An Experimental and Numerical Study for Discrete Element Model Parameters Calibration: Gluten Pellets

Abstract

Discrete element method (DEM) simulation is widely used to calculate the flow characteristics of particles under certain conditions. DEM input parameters are the prerequisite for the accurate modeling and simulation of particles. In order to explore the mechanical properties and breaking behavior of gluten pellets, the pellet material property, the interaction parameters of pellet–stainless steel and pellet–pellet (multi-spheres autofill model), and the bonding parameters (bonded particle model) were calibrated by experiments and simulations. The relative error of the angle of repose, the breaking displacement, and the breaking force between simulated and experimental values were 0.28%, 0.66%, and 1.09%, respectively. Based on the regression analysis in the Design-Expert 12.0 software, the relationships among evaluating indicators (angle of repose, breaking displacement, and breaking force) and their corresponding influencing factors were established, respectively. Meanwhile, the feasibility of applying the interaction parameters of the multi-spheres autofill model to the bonded particle model was verified through the free fall test, the inclined plane sliding test, and the inclined plane tumbling time test. This work can provide a reference for the design of pellet feed processing and transportation machinery.

1. Introduction

Gluten contains fifteen amino acids that the human body needs and it is a natural protein isolated from wheat flour that meets the nutritional needs of modern diets [1,2]. It is widely used in bread [3], noodles [4], instant noodles [5], meat product additives [6], beverage additives [7], and processing high-grade aquatic feed [8]. After extrusion molding, gluten pellets undergo conveying, drying, and packaging processes. When the gluten pellets move in the equipment mentioned above, they collide and squeeze with the inner wall of the equipment and other pellets, resulting in serious problems such as pellet breakage and pulverization, thus reducing the economic value of the product [9,10,11]. Therefore, studying the mechanical interaction parameters of gluten pellets is of great significance.

Cundall proposed the discrete element method (DEM) in 1979 to simulate and analyze particle behavior by establishing a parametric model of a solid particle system [12]. In recent years, commercial software based on the DEM has been widely used in agricultural engineering. The properties of gluten pellets were affected by factors such as hygroscopicity, composition, particle size, etc., so the parameter calibration must be conducted according to specific conditions [13,14,15]. Scholars at home and abroad have conducted much work on the parameter calibration of agricultural materials. The fundamental physical property and mechanical interaction parameters were usually obtained through experiments and simulations. Niu and Peng [10,16] calibrated the discrete element parameters of pellet feed by measuring the angle of repose and used EDEM simulation software to research the calibration of discrete element simulation parameters of pellet feed uniaxial compression damage. Zhao et al. [17] calibrated the discrete element parameters of coconut bran particles and verified the coconut bran’s stacking angle and fluidity by experiments. Kong et al. [18] combined computational fluid dynamics (CFDs) and the DEM to study the effects of the inlet velocity and bending radius on pellet breakage rates and energy loss during the pneumatic conveying of aquafeed pellets. Wang et al. [19] measured the angle of repose of alfalfa with different moisture contents and obtained the interaction parameters by combining it with a discrete element simulation. Li et al. [20] proposed a corn kernel model-building method for Rocky DEM simulations and a calibration method for calibrating the interaction parameters. There are few reports on the research on the discrete element model of gluten pellets both at home and abroad [21].

Pelletization is a polymerization process of raw powder materials by mechanical extrusion, which is most widely used in the feed and biomass energy industry [22,23]. In the process of pelletization, friction between raw materials and mold generates a large amount of heat, which causes the gelatinization of starch and the denaturation of protein to become a binder and bond raw material particles together [24]. At the same time, high temperature and high pressure in the pelletization process can eliminate salmonella and reduce microbial activity, thus improving feed safety [25]. Extrusion molding can significantly improve the bulk density of raw materials, thereby improving their transportation and storage performance and reducing transportation and storage costs [26].

This study takes gluten pellets as the research object, and their physical parameters (size, density, and shear modulus), interaction parameters between pellets and stainless steel plates (static friction coefficient, rolling friction coefficient, and restitution coefficient), and angle of repose (image method) were measured. Based on the Hertz–Mindlin contact model of EDEM software, the multi-spheres autofill model and bonded particle model were established. The contact and bonded parameters between pellets were calibrated by the angle of repose and compression simulation, combined with the response surface optimization test. Finally, the feasibility of using the interaction parameters of the multi-spheres autofill model to the bonded particle model was verified.

2. Materials and Methods

2.1. Material Properties and DEM Model of Gluten Pellets

2.1.1. Material Properties Determination

The gluten pellets were provided by Bilvchun Biotechnology Co., Ltd. And produced in Lai’an County, Chuzhou City, Anhui Province, China. The gluten pellets are pale yellow with a protein content of 85%, and the initial moisture content (dry-basis) is 10.18 ± 0.10% [27].

• Size and density

The electronic digital vernier caliper (DL91150, Shanghai Yuanmai Trading Co., Ltd., Shanghai, China) was used to measure the length and height. The electronic balance (ME55/02, METTLER TOLEDO Instruments (Shanghai) Co., Ltd., Shanghai, China) was used to measure the mass of pellets. The density was calculated by Equation (1).

$$\overline{\rho }=\frac{\overline{m}}{\frac{\pi }{4}{\overline{d}}^2\overline{L}}\times 1000$$

(1)

where $\overline{\rho }$ is the average density of pellets, 10³ kg·m⁻³; $\overline{m}$ is the average mass of pellets, kg; $\overline{d}$ is the average height of pellets, m; and $\overline{L}$ is the average length of pellets, mm.

Fifty pellets were randomly selected, and the final result was expressed as means ± standard deviation. The mass, height, length, and density of gluten pellet are 0.178 × 10⁻³ ± 0.041 × 10⁻³ kg, 4.24 × 10⁻³ ± 0.13 × 10⁻³ m, 11.43 × 10⁻³ ± 2.43 × 10⁻³ m, and 1099 ± 67 kg·m⁻³, respectively.

• Shear modulus

Gluten pellets are brittle materials, and Poisson’s ratio can be obtained from reference [16] as 0.4. A texture analyzer (TA.XT.plus, Stable Micro System, Surrey, UK) with a P/36R probe was used to carry out a compression test on the gluten pellets until the pellets were broken. Gluten pellets were placed horizontally on a rigid base. The test speed and the trigger force were 0.5 × 10⁻³ m·s⁻¹ and 0.049 N, respectively. The texture compression experiment device is shown in Figure 1. The experiment was repeated five times, and the average value was taken. The force–displacement curve is shown in Figure 2. The average compression displacement of gluten pellets was 1.209 × 10⁻³ m, and the average ultimate breaking force was 94.656 N. The slope of the stress–strain curve, which was automatically drawn by the computer, was elastic modulus. The shear modulus was calculated by Equation (2).

$$G=\frac{E}{2(1+\nu )}$$

(2)

where G is shear modulus, Pa; ν is Poisson’s ratio; and E is elastic modulus, Pa.

The gluten pellets’ elastic modulus and shear modulus can be obtained as 2.207 × 10⁸ ± 4.771 × 10⁷ Pa and 7.881 × 10⁷ ± 1.704 × 10⁷ Pa, respectively.

2.1.2. DEM Model

(1)

Gluten pellet model

In this study, a multi-spheres autofill model and a bonded particle model were established and used for the angle of repose simulation and bonding parameters calibration, respectively. Considering factors such as shape fitting, number of particles, or computation time, the two modeling methods are as follows.

• Multi-spheres autofill model: According to the average geometric size of gluten pellets measured by the test above, a three-dimensional model was established in SolidWorks software, and the file (“.stl” format) was exported. Then, the file was imported into EDEM as an autofill template. The smoothing value and the minimum sphere radius were set to 2 and 0.5 × 10⁻³ m, respectively. The generated pellet model with a total of 177 particles is shown in Figure 3a. In order to be as close to the actual size of the pellets as possible in the simulation, a random size distribution with a minimum and maximum value was 0.85 and 1.15, respectively. The pellet’s size was scaled by volume.
• Bonded particle model: The pellet model file (“.stl” format) was imported into EDEM. Then, a box with a length, width, and height of 6 × 10⁻³ m, 6 × 10⁻³ m, and 13 × 10⁻³ m was created to contain the pellet model. The box and pellet model types were set to physical and virtual, respectively. The box was filled with particles, and the box and pellet model types were changed to virtual and physical, respectively. When the particle motion inside the pellet model was stable, and the external particles disappeared under gravity, the contact model of the particles was set to Hertz–Mindlin (no slip), and the connection parameters of the particles were set. Then, we started to generate bonding keys to connect the particles and deleted the box and the pellet model shell to achieve a gluten pellet model. The particle filling effect can be verified by Equation (3).

  $$\alpha ·r^2·L=N·\frac{4}{3}·R^3$$

  (3)

  where α is fill rate; r is the radius of actual gluten pellet, m; L is the length of actual gluten pellet, m; N is the number of particles filled; and R is the radius of particles filled, m.

The bonded particle model is shown in Figure 3b. Particles with a radius of 0.3×10⁻³ m were used for filling, and the number of particles was 839. The filling rate is 0.59, and the filling effect is ideal [28].

(2)

Compression model

After the bonded particle model was established, the indenter and the support plate models were established, respectively. In order to ensure that the quality of the bonding model was equal to the actual pellet, the density amplification method was used to enlarge the density of the pellets from 1099 kg·m⁻³ to 1869 kg·m⁻³, and the other parameters remained unchanged. The compression speed was set to 0.5 × 10⁻³ m·s⁻¹ to simulate the compression process of the pellets, as shown in Figure 3c.

2.2. Test Design and Indicators Determination

2.2.1. Interaction Parameters Determination of Pellet–Stainless Steel

(1)

Static friction coefficient

The static friction coefficient measurement principle is shown in Figure 4b. Five pellets were glued together and placed on a stainless steel inclinometer to prevent the gluten pellets from rolling. The inclination angle gradually increased and was read by a digital protractor (PT180, Shenzhen Red Dragon Instruments Co., Ltd., Shenzhen, China), as shown in Figure 4a. When the pellets just began to slide on the inclined surface, the inclination angle was the static friction angle. Each group of experiments was repeated five times, and the mean value was calculated. The static friction coefficient was calculated by Equation (4) [29].

$${\mu }_s=\tan \alpha$$

(4)

where μ is the static friction coefficient and α is the static friction angle.

The static friction coefficient between the gluten pellets and the stainless steel plate was measured to be 0.51 ± 0.02.

(2)

Rolling friction coefficient

The rolling friction coefficient between the gluten pellets and the stainless steel plate adopted the inclined surface rolling method. The measurement principle of the rolling friction coefficient is shown in Figure 4c. The gluten pellets were placed along the radial directions on the stainless steel plate. The initial velocity of the pellets was zero. When the rolling was stationary, the rolling distance was measured, and the rolling friction coefficient was calculated by Equation (5) [30]. Experiments were repeated five times, and the rolling friction coefficient between the gluten pellets and the stainless steel plate was measured to be 0.27 ± 0.004. Furthermore, the average time for each group of pellets to roll off the stainless steel plate was recorded as 0.45 s.

$${\mu }_r=\frac{{\mathrm{h}}_1-\frac{H^2}{4{cos}^2\beta (h_2-Htan\beta )}}{Lcos\beta }$$

(5)

(3)

Restitution coefficient

The collision test method of pellet to the fixed surface was used to determine the restitution coefficient between gluten pellets and stainless steel plate. The measurement principle of the restitution coefficient is shown in Figure 4d. A stainless steel plate with a horizontal plane was set at 45°. Equations (6) and (7) were used to calculate the restitution coefficient [31,32]. Each group was repeated five times, and the average value was obtained.

$$\left\{\begin{array}{c}{v}_x=\sqrt{\frac{g{s}_1{s}_2(s_1-s_2)}{2(h_1{s}_2-h_2{s}_1)}}\\ v_y=\frac{h_1{v}_x}{s_1}-\frac{g{s}_1}{2v_x}\end{array}\right.$$

(6)

$$e=\frac{\sqrt{v_x^2+v_y^2}·cos\left[45°+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{v_y}{v_x}\right)\right]}{\sqrt{2gH}·sin45°}$$

(7)

where $v_x$ is horizontal speed, m·s⁻¹; g is gravitational acceleration, m·s⁻²; $s_1$ and $s_2$ are horizontal displacement for two measurements, respectively, m; $h_1$ and $h_2$ are vertical displacement for two measurements, respectively, m; $v_y$ is vertical velocity; e is restitution coefficient; and H is the drop height, m.

The restitution coefficient between the gluten pellets and the stainless steel plate was measured to be 0.50 ± 0.04.

(4)

Angle of repose

The angle of repose of gluten pellets was determined by injection [10]. The angle of repose reflects the internal friction and scattering properties of gluten pellets. It is related to pellet shape, size, moisture content, stacking conditions, etc. The larger the angle of repose, the greater the internal friction force and the smaller the scatter.

The angle of the repose measuring device is shown in Figure 5. The outlet diameter of the 304 stainless steel (shear modulus is 2.7 × 10¹⁰ Pa; Poisson’s ratio is 0.3; and density is 7930 kg·m⁻³) funnel is 3.05 × 10⁻² m, the diameter of the plexiglass disc is 0.09 m, the height is 0.0175 m, and the distance between the funnel outlet and the pellets in the plexiglass disc is 0.05 m. During the test, first, a baffle was used to close the outlet of the funnel, and the funnel was filled with gluten pellets. Then, the baffle was quickly removed until the pellets no longer fell and a pellet pile formed in the glass disc. In order to achieve the exact angle of repose, a mobile phone was used to take a vertical photo of the front of the pellets heap, and then Python processed the image. Specific treatment methods were as follows: Firstly, the image was enhanced by Gaussian filtering to remove noise points and histogram homogenization. An appropriate threshold was set for binary segmentation, and then an expansion was carried out. The hole-filling algorithm was used three times to fill the internal holes, remove excess edges and corners, and smooth the contour. Finally, the canny operator was used to detect the edge, and the edge contour was obtained. The edge contour was imported into Origin 2021 and transformed into coordinate data by an image digitization tool, and then the linear fitting was performed. The slope obtained by linear fitting was converted into the angle of repose. The image processing is shown in Figure 6. Approximately 0.1 kg pellets were selected for each group of experiments and repeated five times. The mean value was obtained. The actual measured gluten pellets angle of repose was 28.61 ± 2.65°.

2.2.2. Interaction Parameters Determination of Pellet–Pellet

The Generic EDEM material model (GEMM) database contains thousands of material models representing various materials such as corn, coal powder, sand, etc. It addresses one of the critical challenges of DEM simulation to estimate simulation parameters by inputting material size, density, and angle of repose. The ranges of the static friction coefficient (${\mu }_s^{\prime }$), rolling friction coefficient (${\mu }_r^{\prime }$), and restitution coefficient ($e^{\prime }$) between pellets were determined by inputting the above three parameters of the gluten pellets studied in this paper and combining them with the related research in works from the literature [10,16,33], as shown in

Table 1. The ranges of the simulation parameters of pellet–pellet.

Parameters	Range
${\mu }_s^{\prime }$	0.20–0.80
${\mu }_r^{\prime }$	0.05–0.20
$e^{\prime }$	0.15–0.75

.

The steepest ascent test was carried out to further narrow the factor values range. The simulated angle of repose ${\theta }^{\prime }$ and its relative error $\delta$ with the actual angle of repose $\theta$ were indicators. The relative error was calculated by Equation (8).

$$\delta =\left|\frac{\theta -{\theta }^{\prime }}{\theta }\right|$$

(8)

Based on the above results, the three factors quadratic orthogonal rotation combination test was carried out. The Design-Expert 12.0 software was used to analyze test results by the design principle of CCD (Central Composite Design). Nine center points were set to acquire 23 groups of tests in total. Each group of tests was repeated five times, and the average value was taken as the final simulation result. The factor codes are shown in

Table 2. Factors and codes of quadratic regression orthogonal rotating test.

Factors	Codes
	−1.682	−1	0	1	1.682
${\mu }_s^{\prime }$	0.44	0.49	0.56	0.63	0.68
$e^{\prime }$	0.39	0.44	0.51	0.58	0.63
${\mu }_r^{\prime }$	0.11	0.12	0.14	0.16	0.17

.

2.2.3. Bonding Parameters Calibration Test Design

Normal stiffness per unit area X₁, shear stiffness per unit area X₂, critical normal stress X₃, critical shear stress X₄, and bond disk radius X₅ were taken as test factors, and breaking displacement Y₁ and ultimate breaking force Y₂ were taken as indicators. First, the Plackett–Burman test at 2-level was used, as shown in

Table 3. Plackett–Burman test factors and levels.

Factors	Value
	Low Level	High Level
X1/1010 N·m−3	1.64	2.24
X2/1010 N·m−3	1.8	2.12
X3/106 Pa	7.25	8.5
X4/106 Pa	5.4	6.4
X5/10−3 m	0.34	0.44

, and ANOVA and t-test were performed to determine the significance of the factors. Then, the steepest ascent test was carried out further to reduce the range of parameters of each factor. Finally, the Box–Behnken test was carried out to determine the optimal parameter combination of each factor.

The simulation process of extrusion rupture is shown in Figure 7. With the downward movement of the indenter, when the displacement was 1.23 × 10⁻³ m, the force on the gluten pellet reached the maximum value, and when the displacement was 1.74 × 10⁻³ m, the gluten pellet was broken entirely.

3. Results

3.1. Parameters Calibration of Pellet–Pellet Test

3.1.1. Steepest Ascent Test

The design and results of the steepest ascent tests are shown in

Table 4. Results of the steepest ascent test.

Test No.	Factors			Results
	${\mathit{\mu }}_{\mathit{s}}^{\mathit{\prime }}$	${\mathit{\mu }}_{\mathit{r}}^{\mathit{\prime }}$	${\mathit{e}}^{\mathit{\prime }}$	${\mathit{\theta }}^{\mathit{\prime }}$	$\mathit{\delta }$/%
1	0.20	0.05	0.15	36.06	26.03
2	0.32	0.08	0.27	33.49	17.04
3	0.44	0.11	0.39	26.64	6.88
4	0.56	0.14	0.51	27.68	3.25
5	0.68	0.17	0.63	30.29	5.89
6	0.80	0.20	0.75	30.92	8.07

. With the increase in ${\mu }_s^{\prime }$, ${\mu }_r^{\prime }$, and $e^{\prime }$, the simulation value of the angle of repose rose first and then it decreased. Among them, the relative error of the fourth group of tests was the smallest. Therefore, the third, fourth, and fifth groups of the parameter values were used as the low, center point, and high levels of the quadratic orthogonal rotation combination design test.

3.1.2. Quadratic Orthogonal Rotation Combination Test

The test results of the quadratic orthogonal rotation combination design are shown in

Table 5. Results of quadratic regression orthogonal rotating test.

Test No.	${\mathit{\mu }}_{\mathit{s}}^{\mathit{\prime }}$	${\mathit{e}}^{\mathit{\prime }}$	${\mathit{\mu }}_{\mathit{r}}^{\mathit{\prime }}$	${\mathit{\theta }}^{\mathit{\prime }}$	$\mathit{\delta }$
1	0.49	0.44	0.12	30.56	6.82
2	0.63	0.44	0.12	35.28	23.31
3	0.49	0.58	0.12	28.1	1.78
4	0.63	0.58	0.12	34.53	20.69
5	0.49	0.44	0.16	35.38	23.66
6	0.63	0.44	0.16	39.18	36.95
7	0.49	0.58	0.16	29.16	1.92
8	0.63	0.58	0.16	34.65	21.11
9	0.44	0.51	0.14	30.71	7.34
10	0.68	0.51	0.14	36.36	27.09
11	0.56	0.39	0.14	37.38	30.65
12	0.56	0.63	0.14	34.17	19.43
13	0.56	0.51	0.11	32.65	14.12
14	0.56	0.51	0.17	34.34	20.03
15	0.56	0.51	0.14	29.73	3.91
16	0.56	0.51	0.14	29.56	3.32
17	0.56	0.51	0.14	30.07	5.10
18	0.56	0.51	0.14	28.2	1.43
19	0.56	0.51	0.14	29.09	1.68
20	0.56	0.51	0.14	28.93	1.12
21	0.56	0.51	0.14	29.17	1.96
22	0.56	0.51	0.14	29.79	4.12
23	0.56	0.51	0.14	27.65	3.36

. The simulation results compared different fitting models, showing that the quadratic full model fitting was the best.

The results of the significance analysis and variance analysis of the regression model are shown in

Table 6. Variance analysis of angle of repose regression model.

Source of Variation	Sum of Squares	Degree of Freedom	F-Value	p-Value
Model	235.47	9	21.05	<0.0001 **
${\mu }_s^{\prime }$	65.65	1	52.82	<0.0001 **
$e^{\prime }$	27.44	1	22.08	0.0004 **
${\mu }_r^{\prime }$	11.89	1	9.57	0.0086 **
${\mu }_s^{\prime }{e}^{\prime }$	1.45	1	1.16	0.3005
${\mu }_s^{\prime }{\mu }_r^{\prime }$	0.4325	1	0.3479	0.5654
$e^{\prime }{\mu }_r^{\prime }$	7.11	1	5.72	0.0326 *
${{\mu }_s^{\prime }}^2$	26.96	1	21.69	0.0004 **
${e^{\prime }}^2$	69.71	1	56.09	<0.0001 **
${{\mu }_r^{\prime }}^2$	26.38	1	21.22	0.0005 **
Residual	16.16	13
Lack of fit	11.20	5	3.61	0.0527
Pure error	4.96	8
Cor Total	251.63	22

Note: p < 0.05 (significant, *), p < 0.01 (highly significant, **).

. The fitting degree of the regression model was very significant (p < 0.0001). The interaction terms of the static friction coefficient with the collision restitution coefficient and the rolling friction coefficient had no significant effect on the angle of repose (p > 0.05). All the other effects were significant. The order of influence of each factor was ${\mu }_s^{\prime }$ > $e^{\prime }$ > ${\mu }_r^{\prime }$. The lack of fit of the regression model was p = 0.0527 > 0.05, which was not significant, indicating that there were no other main factors affecting the index in the model. The R² and adjusted R² of the regression model were 0.936 and 0.891, respectively, indicating that the predicted value of the regression model fitted well with the actual value. In order to improve the accuracy of the model, the regression model of the interaction parameters between the pellets and angle of repose was Equation (9), after the non-significant terms were removed.

$$\theta =29.16+2.19{\mu }_s^{\prime }-1.42e^{\prime }+0.93{\mu }_r^{\prime }-0.94e^{\prime }{\mu }_r^{\prime }+1.30{{\mathsf{\mu }}_s^{\prime }}^2+2.09{e^{\prime }}^2+1.29{{\mu }_r^{\prime }}^2$$

(9)

The angle of repose $\mathsf{\theta }$ = 28.61° obtained in the actual test was input into the Optimization-Numerical module of Design-Expert 12.0 software. The parameters were further optimized in the range of 0.44 ≤ ${\mu }_s^{\prime }$ ≤ 0.68, 0.39 ≤ $e^{\prime }$ ≤ 0.63, and 0.11 ≤ ${\mu }_r^{\prime }$ ≤ 0.17. The optimal calibration parameter combinations of the angle of repose were ${\mu }_s^{\prime }$ = 0.52, $e^{\prime }$ = 0.54, and ${\mu }_r^{\prime }$ = 0.12. The simulation angle of repose was 28.69° using the optimal parameter combination for simulation verification, and the relative error between the simulation and the actual test was 0.28%. The comparison between the actual and the simulation test is shown in Figure 8. The results showed that the shape and angle of the simulation test of the angle of repose were consistent with the actual test, which indicated that the simulation parameters obtained by optimization were reasonable.

3.2. Bonding Parameters Calibration Test

3.2.1. Plackett–Burman Test

The 11-factor Plackett–Burman test protocol was selected, and the test results are shown in

Table 7. Plackett–Burman Test Results.

Test No.	X1/1010 N·m−3	X2/1010 N·m−3	X3/106 Pa	X4/106 Pa	X5/10−3 m	Y1/10−3 m	Y2/N
1	2.24	2.12	7.25	6.4	0.44	1.171	82.171
2	1.64	2.12	8.5	5.4	0.44	1.052	69.787
3	2.24	1.8	8.5	6.4	0.34	1.336	102.652
4	1.64	2.12	7.25	6.4	0.44	1.183	85.844
5	1.64	1.8	8.5	5.4	0.44	1.107	74.813
6	1.64	1.8	7.25	6.4	0.34	1.302	94.112
7	2.24	1.8	7.25	5.4	0.44	1.151	68.353
8	2.24	2.12	7.25	5.4	0.34	1.192	103.427
9	2.24	2.12	8.5	5.4	0.34	1.192	95.357
10	1.64	2.12	8.5	6.4	0.34	1.262	110.69
11	2.24	1.8	8.5	6.4	0.44	1.253	75.765
12	1.64	1.8	7.25	5.4	0.34	1.224	90.113

. Design-Expert 12.0 was used to conduct ANOVA and t-test on the test data, and the results are shown in

Table 8. Analysis of significance of parameters in Plackett–Burman test.

Source of Variation	Y1					Y2
	Sum of Squares	Degree of Freedom	Mean Square	F-Value	p-Value	Sum of Squares	Degree of Freedom	Mean Square	F-Value	p-Value
Model	0.0689	5	0.0138	59.93	<0.0001 **	1973.55	5	394.71	14.00	0.0029 **
X1	0.0023	1	0.0023	9.87	0.0200 *	0.4665	1	0.4665	0.0165	0.9018
X2	0.0086	1	0.0086	37.34	0.0009 **	143.30	1	143.30	5.08	0.0650
X3	0.0000	1	0.0000	0.1598	0.7032	2.12	1	2.12	0.0752	0.7931
X4	0.0289	1	0.0289	125.71	<0.0001 **	203.23	1	203.23	7.21	0.0363 *
X5	0.0291	1	0.0291	126.57	<0.0001 **	1624.43	1	1624.43	57.63	0.0003 **
Residual	0.0014	6	0.0002			169.13	6	28.19
Cor Total	0.0703	11				2142.68	11

Note: p < 0.05 (significant, *), p < 0.01 (highly significant, **).

and Figure 9. The influence of each factor on breaking displacement Y₁ was X₅ > X₄ > X₂ > X₁ > X₃, among which X₄, X₅, X₂, and X₁ had significant effects (p < 0.05), and the effective value of X₄ and X₁ on the target was positive. In contrast, the effect value of X₅ and X₂ on the target was negative. Similarly, the influence of each factor on the breaking force Y₂ was X₅ > X₄ > X₂ > X₃ > X₁, among which X₅ and X₄ had significant effects (p < 0.05), and the effective value of X₄ on the target was positive, and the effective value of X₅ on the target was negative. A comprehensive analysis showed that X₃ had no significant effect on the results (p > 0.05), so 8 × 10⁶ Pa was used in the subsequent simulation test.

3.2.2. Steepest Ascent Test

According to the Plackett–Burman test results, the appropriate step size was chosen to carry out the steepest ascent test further, as shown in

Table 9. Results of the steepest ascent test.

Test No.	X1/1010 N·m−3	X2/1010 N·m−3	X4/106 Pa	X5/10−3 m	Y1/10−3 m	${\mathit{\delta }}_1$	Y2/N	${\mathit{\delta }}_2$
1	1.64	2.12	5.4	0.44	1.120	7.438	65.007	31.326
2	1.76	2.06	5.6	0.42	1.143	5.537	66.570	29.675
3	1.88	2.00	5.8	0.40	1.180	2.479	87.037	8.053
4	2.00	1.94	6	0.38	1.233	1.928	90.943	3.926
5	2.12	1.86	6.2	0.36	1.280	5.785	95.747	1.148
6	2.24	1.80	6.4	0.34	1.307	7.989	101.943	7.694

. The breaking displacement Y₁ and the breaking force Y₂ decreased first and then increased, and the errors between the simulation results and the actual values of the fourth and fifth groups were the smallest. Moreover, the actual values Y₁ and Y₂ were between the third and fifth simulated values.

3.2.3. Box–Behnken Test

According to the steepest ascent test results, the third and fifth sets of parameters were chosen as the high and low levels of the Box–Behnken test, and the test results and ANOVA results are shown in

Table 10. Results of the Box–Behnken test.

Test No.	X1/1010 N·m−3	X2/1010 N·m−3	X4/106 Pa	X5/10−3 m	Y1/10−3 m	Y2/N
1	1.88	1.86	6	0.38	1.177	119.803
2	2.12	1.86	6	0.38	1.286	88.246
3	1.88	2	6	0.38	1.322	100.241
4	2.12	2	6	0.38	1.294	114.744
5	2	1.93	5.8	0.36	1.228	99.062
6	2	1.93	6.2	0.36	1.365	92.517
7	2	1.93	5.8	0.4	1.231	91.846
8	2	1.93	6.2	0.4	1.134	83.291
9	1.88	1.93	6	0.36	1.283	102.673
10	2.12	1.93	6	0.36	1.293	94.935
11	1.88	1.93	6	0.4	1.224	92.116
12	2.12	1.93	6	0.4	1.211	96.124
13	2	1.86	5.8	0.38	1.236	99.242
14	2	2	5.8	0.38	1.297	111.627
15	2	1.86	6.2	0.38	1.237	88.913
16	2	2	6.2	0.38	1.337	94.114
17	1.88	1.93	5.8	0.38	1.274	109.722
18	2.12	1.93	5.8	0.38	1.125	89.003
19	1.88	1.93	6.2	0.38	1.142	94.014
20	2.12	1.93	6.2	0.38	1.322	90.792
21	2	1.86	6	0.36	1.301	119.643
22	2	2	6	0.36	1.396	95.811
23	2	1.86	6	0.4	1.218	87.514
24	2	2	6	0.4	1.297	112.987
25	2	1.93	6	0.38	1.226	97.913
26	2	1.93	6	0.38	1.208	93.764
27	2	1.93	6	0.38	1.213	94.407
28	2	1.93	6	0.38	1.212	94.463
29	2	1.93	6	0.38	1.201	92.757

and

Table 11. Box–Behnken test program and results.

Source of Variation		Y1					Y2
	Sum of Squares	Degree of Freedom	Mean Square	F-Value	p-Value	Sum of Squares	Degree of Freedom	Mean Square	F-Value	p-Value
Model	0.1215	14	0.0087	44.33	<0.0001 **	2498.44	14	178.46	15.42	<0.0001 **
A-X1	0.0010	1	0.0010	5.06	0.0411 *	166.69	1	166.69	14.41	0.0020 **
B-X2	0.0198	1	0.0198	101.37	<0.0001 **	57.04	1	57.04	4.93	0.0434 *
C-X4	0.0018	1	0.0018	9.07	0.0093 *	269.43	1	269.43	23.29	0.0003 **
D-X5	0.0253	1	0.0253	129.24	<0.0001 **	138.47	1	138.47	11.97	0.0038 **
AB	0.0047	1	0.0047	23.97	0.0002 **	530.38	1	530.38	45.84	<0.0001 **
AC	0.0271	1	0.0271	138.23	<0.0001 **	76.54	1	76.54	6.61	0.0222 *
AD	0.0001	1	0.0001	0.6755	0.4249	34.49	1	34.49	2.98	0.1062
BC	0.0004	1	0.0004	1.94	0.1851	12.90	1	12.90	1.12	0.3089
BD	0.0001	1	0.0001	0.3269	0.5765	607.75	1	607.75	52.53	<0.0001 **
CD	0.0137	1	0.0137	69.92	<0.0001 **	1.01	1	1.01	0.0873	0.7720
A2	0.0001	1	0.0001	0.4156	0.5296	58.91	1	58.91	5.09	0.0406 *
B2	0.0227	1	0.0227	115.99	<0.0001 **	422.72	1	422.72	36.53	<0.0001 **
C2	1.126 × 10−6	1	1.126 × 10−6	0.0058	0.9406	59.16	1	59.16	5.11	0.0402 *
D2	0.0067	1	0.0067	34.02	<0.0001 **	0.0047	1	0.0047	0.0004	0.9842
Residual	0.0027	14	0.0002			161.99	14	11.57
Lack of Fit	0.0024	10	0.0002	2.88	0.1597	146.88	10	14.69	3.89	0.1013
Pure Error	0.0003	4	0.0001			15.11	4	3.78
Cor Total	0.1242	28				2660.43	28

Note: p < 0.05 (significant, *), p < 0.01 (highly significant, **).

. It can be seen from Table 11 that X₁, X₂, X₄, and X₅ had significant influences on Y₁ and Y₂ (p < 0.05), and the influence sizes were X₅ > X₂ > X₄ > X₁ and X₄ > X₁ > X₅> X₂, respectively. The lack of fit of the regression model p = 0.1597 > 0.05 and p = 0.1013 > 0.05 were not significant. The functional relation between Y₁ and the significant factors was Equation (10). The R² and adjusted R² of the regression model were 0.978 and 0.956, respectively. The functional relation between Y₂ and the significant factors was Equation (11). The R² and adjusted R² of the regression model were 0.939 and 0.878, respectively. It indicated that the fitting model was reliable.

$$Y_1=48.33-12.62X_1-37.38X_2-1.24X_4+25.77X_5-4.08X_1{X}_2+3.43X_1{X}_4-14.63X_4{X}_5+11.95X_2^2+78.53X_5^2$$

(10)

$$Y_2=18485.10-4606.06X_1-12411.53X_2+519.33X_4-17162.46X_5+1370.83X_1{X}_2+182.26X_1{X}_4+8804.46X_2{X}_5+208.93X_1^2+1646.45X_2^2-75.63X_4^2$$

(11)

In order to further determine the optimal parameter combination, the Numerical module of Design-Expert 12.0 software was used to optimize the solution. The solution boundary conditions were set as 1.88 × 10¹⁰ N·m⁻³ ≤ X₁ ≤ 2.12 × 10¹⁰ N·m⁻³, 1.86 × 10¹⁰ N·m⁻³ ≤ X₂ ≤ 2.00 × 10¹⁰ N·m⁻³, 5.8 × 10⁶ Pa ≤ X₄ ≤ 6.2 × 10⁶ Pa, 0.36 × 10⁻³ m ≤ X₅ ≤ 0.40 × 10⁻³ m, Y₁ = 1.209 × 10⁻³ m, and Y₂ = 94.656 N. It can be obtained that X₁, X₂, X₄, and X₅ were 1.988 × 10¹⁰ N·m⁻³, 1.931 × 10¹⁰ N·m⁻³, 6.010 × 10⁶ Pa, and 0.381 × 10⁻³ m, respectively. Simulation tests on the optimized parameters show that Y₁ and Y₂ are 1.217 × 10⁻³ m and 93.623 N, respectively, and the relative errors with the real values are 0.66% and 1.09%, respectively.

3.3. Adaptability Verification of Model Parameters

In this study, the multi-spheres autofill model and the bonded particle model were, respectively, used to carry out the free fall test, inclined plane sliding test, and inclined plane tumbling time test, in order to verify the feasibility of using the calibration parameters of the multi-spheres autofill model on the bonded particle model.

(1) Free fall test: The pellet fell freely from a fixed height and bounced after collision with the stainless steel bottom plate. The ratio between the maximum height of the first rebound and the initial height was calculated.

(2) Inclined plane sliding test: A single pellet was statically placed on the stainless steel plate, and the stainless steel plate rotated at an angular speed of 5°·s⁻¹ until the pellet began sliding. The rotation angle of the stainless steel plate was measured and recorded.

(3) Inclined plane rolling time test: The stainless steel plate was fixed at 45°, a single pellet was placed statically at the end of the stainless steel plate and released, and the total time was recorded until it rolled out of the stainless steel plate.

The simulation test results of the two models are shown in

Table 12. Simulation results of two pellet models.

Tset	Investigation Parameter	Model		Relative Error/%
		Multi-Spheres Autofill Model	Bonded Particle Model
Collision test	Initial height/10−3 m	128.647	128.574	0.578
	Post collision height/10−3 m	64.242	63.873
Sliding test	Initial sliding angle/°	26.733	26.571	0.610
Rolling test	Rolling off time/s	0.431	0.427	0.937

. It can be seen from the test results that the relative errors of the three test results of the two models were 0.578%, 0.610%, and 0.937%, respectively. Furthermore, the motion states of the two models were consistent, indicating that it is feasible to apply the interaction parameters of the multi-spheres autofill model to the bonded particle model.

4. Discussion

The variance analysis results (Table 6) show a strong correlation between the static friction coefficient, the restitution coefficient, and the rolling friction coefficient of pellet–pellet with the angle of repose. It is consistent with the research results of Peng et al. [16] but inconsistent with those of Liao et al. [34,35] and Guo et al. [36]. This is explained by the fact that agricultural materials have complex properties. The same materials have different properties, such as hygroscopic, moisture content, constituent, dimension, etc. The properties of different materials vary greatly. Generally, the greater the friction coefficient between pellets, the more excellent the resistance to downward movement during accumulation. The larger the restitution coefficient, the smaller the energy dissipation of pellets after a collision, which increases the stability time in pellet accumulation. The material upon which the pellets fall has an important influence on the angle of repose. Therefore, under the joint action of the three factors, the angle of repose will be affected to varying degrees.

The breaking displacement and ultimate breaking force were both selected as indicators, which was not similar to that reported by other authors in the bonding parameter’s calibration simulations [10,37]. At the same time, the placement state (along the diameter or axial direction) of the pellet during compression will also affect the calibration results of its bond parameters. However, considering the actual pellet end shape and the pellet stacking state in the stacking experiment (Figure 8a), it is more practical to calibrate the bonding parameters along the radial compression of the pellets. It is assumed that the material properties of the pellet are isotropic in the simulations. The actual situation is that there may be defects in the process of pellet manufacturing, such as cracks, uneven moisture content, etc., resulting in anisotropy of material properties. Therefore, standards for pellet parameter calibration shall be formulated, and uniform samples should be adopted, including size, moisture content, etc.

After extruding the gluten pellets, they must be cooled and dried in the drum dryer. The accurate calibration of the discrete element simulation parameters of the pellet feed is of great significance for optimizing the mechanical parts, contacting the pellets, and determining the drying process. Based on previous research methods and practical tests, this paper calibrates the parameters by a discrete element simulation, but there are still some limitations. In this test, only pellets with a moisture content of 10.18% (d.b.) are studied, and the change in the pellet’s properties with the moisture content is not considered. At the same time, the simulation pellet size is the average value without considering the impact of the pellet’s size. In future research, the above factors should be comprehensively considered to obtain the discrete element parameters of gluten pellets under different conditions.

5. Conclusions

(1) For gluten pellets, the parameters required for constructing the discrete element model and the bonded particle model are yet to be calibrated, such as the pellets’ material parameters, the contact parameters (pellet–pellet and pellet–stainless steel), and the bonding parameters.

(2) It has been confirmed that the interaction parameters (static friction coefficient, restitution coefficient, and rolling friction coefficient) between pellets significantly impact the angle of repose and have a very close correlation (R² = 0.936) with the angle of repose.

(3) It has also been found that the bonding parameters (normal stiffness per unit area, shear stiffness per unit area, critical shear stress, and bond disk radius) significantly influence the breaking displacement and ultimate breaking force. Moreover, the correlation coefficients are R² = 0.978 and R² = 0.939, respectively.

(4) The simulation results show that the motion states of the multi-spheres autofill model and the bonded particle model with the same interaction parameters are the same. The simulations can reference pellet breakage in production, processing, and transportation.