- (3)
- The piling tests and the bulk density test were both adopted for further modeling verification. With the increase in the number of filing spheres, the simulated results were consistent with those obtained experimentally in the piling test and the bulk density test. Except for the 7-sphere of Jinong 1 and 9-sphere of Jinong 2 in the piling test, and the 9-sphere of Jinong 2 in the bulk density test, the mean value of the simulated results fluctuated within the standard deviation of the experimental results. The mean values of the simulated results were all within the margin of the standard errors of the experimental results. Thus, the feasibility and rationality of the Cyperus esculentus seed models established and the parameters’ selection in this paper were further verified;
- (4)
- Future research will be conducted as follows: The established seed model and the simulation parameters selected will be applied to the analysis of the working process of the seed metering device and the cleaning apparatus of the Cyperus esculentus seeds in simulations. In addition, other types of irregular seed modeling will be studied to enrich the theory of irregular seed modeling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

DEM | Discrete element method |

MS | Multisphere method |

PB | Plackett–Burman |

ANOVA | Analysis of Variance |

L | Length, mm |

W | Width, mm |

T | Thickness, mm |

E^{*} | Young’s modulus, MPa |

F | Normal force to the seed, N |

µ | Poisson’s ratio, dimensionless |

D | Deformation of the seed, mm |

R | Minimum curvature radii of the seeds with the compression probe and undersurface, mm |

R′ | Maximum curvature radii of the seeds with the compression probe and undersurface, mm |

K_{U} | Constant |

H′ | Thickness of the seed when compressed, mm |

L′ | Length of the seed when compressed, mm |

G^{*} | Shear modulus, MPa |

e^{*} | Coefficient of restitution, dimensionless |

H | Rebound height of the seed, mm |

H | Release height of the seed, mm |

v_{0} | Release velocity of seed No. 1, m/s |

v_{1} | Velocity of seed No. 2 after collision, m/s |

v_{2} | Velocity of seed No. 1 after collision, m/s |

h_{1} | Rebound height of seed No. 2 after collision, mm |

h_{2} | Rebound height of seed No. 1 after collision, mm |

G | Gravitational acceleration, m/s^{2} |

m | Mass, g |

θ | Angle between the inclined apparatus and horizontal plane, ° |

μ_{s} | Coefficient of static friction, dimensionless |

E_{z}, E_{y} | Young’s moduli of particle z and y, MPa |

µ_{z}, µ_{y} | Poisson ratios of particle z and y, dimensionless |

R^{*} | Equivalent radius, mm |

R_{za}, R_{yb} | Radii of elemental sphere a and b, mm |

δ_{zap} | Normal overlap |

m^{*} | Equivalent mass, g |

m_{za}, m_{yb} | Masses of elemental sphere a and b, g |

x_{za}, x_{zap} | Position vectors of the center of elemental sphere a and the contact point p |

ξ_{zap} | Total tangential displacement of elemental sphere a, mm |

v^{n}_{zap}, v^{t}_{zap} | Relative normal and tangential velocities of elemental sphere a at the contact point p, m/s |

N, P | Numbers of elemental spheres and contact points |

x_{z} | Position vector of the center of particle z |

μ_{r} | Coefficient of rolling friction, dimensionless |

${\widehat{\mathsf{\omega}}}_{z}$ | Unit angular velocity of particle z, rad/s |

V | Volume, m^{3} |

Ix, Iy, Iz | Moment of inertia, kg·m^{2} |

a | Half of the length of the seed, mm |

b | Half of the thickness of the seed, mm |

c | Half of the width of the seed, mm |

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**Figure 5.**Relationships between two dimensions of Jinong 1 seeds: (

**a**) width–length and (

**b**) width–thickness.

**Figure 6.**Relationships between two dimensions of Jinong 2 seeds: (

**a**) width–length and (

**b**) width–thickness.

**Figure 7.**Relationships between two dimensions of Jinong 3 seeds: (

**a**) width–length and (

**b**) width–thickness.

**Figure 9.**a–b Collision experiments: (

**a**) Free fall test, 1—coordinate paper, 2—copper plate, 3—mirror, 4—air pump nozzle; (

**b**) Single pendulum impact test, 1—coordinate paper, 2—high-speed camera, 3—self-made test bench.

**Figure 10.**(

**a**,

**b**) Static friction coefficient measurement tests: (

**a**) Measurement device: 1—inclinometer, 2—angle transducer, 3—seed plate, 4—electromotor; (

**b**) Seed plate.

**Figure 17.**Snapshots of the simulation of the direct shear test of Jinong 1 seeds at different times using the 9-sphere model: (

**a**) t = 1.5 s; (

**b**) t = 3.5 s; (

**c**) t = 6.5 s; and (

**d**) t = 12 s.

**Figure 19.**Packing test: (

**a**) photograph of the piling test captured by a high-speed camera; (

**b**) image binarization.

**Figure 21.**Snapshot of the simulation of the piling test of Jinong 2 seeds using the 11-sphere model.

**Figure 22.**(

**a**–

**c**) Variations of the simulated static angle of repose versus the number of filing spheres for different varieties of seeds.

**Figure 23.**Snapshots of the simulation of the bulk density test of Jinong 3 seeds at different times using the 9-sphere model: (

**a**) t = 1.4 s; (

**b**) t = 1.6 s; (

**c**) t = 2 s.

**Figure 24.**(

**a**–

**c**) Variations of the simulated bulk density versus the number of filing spheres for different varieties of seeds.

Variety | Density | Moisture Content | Thousand Seed Weight |
---|---|---|---|

Jinong 1 | 1.34 g/cm^{3} | 28.8% | 427 g |

Jinong 2 | 1.27 g/cm^{3} | 28.4% | 406 g |

Jinong 3 | 1.19 g/cm^{3} | 35.8% | 809 g |

Variety | Size | Mean/mm | Standard Deviation/mm |
---|---|---|---|

Jinong 1 | Length (L) | 9.63 | 0.54 |

Width (W) | 9.11 | 1.19 | |

Thickness (T) | 7.94 | 1.43 | |

Jinong 2 | Length (L) | 8.02 | 0.62 |

Width (W) | 13.70 | 1.71 | |

Thickness (T) | 5.80 | 0.65 | |

Jinong 3 | Length (L) | 11.86 | 1.37 |

Width (W) | 11.45 | 1.56 | |

Thickness (T) | 9.49 | 1.56 |

**Table 3.**Relationships between the primary dimension and the secondary dimensions of different varieties of Cyperus esculentus seeds.

Variety | Expression | R^{2} |
---|---|---|

Jinong 1 | L = (W − 5.9814)/(0.0004 × W^{3}) | 0.9516 |

T = (W − 5.984)/(0.0005 × W^{3}) | 0.892 | |

Jinong 2 | L = (W − 9.3913)/(0.0002 × W^{3}) | 0.9334 |

T = (W − 9.4943)/(0.0003 × W^{3}) | 0.8844 | |

Jinong 3 | L = (W − 7.9676)/(0.0002 × W^{3}) | 0.893 |

T = (W − 8.1707)/(0.0002 × W^{3}) | 0.8217 |

Variety | Collision Material | Restitution Coefficient |
---|---|---|

Jinong 1 | Copper | 0.58 |

Steel | 0.65 | |

Polymethyl methacrylate | 0.41 | |

Seed | 0.28 | |

Jinong 2 | Copper | 0.64 |

Steel | 0.75 | |

Polymethyl methacrylate | 0.42 | |

Seed | 0.34 | |

Jinong 3 | Copper | 0.68 |

Steel | 0.79 | |

Polymethyl methacrylate | 0.55 | |

Seed | 0.50 |

Material | Jinong 1 | Jinong 2 | Jinong 3 |
---|---|---|---|

Copper | 0.41 | 0.39 | 0.38 |

Steel | 0.40 | 0.39 | 0.35 |

Polymethyl methacrylate | 0.42 | 0.40 | 0.34 |

Variety | Maximum Shear Strength /kPa | Internal Friction Angle /° | Cohesive Force /kPa |
---|---|---|---|

Jinong 1 | 118.23 | 27.03 | 14.11 |

Jinong 2 | 136.67 | 31.43 | 6.88 |

Jinong 3 | 105.76 | 25.83 | 14.49 |

Symbol | Factor | Low Level (−1) | High Level (+1) |
---|---|---|---|

A | Poisson’s ratio of seed | 0.3 | 0.5 |

B | Shear modulus of seed/MPa | 30 | 300 |

C | Coefficient of static friction of seed–seed | 0.15 | 0.55 |

D | Coefficient of static friction of seed–polymethyl methacrylate | 0.2 | 0.6 |

E | Coefficient of rolling friction of seed–seed | 0 | 0.1 |

F | Coefficient of rolling friction of seed–polymethyl methacrylate | 0 | 0.1 |

G | Restitution coefficient of seed–seed | 0.15 | 0.75 |

H | Restitution coefficient of seed–polymethyl methacrylate | 0.2 | 0.8 |

I1, I2, I3 | Virtual parameters | — | — |

Factor | A | B | C | D | E | F | G | H | I1 | I2 | I3 | Y Maximum Shear Strength (kPa) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | |||||||||||||

1 | 0.5 | 300 | 0.15 | 0.6 | 0.1 | 0.1 | 0.15 | 0.2 | −1 | 1 | −1 | 97.9953 | |

2 | 0.3 | 300 | 0.55 | 0.6 | 0 | 0 | 0.15 | 0.8 | −1 | 1 | 1 | 127.114 | |

3 | 0.3 | 30 | 0.15 | 0.2 | 0 | 0 | 0.15 | 0.2 | −1 | −1 | −1 | 51.6994 | |

4 | 0.3 | 300 | 0.55 | 0.2 | 0.1 | 0.1 | 0.75 | 0.2 | −1 | −1 | 1 | 88.6227 | |

5 | 0.5 | 30 | 0.55 | 0.6 | 0 | 0.1 | 0.75 | 0.8 | −1 | −1 | −1 | 99.1627 | |

6 | 0.3 | 30 | 0.55 | 0.2 | 0.1 | 0.1 | 0.15 | 0.8 | 1 | 1 | −1 | 77.4156 | |

7 | 0.5 | 30 | 0.15 | 0.2 | 0.1 | 0 | 0.75 | 0.8 | −1 | 1 | 1 | 46.1292 | |

8 | 0.3 | 300 | 0.15 | 0.6 | 0.1 | 0 | 0.75 | 0.8 | 1 | −1 | −1 | 61.1053 | |

9 | 0.5 | 300 | 0.15 | 0.2 | 0 | 0.1 | 0.15 | 0.8 | 1 | −1 | 1 | 80.4509 | |

10 | 0.5 | 30 | 0.55 | 0.6 | 0.1 | 0 | 0.15 | 0.2 | 1 | −1 | 1 | 194.423 | |

11 | 0.5 | 300 | 0.55 | 0.2 | 0 | 0 | 0.75 | 0.2 | 1 | 1 | −1 | 122.911 | |

12 | 0.3 | 30 | 0.15 | 0.6 | 0 | 0.1 | 0.75 | 0.2 | 1 | 1 | 1 | 81.3848 |

Factor | Sum of Squares | F Value | p Value | Significance |
---|---|---|---|---|

A | 1969.42 | 3.01 | 0.1813 | 3 |

B | 65.26 | 0.100 | 0.7729 | 7 |

C | 7051.12 | 10.77 | 0.0464 | 1 |

D | 3134.91 | 4.79 | 0.1165 | 2 |

E | 0.73 | 1.121 × 10^{−3} | 0.9754 | 8 |

F | 511.55 | 0.78 | 0.4419 | 6 |

G | 1403.62 | 2.14 | 0.2394 | 5 |

H | 1768.04 | 2.70 | 0.1989 | 4 |

No. | Coefficient of Static Friction of Seed–Seed | Maximum Shear Strength /kPa | Relative Error |
---|---|---|---|

1 | 0.15 | 50.73 | 57.09% |

2 | 0.25 | 62.54 | 47.10% |

3 | 0.35 | 95.80 | 18.98% |

4 | 0.45 | 108.67 | 8.09% |

5 | 0.55 | 122.88 | 3.93% |

Parameter | Jinong 1 | Jinong 2 | Jinong 3 | Polymethyl Methacrylate |
---|---|---|---|---|

Poisson’s ratio | 0.4 | 0.4 | 0.4 | 0.32 |

Density kg/m^{3} | 1340 | 1270 | 1190 | 1190 |

Shear modulus MPa | 165 | 165 | 165 | 1197 |

Coefficient of restitution | 0.45 | 0.45 | 0.45 | 0.55 |

Coefficient of static friction | 0.55 | 0.35 | 0.35 | 0.34 |

Coefficient of rolling friction | 0.05 | 0.05 | 0.05 | 0.05 |

Variety | Volume *^{1}/m^{3}V | Moment of inertia /kg·m^{2} | ||
---|---|---|---|---|

I_{x} *^{2} | I_{y} *^{3} | I_{z} *^{4} | ||

Jinong 1 | 2.91722 × 10^{−6} | 1.37398 × 10^{−10} | 1.21829 × 10^{−10} | 1.14194 × 10^{−10} |

Jinong 2 | 2.66842 × 10^{−6} | 1.70719 × 10^{−10} | 6.64186 × 10^{−11} | 1.49946 × 10^{−10} |

Jinong 3 | 5.39506 × 10^{−6} | 3.49005 × 10^{−10} | 2.96202 × 10^{−10} | 2.83974 × 10^{−10} |

^{1}V = 4/3πabc; m = ρV; *

^{2}I

_{x}= 1/5m(b

^{2}+ c

^{2}); *

^{3}I

_{y}= 1/5m(a

^{2}+ c

^{2}); *

^{4}I

_{z}= 1/5m(a

^{2}+ b

^{2}); where a, b, and c are half of the length, thickness, and width of the seed.

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## Share and Cite

**MDPI and ACS Style**

Xu, T.; Zhang, R.; Zhu, F.; Feng, W.; Wang, Y.; Wang, J.
A DEM-Based Modeling Method and Simulation Parameter Selection for *Cyperus esculentus* Seeds. *Processes* **2022**, *10*, 1729.
https://doi.org/10.3390/pr10091729

**AMA Style**

Xu T, Zhang R, Zhu F, Feng W, Wang Y, Wang J.
A DEM-Based Modeling Method and Simulation Parameter Selection for *Cyperus esculentus* Seeds. *Processes*. 2022; 10(9):1729.
https://doi.org/10.3390/pr10091729

**Chicago/Turabian Style**

Xu, Tianyue, Ruxin Zhang, Fengwu Zhu, Weizhi Feng, Yang Wang, and Jingli Wang.
2022. "A DEM-Based Modeling Method and Simulation Parameter Selection for *Cyperus esculentus* Seeds" *Processes* 10, no. 9: 1729.
https://doi.org/10.3390/pr10091729