Parameter Optimization Method for Centrifugal Feed Disc Discharging Based on Numerical Simulation and Response Surface

Abstract

In this study, a centrifugal feeding disc device is proposed. To investigate the influence of the process parameters on the discharging efficiency and the lifting of the discharging efficiency, the centrifugal feeding disc device was dynamically simulated based on the discrete element method (DEM), and the simulation results were experimentally verified. Based on the quadratic regression orthogonal test method, a significant lossless regression model of process parameters and discharging efficiency was established, and the response surface of the interaction of process parameters was obtained. The results indicated that the order of influence of the process parameters on the discharging speed of the centrifugal feeding disc was as follows: outer turntable speed > inner turntable speed > inner turntable tilt angle > conical turntable angle. The interaction of the conical turntable angle and the inner turntable tilt angle had the greatest influence on the centrifugal feed disc discharge efficiency. The response surface method (RSM) was used to optimize the process parameters, and the optimal combination of process parameters included an outer turntable speed of 135 r/min, an inner turntable speed of 64 r/min, an inner turntable tilt angle of 7°, and a conical angle of 15°. The discharged efficiency of the optimized centrifugal feeding disc device was increased by 31.9%.

1. Introduction

Centrifugal feeding disc equipment primarily functions as a device that provides organized and precise upward feeding through a combination of mechanical and electromechanical systems, utilizing rotating discs, sensors, and other components. This equipment is characterized by its low noise, adjustable speed, and adaptability to varying workloads, making it widely applicable in packaging production lines across industries such as chemical manufacturing, pharmaceuticals, food processing, agriculture, and light industrial machinery [1,2,3,4]. Pre-filled syringes [5,6,7] are extensively used in fields like pharmacy, biology, and healthcare [8,9,10]. The feeding process of the push rod device for pre-filled syringes is a critical step in the packaging production line. Currently, centrifugal feeding discs are commonly employed to supply small materials with distinct physical characteristics. Therefore, studying the centrifugal feeding disc to improve discharge efficiency is essential. The centrifugal feeding disc device was developed from the centrifugal vibrating disc [11,12,13], but there is limited research on it both domestically and internationally.

Regarding feeding performance, Matteo Bottin [14] developed and validated a dynamic model that describes the interactions between components of the blade feeding system in a rotary feeder. Xingjian Huang [15] proposed a shape optimization method, where the optimized curve design increased the mass flow region in the hopper by over 90%. The optimized hopper also significantly reduced particle segregation by approximately 70% during the discharge process. Sun et al. [16] focused on an efficient feeding head as the research object and conducted a dynamic analysis of the material within it. Through EDEM+FLUENT simulation analysis, they derived an equation for the conveying volume of the feeding head.

In terms of centrifugal spreading performance, Yuan Hao [17] analyzed the feeding characteristics of sticky miscellaneous fish pellet feed within the silo through feeding experiments and discrete element numerical simulations. The discharge device was optimized, including improvements in the silo structure, the design of the stirring device, and the optimization of the stirring device’s rotational parameters. Liedekerke et al. [18] used EDEM to simulate and verify the fertilizer distribution of a centrifugal fertilizer spreader. Liu et al. [19] optimized the structural parameters of the fertilizer diverter plate using EDEM software(2022) and orthogonal experiment analysis.

Regarding screw feeding performance, Lian [20] conducted a numerical study on the complex mixing and feeding process of binary mixtures of coal and cylindrical biomass particles in a screw feeder. The study investigated the effects of the biomass feed ratio, feed rate, and screw speed on feeding performance. Bell [21] established a quantitative relationship between the material properties, process settings, and screw feeder response in high-throughput feeders using a multivariate model (PLS). The study also investigated the relationships between the torque, screw pitch, and material stack height above the inlet, as well as the impact of the drive power and screw pitch on productivity.

In other studies, Lee et al. [22] utilized the population balance model and discrete element method (DEM) to investigate the grinding characteristics of a centrifugal mill with varying G/D (gyration/mill diameter) ratios. A series of grinding tests was conducted on illite samples using a centrifugal mill under various conditions, and the breakage parameters were calculated. Xie et al. [23] optimized the robot polishing process parameters, including the polishing pressure, feed rate, and tool speed.

In this study, a centrifugal feeding disc is proposed. To investigate the influence of the feeding disc’s process parameters on its feeding efficiency, a numerical simulation model of the centrifugal feeding disc was first established and experimentally verified. A single-factor analysis was performed on the primary factors of the centrifugal feeding disc, followed by the application of the response surface method (RSM) to explore the effects of the process parameters on the feeding efficiency and optimize these parameters.

2. Structure and Modeling of Centrifugal Feeding Disc

2.1. Structure of Centrifugal Feeding Disc Device

The centrifugal feed disc (Wuxi, China) assembly primarily consists of a frame, a three-phase asynchronous motor, an inner turntable, an outer turntable, a discharge conduit, a feed baffle, and a transmission system, as illustrated in Figure 1. The two motors independently control the rotational speeds of the inner and outer turntables, with speed adjustments made via the transmission system. The feeding device allows the push rod material to descend by free fall onto the inner turntable of the centrifugal feed disc, from which it is discharged through the outlet into the discharge pipe, achieving automatic directional sorting and feeding. The most critical components of the centrifugal feed disc are the outer turntable and the inner turntable.

2.2. Working Principle

The centrifugal feeding disc is driven by two three-phase asynchronous motors. Once the motors are activated and the turntables begin rotating, the push rod material is fed directly onto the centrifugal turntable. Through free-fall motion, the material descends from the inlet onto the inner turntable, moving toward its edge due to the rotation. Gradually, the material transitions from the bottom to the gap between the inner and outer turntables. Owing to the physical properties of the push rod, it becomes lodged in the gap, creating a directional arrangement on the centrifugal turntable. The outer turntable then provides the horizontal circular motion, guiding the material to the discharge port in an orderly and directed manner.

2.3. Particle Mechanics Model Analysis

The essence of the feeding process is to transfer energy through the contact and collision between particles and turntable, particles and shell, particles and particles, and finally achieve the purpose of moving particles and feeding. The Hertz–Mindlin model and the JKR model (Johnson–Kendall–Roberts model) are used to simulate the collision process between particles and geometry and between particles [24]. The theoretical diagram is shown in Figure 2. Through the contact mechanics analysis, the following results can be obtained:

$$F_{n,d}=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{k_n{m}^{*}}{v}_{n,rel}$$

(1)

$$F_t=-k_t{a}_t$$

(2)

$$F_{t,d}=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{k_t{m}^{*}}{v}_{t,rel}$$

(3)

where $F_{n,d}$ is the normal damping force, N; $F_{t,d}$ is the tangential damping force; $v_{n,rel}$ is the normal relative velocity; $v_{t,rel}$ is the tangential relative velocity, m/s; $\beta$ is the dimensionless factor related to the recovery coefficient; $a_t$ is the tangential overlap, m; and $m^{*}$ is the equivalent mass, kg.

In the calculation of the directional damping force, tangential elastic force, and tangential damping force, the Hertz–Mindlin model combined with the JKR model is consistent with the Hertz–Mindlin (no slip) model. However, the normal elastic force of the JKR model depends on the overlap between particles, the surface energy of particles, and the interaction parameters.

The specific calculation formula is re-listed below:

$$F_{JKR}=-4\sqrt{\pi \cdot \gamma \cdot E^{*}}{a}^{\frac{3}{2}}+{\displaystyle \frac{4E^{*}}{3R^{*}}}{a}^3$$

(4)

$k_n$, $\beta$, and $k_t$ can be calculated using the following formula:

$$k_n=2E^{*}\sqrt{R^{*}a}$$

(5)

$$\beta ={\displaystyle \frac{\ln e}{\sqrt{{\ln }^{2}e+{\pi }^2}}}$$

(6)

$$k_t=8G^{*}\sqrt{R^{*}a}$$

(7)

where $e$ is the recovery coefficient; $G^{*}$ is the equivalent shear modulus, MPa; $R^{*}$ is the equivalent particle radius; $E^{*}$ is the equivalent elastic modulus, Pa; $a$ is the overlap between particles, m; and $\gamma$ is the surface energy of particles.

2.4. Establishment of Push Rod Model

The structure of the centrifugal feed disc is discretized into numerous small elements, which are then subjected to mechanical analysis. By simulating these discrete elements, the motion and deformation of the centrifugal feed disc under various working conditions can be assessed, facilitating design optimization. The physical representation is shown in Figure 3a. Initially, the three-dimensional simulation data are generated using SolidWorks and saved as a 3D model in IGS format, with its shape characteristics and basic structure depicted in Figure 3b. Since it is necessary to investigate the impact of key material parameters on the degree of centrifugation, and given that the material essentially resembles a nail-shaped column, the 3D model is simplified into a cylindrical form. By measuring the dimensions of the material particles, the positions of the basic spherical particles are adjusted in the DEM software (2022) to form nail-shaped columnar particles. The appearance of the material during the simulation is illustrated in Figure 3c.

2.5. Contact Model Selection and Parameter Setting

In the simulation, the texture of the push rods is assumed to be uniform, with identical physical parameters—including Poisson’s ratio, shear modulus, and density—applied across all push rods. The inner and outer turntables are made of polyoxymethylene (POM), while the push rod material is polypropylene (PP). The shell of the centrifugal feeding plate and the discharge conduit are constructed from Q235 steel. The material properties are detailed in

Table 1. Material attribute parameters of simulation model.

Materials	Poisson’s Ratio	Shear Modulus/GPa	Density/g·cm−3
PP	0.42	0.89	900
POM	0.38	2.60	1420
Q235 Steel	0.27	70.0	7800

, and the contact parameters between the materials are presented in

Table 2. Simulation model contact parameters.

Contact Form	Recovery Coefficient	Static Friction Factor	Dynamic Friction Factor
Push rod–Push rod	0.3	0.36	0.05
Push rod–Turntable	0.2	0.4	0.08
Push rod–Shell body	0.2	0.3	0.01

.

The whole simulation process is 8 s, the time step is 0.68778 μs, the data storage time interval is 0.05 s, the calculation grid size is 2 mm, and the number of material feed per second is 10.

3. Numerical Simulation Validation and Analysis

3.1. EDEM Feeding Simulation and Test

The model was then imported into the DEM software, where the relevant parameters, including the inner and outer rotational speeds, were set. During DEM post-processing, the calculation domain was established at the discharge port, and the number of discharges within the first 8 s of the simulation was recorded. This discharge count within the specified timeframe serves as the evaluation index for measuring the discharge efficiency. The average discharge count was obtained from three simulation runs. In the company workshop, the inner and outer turntable speeds, as well as the feed speed, were adjusted via transmission to closely match the parameters used in the simulation. The system was then run for 8 s, and the process was repeated multiple times to obtain an average value. The simulation and testing of the centrifugal feeding turntable’s discharge process are illustrated in Figure 4. The measured data are presented in

Table 3. Workshop measured data.

Outer Turntable Speed/r/min	Number of Discharges/Pieces
	The First Time	The Second Time	The Third Time	Average Value
40	46	49	47	47
60	50	51	53	51
80	54	52	53	54
100	59	58	58	58
120	63	60	63	62

. Figure 5a depicts the transmission system that controls the turntable speed, Figure 5b shows the irregular feeding state of the material, Figure 5c illustrates the material entering the turntable frame and beginning the discharge process, and Figure 5d displays the statistics after discharge.

Figure 6 shows that the data trends from both the test and the simulation are largely consistent, with a maximum error of 9.6%. This indicates that the simulation test is reliable and effective, demonstrating the accuracy of the results obtained through discrete element simulation. The observed error may be attributed to factors such as losses in motor transmission during the test and the inability to precisely control the feed rate to exactly 10 units per second.

3.2. Analysis of Discharge per Second

To verify the stable operation of the centrifugal feeding plate, a long-term simulation was conducted for 50 s, with the results displayed in Figure 7. As shown in Figure 7, the number of material discharges exhibits a linear upward trend after the initial 3 s. Figure 7 further illustrates that once the device reaches stable operation, the discharge rate stabilizes between 4 and 12 discharges per second, with the majority of discharges concentrated in the range of 7 to 11 per second. These findings indicate that the device achieves stable operation after approximately three seconds.

3.3. One-Factor Simulation Analysis

3.3.1. Influence of Inner and Outer Turntable Speed on Discharge Efficiency

The original speed of the inner turntable was 49.0 r/min, and the outer turntable speed was 41.4 r/min. The outer turntable speed was then set to 100.0 r/min, while the inner turntable speed was varied at 40.0 r/min, 60.0 r/min, 65.0 r/min, 70.0 r/min, 80.0 r/min, and 120.0 r/min, respectively, to simulate the pusher during the centrifugal feeding process. The discharge quantities under these conditions were compared, and the simulation results are shown in Figure 8a. Similarly, with the inner turntable speed fixed at 50.0 r/min, the outer turntable speed was set at 60.0 r/min, 80.0 r/min, 100.0 r/min, 120.0 r/min, and 140.0 r/min to simulate the process, and the discharge quantities were compared, as illustrated in Figure 8b.

From the simulation results in Figure 8a,b, it can be observed that when the inner turntable speed is constant, the number of discharges increases sharply and then stabilizes as the outer turntable speed increases. Conversely, when the outer turntable speed is constant, the number of discharges increases steadily and then stabilizes as the inner turntable speed rises. Figure 8a shows that when the inner turntable speed reaches 65.0 r/min, further increases in speed have a minimal impact on the discharge quantity. Figure 8b indicates that, with a fixed inner turntable speed, the number of discharges continues to rise as the outer turntable speed increases up to 140.0 r/min. The discharge quantity is highest when the outer turntable speed ranges from 120.0 r/min to 160.0 r/min. However, when the outer turntable speed exceeds 160.0 r/min, although the discharge quantity increases further, the higher speed leads to significantly increased energy consumption.

In summary, the optimal range for the inner turntable speed is 60.0 r/min to 70.0 r/min, and for the outer turntable speed it is 120.0 r/min to 160.0 r/min.

3.3.2. Influence of Tilt Angle of Inner Turntable on Discharge Efficiency

In the original model, the tilt angle δ of the inner turntable was 11.3°. To assess its impact on the feeding process, the tilt angle δ was varied at 5.0°, 7.5°, 10.0°, 12.5°, and 15.0°, while maintaining the inner turntable speed at 50.0 r/min and the outer turntable speed at 60.0 r/min. The simulation aimed to evaluate the feeding process of the pusher in the centrifugal feeding turntable, comparing the number of discharges while ensuring the efficiency and stability of the turntable settings. The simulation results are shown in Figure 8c.

As depicted in Figure 8c, the number of discharges initially increases and then decreases as the tilt angle δ increases. When the tilt angle δ was 15.0°, the number of discharges was at its lowest, with an average of 45 discharges over three simulations, each lasting 8 s. Conversely, when δ was 7.5°, the discharge count was at its highest, with an average of 56 discharges over the same duration and number of simulations. Therefore, the optimal range for the inner turntable’s tilt angle δ was determined to be between 5.0° and 10.0°.

3.3.3. Influence of Conical Turntable Angle on Discharge Efficiency

The conical turntable of the inner turntable was mounted on top of the inner turntable to secure it in place, with the angle formed between the turntable and the horizontal plane of the inner turntable referred to as the conical angle θ. In the original model, the conical angle θ was 11.5°. To evaluate the impact of varying this angle on the feeding process, simulations were conducted with conical angles of 9.0°, 12.0°, 15.0°, 18.0°, and 21.0°. The inner turntable speed was set at 50.0 r/min, and the outer turntable speed at 60.0 r/min. The simulation results, showing the number of discharges per second, are illustrated in Figure 8d. As observed in Figure 8d, the number of discharges initially increases and then decreases with the increase in conical angle θ during the 8 s simulation. When θ was 15.0°, the discharging performance was optimal, with an average of 53 discharges across three simulations. Conversely, when θ was 9.0°, the discharging performance was the least effective, with an average of 47 discharges. Therefore, the optimal conical angle θ for achieving the best discharging effect lies between 12.0° and 18.0°.

3.3.4. Influence of Outer Turntable Radius on Discharging Efficiency

In the original model, the radius of the outer turntable in the centrifugal feeding turntable device was denoted as r. To assess the effect of varying this radius, simulations were conducted with the outer turntable radius set to 0.4 r, 0.6 r, 0.8 r, 0.9 r, 1.1 r, 1.2 r, 1.4 r, and 1.6 r. To ensure the normal operation of the centrifugal turntable device, the radius of the inner turntable was also adjusted accordingly. However, due to its angled placement, the inner turntable’s radius does not change in direct proportion to that of the outer turntable. The simulation was conducted with an inner turntable speed of 50.0 r/min, an outer turntable speed of 60.0 r/min, and a feed rate of 10 feeds per second, over a simulation time of 12 s. The results are presented in Figure 8e.

As shown in Figure 8e, the outer turntable radius of 0.4 r and 0.6 r had a significant impact on the simulation results. The reason for this is that as the radius decreases, materials are more likely to collide within the turntable, leading to a lower efficiency in correctly entering the trough at the feed opening. When the outer turntable radius was between 0.8 r and 1.6 r, the centrifugal turntable was large enough to reduce the likelihood of material collisions at the upper feed opening, resulting in a minimal impact on the discharge results.

In conclusion, the effects of various factors, including the inner and outer turntable speeds, the inner turntable tilt angle, the conical turntable angle, and the outer turntable radius, on the discharging process were studied. Among these, the turntable speeds, inner turntable tilt angle, and conical turntable angle were found to have a significant influence on the discharging efficiency. For further investigation, a response surface optimization design was conducted to analyze the primary and secondary influences of these four factors.

3.4. Response Surface Optimization Design

RSM is a comprehensive optimization method used in experimental design [25,26,27]. RSM fits a function by conducting experiments on multiple influencing factors that describes the relationship between these factors’ interactions and the outcome, identifying the optimal levels for each factor to achieve the best possible result. The optimal working ranges for the inner turntable speed, outer turntable speed, inner turntable tilt angle, and conical turntable angle were initially determined through single-factor tests. Based on these optimal ranges, a four-factor, three-level responsive surface design was conducted, with the factor level table presented in

Table 4. Experimental factor level.

Level	Outer Turntable Speed A (r/min)	Inner Turntable Speed B (r/min)	Tilt Angle of Inner Turntable C (°)	Conical Turntable Angle D (°)
1	120	60	5	12
2	140	65	7.5	15
3	160	70	10	18

. The responsive surface design compared the number of discharges for each factor parameter combination over an 8 s simulation, determining the optimal parameter for the centrifugal feeding turntable device and the relative influence of each factor on discharge efficiency.

3.4.1. Parameter Selection and Experimental Design

Based on preliminary single-factor experiments, four factors were identified as influencing the discharge efficiency of the centrifugal feeding turntable device: the inner turntable speed, outer turntable speed, inner turntable tilt angle, and conical turntable angle. Working ranges for these factors were established. Within these optimal ranges, the Box–Behnken design was employed to plan the experiments. The selected factors were the inner turntable speed (A), outer turntable speed (B), inner turntable tilt angle (C), and conical turntable angle (D), each at three levels. The factor levels are shown in Table 4. By comparing the number of discharges over an 8 s simulation for each factor combination, the optimal parameter range for the centrifugal feeding turntable device was determined, along with the degree to which each factor influenced discharge efficiency.

For the 27 experimental designs, simulations were conducted under the previously specified conditions, with all other parameters set to their default values. Each simulation ran for 8 s, during which the number of discharges was recorded. The data obtained are presented in the following table. The response surface experimental design included four factors and three levels, and the corresponding results are shown in

Table 5. Test plan and results.

No.	Factors				Number of Discharges
	Outer Turntable Speed A (r/min)	Inner Turntable Speed B (r/min)	Tilt Angle of Inner Turntable C (°)	Conical Turntable Angle D (°)
1	2	2	2	2	62
2	3	1	2	2	44
3	3	3	2	2	35
4	3	2	2	3	40
5	2	3	3	2	39
6	2	2	3	3	50
7	3	2	3	2	36
8	1	2	2	3	53
9	2	2	3	1	44
10	2	1	3	2	44
11	2	2	1	3	42
12	2	1	2	3	57
13	2	1	1	2	55
14	2	2	2	2	64
15	1	1	2	2	48
16	2	1	2	1	52
17	2	2	2	2	61
18	3	2	2	1	50
19	2	3	2	3	41
20	1	3	2	2	46
21	1	2	2	1	51
22	2	3	2	1	52
23	2	3	1	2	41
24	2	2	1	1	54
25	1	2	3	2	45
26	1	2	1	2	54
27	3	2	1	2	34

.

3.4.2. Establishment and Validation of the Regression Equation

An analysis of variance (ANOVA) and significance tests were conducted on the results presented in Table 5. A quadratic polynomial was used to fit the model, as shown in

Table 6. Analysis of extreme results.

Source of Variation	Sum of Squares	Degree of Freedom	Mean Square	F Value	p Value
model	1674.25	14	119.59	20.41	<0.0001
A	283.32	1	283.32	48.36	<0.0001
B	176.57	1	176.57	30.14	0.0001
C	41.85	1	41.85	7.14	0.0203
D	29.52	1	29.52	5.04	0.0444
AB	12.32	1	12.32	2.10	0.1726
AC	32.34	1	32.34	5.52	0.0367
AD	33.11	1	33.11	5.65	0.0349
BC	18.77	1	18.77	3.20	0.0987
BD	67.13	1	67.13	11.46	0.0054
CD	83.72	1	83.72	14.29	0.0026
A2	571.21	1	571.21	97.50	<0.0001
B2	334.67	1	334.67	57.13	<0.0001
C2	529.12	1	529.12	90.32	<0.0001
D2	84.91	1	84.91	14.49	0.0025
Residual error	70.30	12	5.86
Lack of fit	64.64	10	6.46	2.29	0.3426
Pure error	5.66	2	2.83
Total value	1744.55	26

. The F-value and p-value represent the significance of the relevant influencing factors. The F-value of this model was 20.41, indicating that the model was significant. A p-value of less than 0.05 suggests that the model is statistically significant, while a p-value of less than 0.01 indicates a highly significant model. From the ANOVA results, the p-values of factors A, B, A², B², and C² were found to be significant. The larger the F-value, the greater the influence of the corresponding factor on the response variable. The ranking of factors affecting the discharge efficiency from highest to lowest was as follows: A >B >C > D (primary term), CD > BD > AD > AC > BC > AB (interaction term).

The regression equation that models the discharge efficiency of the centrifugal feeding turntable device, considering the combined effects of outer turntable speed, inner turntable speed, inner turntable tilt angle, and conical angle, is as follows:

$$\begin{array}{l}{R}_a=62.25-4.86A-3.84B-1.87C-1.57D\\ -1.76AB+2.84AC-2.88AD+2.17BC-4.10BD+4.58CD\\ -10.35A^2-7.92B^2-9.96C^2-3.99D^2\end{array}$$

After the regression model was obtained, the regression diagnosis function was further used to judge the fitting results of the regression model, as shown in the Figure 9. Figure 9a shows whether the residuals conformed to a normal distribution. If the points are roughly distributed along this red diagonal, the distribution of the residuals is approximately normal, and all the points in the graph should be arranged close to this line to meet the normal distribution requirement. Figure 9b shows the relationship between the residuals and the predicted values. Ideally, the residuals should be randomly distributed and should not exhibit any patterns or trends. No obvious pattern is seen in this plot, indicating that the linear regression assumption is satisfied.

Based on the response surface experimental data and software analysis, response surfaces and contour plots were generated to illustrate the interactions between various factors. These visualizations show how changes in these factors influence the number of discharges. In the response surface plots, a steeper slope indicates a more significant interaction between factors, while a flatter slope suggests a weaker interaction effect. In the contour plots, an elliptical shape signifies a significant interaction between factors, whereas a circular shape suggests a weaker interaction. The significance of the interaction can also be inferred from the rate of color change: a faster color change indicates a stronger interaction effect, meaning a greater impact on the number of discharges.

By analyzing the contour plots with these properties in mind, significant interaction effects were observed between the outer turntable speed and the inner turntable tilt angle, as well as between the conical angle and the inner turntable speed, and between the inner turntable tilt angle and the conical angle. Other factors showed no significant interaction effects.

From the response surfaces in Figure 10a,c, it can be seen that in Figure 10a the slope of the inner turntable speed is relatively moderate compared to the slope of the outer turntable speed, indicating that the quadratic term of the inner turntable speed had a smaller effect on the number of discharges than that of the outer turntable speed. In Figure 10c, the slope of the outer turntable speed is steeper compared to that of the conical angle, indicating that the quadratic term of the outer turntable speed had a significant impact on the number of discharges, while the conical angle had a relatively smaller effect. Additionally, the contour plot in Figure 11c is elliptical, while the one in Figure 11a tends towards a circular shape, indicating that the interaction between outer turntable speed and the conical angle had a greater impact on the number of discharges from the centrifugal feeding turntable than the interaction between the outer turntable speed and inner turntable speed. This observation is consistent with the results from the analysis of variance (ANOVA). Similar analyses can be applied to other response surfaces and contour plots.

3.5. Optimization Design Analysis

Taking the discharge efficiency of the centrifugal feeding tray as the optimal comprehensive response index, the response surface method was used to find the best combination of process parameters in the design test interval, and the results are shown in Figure 12.

From Figure 12, based on the results of the response surface analysis, the optimal process parameters for the centrifugal feeding turntable device were predicted to be an outer turntable speed of 135.481 r/min, an inner turntable speed of 63.934 r/min, an inner turntable tilt angle of 7.068°, and a conical angle of 14.688°. The predicted number of discharges in 8 s was 63.

According to the optimal process parameters, the structural model of the feeding tray was re-established and simulated; the number of discharges in 8 s was 62, and the simulation and test error of the centrifugal feeding tray was only 1.5%, which proves that the optimization results are reliable and the optimization method is feasible.

4. Conclusions

In this study, a centrifugal feeding tray was proposed, and the performance of the feeding tray was simulated and experimentally studied. A DEM simulation and the response surface method were used to statistically analyze the discharging characteristics of the feeding tray. In addition, the response law of different process parameters to the discharge efficiency was discussed, and the process parameters were optimized. The results of the study showed the following:

• Based on the discrete elements method, DEM simulation software was used to carry out a numerical simulation analysis of the centrifugal feed tray device, and the simulation results were verified; the error was about 9.6%, and the simulation results were basically consistent with the prototype test results, which verified the feasibility of using DEM software for a numerical simulation analysis.
• The simulation analysis of five process parameters showed that the speed of the inner turntable, the speed of the outer turntable, the inclination angle of the inner turntable, and the angle of the conical turntable had a significant influence on the discharge result of the centrifugal feeding tray, and the radius of the outer turntable had no significant effect on the discharging result of the centrifugal feeding tray.
• The significant order of the influence of process parameters on the discharging speed of the centrifugal feeding tray was as follows: outer turntable speed > inner turntable speed > inner turntable tilt angle > conical turntable angle. The interaction of the conical turntable angle and the inner turntable tilt angle had the greatest influence on the centrifugal feed tray discharge efficiency.
• The optimal combination of process parameters obtained using the response surface method was an outer turntable speed of 135 r/min, an inner turntable speed of 64 r/min, an inner turntable tilt angle of 7°, and a conical angle of 15°. The discharged efficiency of the optimized centrifugal feeding tray device increased by 31.9%.