Influence of Rotor Cage Structural Parameters on the Classification Performance of a Straw Micro-Crusher Classifying Device: CFD and Machine Learning Approach

Abstract

The rotor cage is a key component of the classifying device, and its structural parameters directly affect classification performance. To improve the classification performance of the straw micro-crusher classifying device, this paper proposes a CFD-ML-GA (Computational Fluid Dynamics-Machine Learning-Genetic Algorithm) method to quantitatively analyze the coupled effects of rotor cage structural parameters on classification performance. Firstly, CFD and orthogonal experimental methods are used to qualitatively investigate the effects of the number of blades, length of rotor blades, and blade installation angle on the classification performance. The conclusion obtained is that the blade installation angle exerts the greatest effect on classification performance, while the number of blades has the least effect. Subsequently, four machine learning algorithms are used to build a cut size prediction model, and, after comparison, the Random Forest Regression (RFR) model is selected. Finally, RFR is integrated with a Genetic Algorithm (GA) for quantitative parameter optimization. The quantitative analysis results of GA indicate that with 29 blades, a blade length of 232.8 mm, and a blade installation angle of 36.8°, the cut size decreases to 47.6 μm and the classifying sharpness index improves to 0.62. Compared with the optimal solution from the orthogonal experiment, the GA solution reduces the cut size by 9.33% and improves the classifying sharpness index by 9.68%. This validates the feasibility of the proposed method.

1. Introduction

Given the exacerbation of the energy crisis, there has been an increasing focus on utilizing renewable resources to mitigate resource scarcity and alleviate environmental pollution [1,2]. Biomass resources, particularly crop residues, find extensive applications in various sectors, including composites [3,4], biodegradable stalks [5,6], hydrogen production [7,8], and battery materials [9,10], due to their abundance, reproducibility, cost-effectiveness, and carbon neutrality. The high-value utilization of crop straw relies on its crushing treatment. Micro-crushed straw powder offers numerous advantages, including having a large specific surface area, low melting point, good activity, and rapid dissolution speed [11]. The uniformity of straw powder particle size is a crucial indicator for assessing powder quality; however, the particle size distribution of straw powder processed by mechanical micro-crushing methods tends to be wide [12]. Comminution combined with particle classification is a crucial method for transforming raw materials into the desired particle size [13,14]. Thus, during crop straw crushing, it is essential to promptly separate straw powder that meets standards, while unqualified products are returned to the crushing device for secondary processing, preventing over-crushing and particle agglomeration [15].

In order to achieve efficient particle classification, the design and optimization of the classifying device are particularly important. The optimization of the structural parameters of the rotor cage, as the only rotating part in the classifying chamber, is a crucial method to enhance the classification performance of the classifying device. Compared to physical tests, the CFD method is suitable for the structural optimization study of the rotor cage in a classifying device due to its cost-effectiveness and time efficiency. Many scholars have conducted related studies. For instance, Liu et al. [16] investigated the effect of the number of rotor cage blades on classification performance using the CFD method. The results showed that if the number of blades is too small, the coarse particles in the product increase. Conversely, if the number of blades is too large, the device cannot separate qualified products in time, and both internal resistance and energy consumption increase. Yu et al. [17] compared and analyzed the effect of the inner and outer radius of the rotor cage on the flow field in the classifying chamber using the CFD method. The results showed that the inner and outer radius of the rotor cage directly affected the radial velocity distribution within the rotor cage channel and that they should not be excessively large or small. Jia et al. [18] analyzed the effect of an inverted conical rotor cage on the flow field and classification performance in the classifying chamber using the CFD method. The results indicated that the optimal structure is achieved when the ratio of the top surface diameter of the rotor cage to the bottom surface diameter is 1:0.8 and 1:0.7, meeting the requirements of industrial classification efficiency and reducing energy consumption. Ismail et al. [19] analyzed the effect of the length of rotor cage blades on classification efficiency using the CFD method. The results indicated that increasing blade length can enhance the production of fine particles. However, excessively long blades may hinder timely particle discharge from the classifying chamber. Ren et al. [20] designed a rotor cage with non-radial arc blades by analyzing the effect of the blade mounting angle on the flow field. CFD simulations of the structure revealed a significant reduction in the incidence angle at the entrance of the rotor cage and the absence of air vortex in the blade flow path.

The above literature indicates that the classification performance of a classifying device is influenced by various factors, and geometric optimization of the rotor cage is an effective approach to enhance its performance. However, most current studies adopt the control variable method, optimizing only a single structural parameter and ignoring the effect of multi-feature parameter coupling. Additionally, these studies primarily focus on simple empirical models and traditional CFD simulations, which have limited optimization effects. Therefore, existing optimization methods still have significant room for improvement in enhancing classification performance.

In recent years, Machine Learning Algorithms (MLAs) have flourished, with many studies utilizing these algorithms to predict and optimize experimental parameters. Machine learning offers new methods for parameter optimization by training predictive models with CFD data [21,22]. Wu et al. [23] utilized CFD computational data to train the GWO-SVR model, predicting nitrate decomposition rates with response times that were 259,200 times shorter than industrial-scale CFD simulations. Bakhtiari et al. [24] trained a neural network using CFD data to predict the hydrodynamic performance of a marine cycloidal propeller. Mohammadpour et al. [25] combined CFD and ML to predict nanofluid heat transfer in microchannel heat sinks, optimizing the microchannel geometry using four regression models: K-nearest neighbor, random forest, Gaussian process regression, and multilayer perceptron. The results showed that the K-nearest neighbor model provided more accurate predictions than the other MLAs. Liu et al. [26] combined CFD and ML to predict the multi-objective in-furnace combustion characteristics associated with pulverized coal injection operation. By comparing the accuracy of six ML models, the Random Forest model was selected as the predictive model, reducing the response time by nearly 16,000 times compared to CFD simulation. In summary, MLAs have powerful regression and classification prediction capabilities and are widely used in various engineering fields, but few studies apply MLAs for the structural optimization of rotor cages. Additionally, most studies use a single algorithm to predict and optimize structural parameters, which lacks quantitative optimal solutions.

To address the challenges in quantitatively analyzing rotor cage structural parameters, coupling effects on classification performance, and the high computational costs of industrial-scale simulations, this paper proposes a novel CFD-ML-GA (Computational Fluid Dynamics-Machine Learning-Genetic Algorithm) methodology. The methodology aims to optimize rotor cage structural parameters and enhance the classification performance of the classifying device. Initially, the rotor cage classifying devices with varying structures is modeled using CFD and orthogonal experiments, leading to qualitative conclusions via analysis of variance. Subsequently, four machine learning algorithms—Decision Tree Regression (DTR), Support Vector Regression (SVR), Random Forest Regression (RFR), and Artificial Neural Network (ANN)—are used to develop a cutting particle size prediction model. Among these, RFR is identified as the most effective model, and its accuracy is further validated through correlation analysis, SHAP, and PDP methods. Finally, a Genetic Algorithm (GA) combined with the RFR model is used for the quantitative optimization of rotor cage geometric parameters. The integration of CFD and machine learning not only reduces computational resources and experimental costs but also provides a scientific foundation for designing key engineering equipment. The multidisciplinary approach used in this paper can inform and inspire research in related fields.

2. Materials and Methods

2.1. Numerical Modeling of the Straw Micro-Crusher Classifying Device

2.1.1. Physical Model

The straw micro-crusher designed and manufactured by our research team can perform both crushing and classifying operations. Its composition, primarily consisting of a crushing device and a classifying device, is shown in Figure 1. After coarse crushing, straw material approximately 10 mm in size is fed into the crushing chamber through the feeding pipe. In the crushing chamber, straw materials are subjected to the impact of the crush cutter and shear forces. Simultaneously, the rotating airflow propels the materials against the crushing chamber wall, resulting in crushing through the combined effects of impact, shear, friction, and collision. With the air inlet positioned below the crushing chamber, the crushed material enters the classifying chamber along the conveying channel due to the negative pressure airflow. The rotor cage is located in the classifying chamber, and its primary function is to separate coarse and fine particles. The rotor cage rotates at high speed to generate centrifugal force opposing the negative pressure. The straw powder entering the flow area between the blades is subjected simultaneously to the centripetal force of the negative pressure airflow, gravity, and the centrifugal force generated by the rotor cage. The particles smaller than the cut size enter the cyclone collection device through the discharge pipe and valve, while the particles larger than the cut size are flung back to the crushing area to be further crushed along the inner side of the flow guiding loop due to their larger mass.

2.1.2. Computational Model

In this paper, the structural parameters of the classifying device are optimized with the objective of improving its classification performance. Based on the mechanism and working principle of the straw micro-crusher, the fluid space inside the classifying chamber is selected as the calculation area. SolidWorks software 2020 is used to model the classifying device, with its main structural dimensions shown in Figure 2. The bottom surface of the rotor cage is the XOY plane (Z = 0 mm), with the center of this bottom surface as the coordinate origin. The original structure has 48 rotor blades, each with a length of 272 mm and a blade installation angle of 0°, and they are evenly distributed along the outer edge of the rotor cage.

2.1.3. Mathematical Models

CFD can be used to analyze the performance of the air classifier and predict the internal flow field [18,27]. The internal flow field of the classifying device is cyclonic, with the state of motion characterized by turbulent flow [28]. The RNG k-ε model is more capable of simulating high-strain flow and accurately simulates vortex flow; the use of this model is well documented in the literature [29,30]. Therefore, the RNG k-ε model is selected for numerical simulation in this paper.

The classifying device is simulated in three dimensions using ANSYS Fluent software 2022 R1. The gas-solid phases are treated separately for numerical simulations. Initially, the Eulerian method is used for the gas-phase solution, with the mathematical model primarily comprising the mass conservation equation and the energy conservation equation [31]. Secondly, the Lagrangian method is applied for the discrete phase model. This involves calculating the gas-phase flow field, solving for each particle’s velocity by combining the flow field variables, and then tracking the trajectory of each particle [32]. Considering the particle phase volume fraction at the inlet to be below 10% and thus neglecting the role of particles relative to the gas phase, a single-coupled discrete phase model is adopted to simulate the particle motion characteristics in the classifying chamber [33].

The equation of motion for discrete phase particles is:

$${\displaystyle \frac{d{u}_P}{dt}}=F_D(u_e-u_P)+{\displaystyle \frac{g({\rho }_s-{\rho }_e)}{{\rho }_s}}+F_x$$

(1)

where $F_D$ is the drag coefficient, given by

$$F_D={\displaystyle \frac{18\mu }{{\rho }_s{d}_P^2}}{\displaystyle \frac{C_{D}Re}{24}}$$

(2)

$$R{e}_P={\displaystyle \frac{{\rho }_e{d}_P|u_P-u_e|}{\mu }}$$

(3)

where $g$ is the gravitational acceleration; $\mu$ is the gas kinematic viscosity; $u_P$ is the particle velocity; $u_e$ is the gas velocity; ${\rho }_s$ is the particle density; ${\rho }_e$ is the gas density; $R{e}_P$ is the relative Reynolds number; $d_P$ is the particle diameter; $F_x$ is the additional acceleration; $C_D$ is the drag coefficient.

2.1.4. Boundary Conditions

To gain deeper insight into the effect of blade shape on classification performance, the other structural parameters of the classifying device are assumed to remain unchanged in the subsequent simulation analysis. Based on actual operating conditions, the process parameters of the classifying device are set, including inlet air velocity $v=8\mathrm{m}/\mathrm{s}$ and rotor cage speed $n=1200\mathrm{rpm}$. The material particle density is 2750 kg/m³, and single particles ($d_P=10,20,30,40,50,60,70,80,90,100\mathsf{\mu }\mathrm{m}$) are injected from the inlet surface. The Tromp curve is obtained by tracking the particle trajectory.

To couple the moving rotor cage with the stationary surrounding components, the Multiple Reference Frame (MRF) method is introduced, and the inner and outer surfaces of the cage are used as interfaces. The “velocity-inlet” boundary condition is applied to the air inlet, and the “pressure-outlet” boundary condition is used for the air outlet. The wall boundaries are defined using no-slip boundary conditions, and the near-wall surfaces are treated with standard wall functions. The detailed parameters used in the CFD modeling are listed in

Table 1. Simulation conditions.

Name	Conditions
Gas inlet	Velocity inlet, Escape
Gas outlet	Pressure outlet, Escape
Chassis walls	Reflect
Flow guiding loop walls	Reflect
Rotor cage walls	Reflect
Bottom of Classifying chamber walls	Trap
Pressure-velocity coupling	SIMPLE
Pressure discretization	Second Order
Momentum discretization	Second Order Upwind
Turbulent kinetic energy	Second Order Upwind
Turbulent dissipation rate	Second Order Upwind

.

2.1.5. Performance Evaluation Parameters

Cut size ($d_{50}$) and classifying sharpness index ($K$) are the standard evaluation indices of air classifiers [34,35]. After each CFD experiment, the number of particles captured for each particle size is counted, and the classification efficiency is then calculated according to the following equation:

$$\eta ={\displaystyle \frac{m}{m_0}}$$

(4)

where $\eta$ is the classification efficiency; $m$ is the number of particles of a given size in the trap; $m_0$ is the number of particles of a given size in the total number of particles.

$d_{25}$ and $d_{75}$ represent the corresponding particle sizes when the classification efficiency is 25% and 75%, respectively. According to the definition of the classifying sharpness index, the classifying sharpness index can be calculated using the following formula:

$$K=d_{25}/d_{75}$$

(5)

The classifying sharpness index ($K$) reflects the accuracy of particle size distribution after the classifying experiment of the classifying device on the material. The larger the value of $K$, the higher the classifying sharpness index; for ideal particle grading, the value of $K$ is 1. $d_{50}$ means that the probability of particles of this size entering both the fine and coarse powder outlets is 50%, meaning the classification efficiency is 50% at this point. As the most important indicator, obtaining a smaller $d_{50}$ is the current research hotspot [36,37].

2.1.6. Grid Independence Analysis

The simplified model of the classifying device is meshed using ANSYS Workbench Meshing software 2022 R1. Appropriate cell sizes are selected to generate the mesh. An overview of the meshes after delineation is shown in Figure 3a, where the entire area is divided into 3,044,700 grid cells. Due to the relatively narrow spacing between the rotor cage blades, the rotor cage region is treated with local mesh refinement.

The accuracy of the simulation calculations is ensured by the mesh sensitivity study, as shown in Figure 3b,c. When the number of meshes exceeds 3,044,700, the tangential velocity and pressure drop do not change significantly, continuing to increase the number of meshes has little effect on the computational results. Meanwhile, four measurement points at various positions were taken at the outlet of the rotor cage, and the axial velocities of the corresponding models under different grid numbers were measured, as shown in Figure 3d. It is observed that the axial velocities in the outlet region of the rotor cage remained essentially unchanged when the number of grids exceeded 3,044,700, indicating that the simulation results are independent of the mesh size. Therefore, the maximum total number of meshes is determined to be 3,044,700 to ensure the accuracy of the calculation results while saving calculation time and computational resources.

2.2. Orthogonal Experiment

2.2.1. Determination of Optimization Parameter Range

The blade is the core component of the rotor cage, and its structure determines the flow field distribution within the classifying device. In this paper, the number of rotor blades, the length of rotor blades, and the rotor blade installed angle are selected as three parameters for the orthogonal test.

The flowing zone between the blades is crucial for classifying and conveying fine powder. An excessive or insufficient number of blades can negatively impact classification; a reasonable number of rotor blades can improve the classification performance of the classifying device and reduce energy consumption. Based on previous scholarly research and engineering practice [16,38,39], the number of blades is selected between 24 and 64 for the study. Considering the height of the classifying chamber and the crushing chamber below, the length of rotor blades in this study is selected to be between 224 and 304 mm. When the airflow enters the rotor cage, it impacts the blades, forming a vortex between them and causing blade wear. The stability of the flow field can be effectively improved by adjusting the blade installation angle. Based on previous scholarly research on the airflow impact angle and blade installation angle [20,38], the blade installed angle is varied between 0° and 40° for this study.

2.2.2. Orthogonal Experimental Protocol Design

The orthogonal design of experiments is a fast and effective mathematical and statistical method for addressing complex multifactorial problems in mechanical engineering [40]. Using predetermined orthogonal tables significantly reduces the number of required experiments while maintaining the integrity of the study [41].

This section discusses the effect of blade geometry on the classification performance. The selected characteristic data include the number of rotor blades, the length of rotor blade, and the installation angle of rotor blade, denoted as A, B, and C, respectively. The performance of the classifying device is evaluated based on the cut size $d_{50}$ and classifying sharpness index $K$. The labeled data include cut size and classifying sharpness index. The objective is to ensure that the cut size ($d_{50}$) is as small as possible while maintaining a high classifying sharpness index ($K$), thereby identifying the blade geometry with the best classification performance. An orthogonal experimental design is employed to investigate the effect of three factors on the performance of the classifier. Six levels are selected for each factor, and the experiment is arranged according to the commonly used orthogonal design form, the L36 table. The factor levels are displayed in

Table 2. Table of orthogonal tests.

Level	Factors of Influence
	Number of Blades	Length of Blades/mm	Installed Angle of Blades/mm
1	24	224	0
2	32	240	8
3	40	256	16
4	48	272	24
5	56	288	32
6	64	304	40

.

2.3. Machine Learning Algorithms

Machine learning method is a data-driven data mining method based on data. The research idea of this paper using a machine learning algorithm is shown in Figure 4.

Implementation of machine learning algorithms such as DTR, SVR, RFR, ANN, etc., using Python language, details of these machine learning algorithms are given in the literature [42]. Given the extensive prior research on cut size prediction in classifying devices [36,37,43,44], it is established that cut size serves as an indicator of classification performance. In this paper, we apply four machine learning algorithms to construct a cut size prediction model and compare them to find the most accurate and stable prediction model. Subsequently, we utilize the selected model to predict the classifying sharpness index.

DTR, a classic algorithm in machine learning, differs from others by not relying on a fitting function in favor of constructing a mapping between inputs and outputs using logical language. Essentially, it segments the feature space into cells, each with its own output, with boundaries aligned parallel to coordinate axes. Test data is assigned to these cells based on features, yielding corresponding output values. Unlike classification tasks, DTR minimizes differences between sets using a squared error metric. The final prediction is based on the continuous variables derived from the decision tree.

SVR is a machine learning method based on statistical learning theory. One of its distinctive features is that the computation of model complexity does not depend on the dimensionality of the input data. SVR uses a nonlinear mapping to abstract the data into an n-dimensional feature space, and linear regression fits in the n-dimensional space by finding a hyperplane that can achieve the best fit to the data points.

RFR, a supervised machine learning algorithm, harnesses ensemble learning. It excels in combining various algorithms or iterations of the same algorithm to create more potent predictive models. Specifically, each tree within a random forest is trained on a subset of data, amalgamating multiple decision trees to determine the final output rather than relying on a single one. This process enhances stability and robustness. In this study, a random forest model comprising 80 decision trees predicts the cut size. Feature information guides the selection of optimal features and their respective thresholds for node splitting during decision tree construction.

ANNs play a significant role in machine learning algorithms, utilizing back-propagation of errors to minimize prediction errors through iterative weight updates using the steepest descent method. In this study, a four-layer neural network with two hidden layers (3-19-14-1) is used to establish a nonlinear mapping between input and output terms. The training process involves two nested loops: in the inner loop, forward propagation computes input and activation values for each hidden and output layer, followed by backpropagation to compute errors for both output and hidden layers, facilitating weight updates based on these errors.

The hyperparameter settings for the four prediction models are shown in

Table 3. The main design parameters of the MLAs.

Machine Learning Algorithms	Main Parameters
DTR	criterion: “mse”, splitter: “best”, max depth = 5
SVR	max terations:1000, learning rate = 0.01, C = 1
RFR	criterion: “mse”, n-estimator = 80, 6, max depth: None
ANN	n_epochs = 80, learning rate = 0.01

, and the optimal parameters are found by examining the learning curves.

In model evaluation, the performance of the prediction model is evaluated in this paper using two key metrics: the coefficient of determination ($R^2$) and the mean squared error ($MSE$). $R^2$ is used to assess the degree of fit between the predicted values and the true values, measuring the proportion of the variance in the dependent variable that is predictable from the independent variables. Meanwhile, the magnitude of deviation of the predicted values from the true values is expressed as $MSE$, computed by averaging the squares of the differences between predicted and true values. Thus, a higher $R^2$ value indicates a better fit of the model to the data, while a lower $MSE$ indicates higher accuracy. $R^2$ and $MSE$ are calculated as [45]:

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=0}^m{\left(y_i-\widehat{y_i}\right)}^2}}{{\displaystyle \sum _{i=0}^m{\left(y_i-\overline{y}\right)}^2}}}$$

(6)

$$MSE={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{(y_i-\widehat{y_i})}^2}$$

(7)

where $y_i$ is the true value; $\widehat{y_i}$ is the predicted value; $\overline{y}$ is the average value.

3. Results and Discussion

3.1. Analysis on the Results of Orthogonal Experiments

The results of orthogonal experiments can be analyzed using both extreme value analysis of variance and analysis of variance. Extreme value analysis of variance can reduce the influence of error through data transformation, improve the independence of factors, and accurately identify the primary and secondary factors affecting the classification performance of the rotor cage, as well as the optimal level and combination of each factor. In this section, the results of orthogonal experiment are processed and analyzed using extreme value analysis of variance. The calculation steps are outlined as follows:

• As an example, the mean value of the same class of experimental indicator $Ki$ is calculated for both the cut size $d_{50}$ indicator and the number of rotor blades.

  $$K1=\left(50.9+57.9+59.3+51.1+43.5+50.9\right)/6=52.3$$

  $$K2=\left(51.9+47.4+58.3+47.9+59.4+51.5\right)/6=52.7$$

  $$K3=\left(54.1+51.5+47.5+60.2+52.7+53.2\right)/6=53.2$$

  $$K4=\left(50.2+46.0+55.9+51.0+55.1+55.2\right)/6=52.2$$

  $$K5=\left(53.7+48.0+47.0+58.6+53.4+56.0\right)/6=52.8$$

  $$K6=\left(52.0+56.7+53.6+50.4+47.2+52.8\right)/6=52.1$$
• The extreme differences are calculated, which reflects the fluctuation of the evaluation indicators when the factor level changes. The formula for calculating the extreme value difference for each factor is as follows:

  $$R=\max (Ki)-\min (Ki)$$

  (8)

According to the above calculation method, the $Ki$ and $R$ values of the other two factors are calculated separately to obtain the extreme value difference analysis table of different experimental indicators, as shown in

Table 4. Results of the extreme variance analysis.

Test Indicators		A	B	C
Cut size	K1	52.3	49.4	56.5
	K2	52.7	49.6	56.0
	K3	53.2	51.6	54.3
	K4	52.2	54.9	50.4
	K5	52.8	54.8	49.5
	K6	52.1	55.1	48.6
	extreme variance R	1.1	5.7	7.9
	Superior level	A6	B1	C6
	Order of influence	C > B > A
Classifying sharpness index	K1	0.57	0.54	0.61
	K2	0.57	0.52	0.62
	K3	0.58	0.52	0.60
	K4	0.53	0.63	0.58
	K5	0.56	0.57	0.51
	K6	0.55	0.58	0.47
	extreme variance R	0.05	0.11	0.15
	Superior level	A3	B4	C1
	Order of influence	C > B > A

.

The larger the extreme value variance $R$, the greater the influence of the factors on the index. As can be seen from Table 4, for the cut size and classifying sharpness index, the factors have the same order of influence: the influence of rotor blade installed angle is the largest, followed by the length of rotor blades, and the number of rotor blades has the smallest influence. Therefore, when designing and optimizing the rotor cage of the classifying device, the appropriate rotor blade installed angle should be determined first, followed by the length of rotor blades, and finally the number of rotor blades, according to the actual demand.

To facilitate the observation of trends, the effects of different factor levels on performance evaluation indices are plotted in Figure 5. The trend of change in cut size with the increase in the number of rotor blades is not obvious; the cut size increases initially with the length of rotor blades and then tends to stabilize, showing a general positive correlation; the cut size decreases with the increase in the rotor blade installation angle, generally showing a negative correlation. The classifying sharpness index fluctuates with the increase in the number of rotor blades, reaching its highest at the K3 level and its lowest at the K4 level. The classifying sharpness index initially tends to stabilize with the increase in the length of rotor blades and then increases significantly, reaching its highest at the K4 level. The classifying sharpness index initially increases with the increase in the rotor blade installation angle and then decreases significantly, generally showing a negative correlation. In conclusion, the numerical simulation results indicate a significant nonlinear dependence between the input parameters (number of rotor blades, length of rotor blades, and rotor blade installation angle) and the output values (cut size and classifying sharpness index), primarily due to the presence of complex vortex behaviors within the classifying device. In addition, as seen from Figure 5, for different evaluation indicators, when one indicator is optimal, the other indicator is not necessarily optimal, and the combination of factors at the optimal level of each indicator differs. To obtain the optimal combination of rotor cage blade shapes, the evaluation indicators at different levels of each factor are considered together to derive the influence weights of the different levels. To facilitate the evaluation, the different evaluation indicators are made dimensionless, and then comprehensively considered to derive the integrated value at the corresponding level, as shown in

Table 5. The comprehensive evaluation value of each factor.

Factor	Overall Assessment Value	Factor	Overall Assessment Value
A1	0.9952 + 0.9816 = 1.9768	B4	1.0444 + 0.8848 = 1.9292
A2	1.0029 + 0.9816 = 1.9845	B5	1.0425 + 0.9779 = 2.0204
A3	1.0124 + 0.9647 = 1.9770	B6	1.0482 + 0.9611 = 2.0093
A4	0.9933 + 1.0557 = 2.0490	C1	1.0752 + 0.9165 = 1.9917
A5	1.0048 + 0.9991 = 2.0039	C2	1.0657 + 0.9017 = 1.9674
A6	0.9914 + 1.0173 = 2.0087	C3	1.0333 + 0.9318 = 1.9651
B1	0.9398 + 1.0322 = 1.9720	C4	0.9591 + 0.9640 = 1.9231
B2	0.9436 + 1.0719 = 2.0155	C5	0.9420 + 1.0962 = 2.0382
B3	0.9816 + 1.0720 = 2.0536	C6	0.9248 + 1.1896 = 2.1144

. Therefore, according to the analysis, considering the rotor blade installed angle first, then the length of rotor blades, and finally the number of rotor blades, it can be observed from Table 5 that the combined evaluation of A1, B4, and C4 is the smallest among all the factors. At this point, the number of rotor blades is 24, the length of rotor blades is 272 mm, and the rotor blade installed angle is 24°.

The performance index under the optimal scheme of orthogonal experiments (A1, B4, C4) is obtained through numerical simulation, as shown in

Table 6. Comparison of optimization results between orthogonal experimental scheme and ML-GA scheme.

Scheme	Factors of Influence			Performance Parameters
	A	B	C	Cut Size	Classifying Sharpness Index
Original	48	272	0	56.8	0.54
Orthogonal experiment	24	272	24	52.5	0.56

. In this case, the cut size is 52.5 μm, and the classifying sharpness index is 0.56. Compared with the original scheme, it is found that the cut size is reduced by 7.57%, and the classifying sharpness index is improved by 3.57%.

It is evident that the optimized rotor cage structure, as determined by orthogonal experiments, results in improved classification performance. Compared to the original structure, the optimized rotor cage parameters show no change in blade length, but the number of blades is reduced by 50% to 24, and the blade installation angle is increased to 24°.

To analyze the intrinsic mechanism behind the improved classification performance, the turbulent dissipation energy in the rotor cage region of the original and optimized structures, as determined by orthogonal experiments, is compared, as shown in Figure 6. The turbulent energy dissipation value of the original structure varies from 354 to 43,652 m²/s³, while that of the optimized structure from the orthogonal experiment ranges from 791 to 28,483 m²/s³. This indicates that the flow field in the rotor cage region of the original structure is unstable. Smaller turbulent energy dissipation can enhance classification performance [46,47,48]. Therefore, an optimal number of blades can stabilize the flow field in the rotor cage area, and the presence of blade installation angles can reduce the velocity gradient between the blades [20], thereby improving classification performance.

3.2. Model Evaluation

Machine learning models are constructed using data from 36 orthogonal experiments. To maintain generality, the dataset is divided so that 80% of the data are used for model training, and 20% are used to test the model performance. Four machine learning algorithm models are implemented using Python to train on the training set and predict the test set, and the results are shown in Figure 7. The green and blue solid circles in the figure represent the data from the training set and the test set, respectively. The points on the dashed line indicate that the true value of the cut size and the predicted value match exactly. The closer the data points are to this dashed line, the smaller the absolute error between the predicted value and the true value.

As can be seen from Figure 7, the four algorithmic models show most of the training set data falling around the dashed line, indicating better prediction results. In comparison, the DTR, RFR, and ANN models achieve more than 92% fit to the training set. Among them, the DTR model even achieves a perfect fit with an $R^2$ value of 1, indicating the best training performance. This is followed by the RFR and ANN models. The SVR model, however, showed relatively poorer performance, with an $R^2$ value of only 0.8446.

For machine learning algorithms, performing well on the training dataset does not ensure absolute accuracy, which must be tested with the test dataset. As seen in Figure 7, for the prediction effect on the test set, the ANN model has the highest prediction accuracy with an $R^2$ value of 0.9612, outperforming its performance on the training set. This is followed by the RFR and DTR models, while the SVR model has the worst prediction results. It is also evident that the degree of fitting of the DTR model on the test set is significantly lower than on the training set, indicating an obvious overfitting phenomenon.

To measure the accuracy and stability of the algorithmic models more comprehensively, the MSEs of the training set and validation set for the four algorithmic models are analyzed, as shown in Figure 8. Additionally, the model’s error on the training set is called the empirical error, and the error on the test set is called the generalization error. It can be seen that the generalization error of the DTR model is much larger than the empirical error, indicating that the dataset contains noise. This leads to inaccurate interpolation of the true correlation of the dataset, resulting in overfitting, and the overfitted model has lower generalization ability. The SVR model performs below 85% on both the training set and the test set, and its empirical and generalization errors are larger, suggesting that the model may be overly simple and insufficient in feature extraction. The RFR model performs well not only on the training set but also on the test set, with empirical and generalization errors of only 0.4794 and 0.4632, respectively. The ANN model has a generalization error of 0.8776, which is lower than the empirical error, indicating good prediction results.

In summary, although the ANN model fits better than the RFR model on the test set, the empirical and generalization errors of the RFR model are smaller than those of the ANN model. The RFR model performs well on both the training and test datasets, exhibiting the lowest error and the highest accuracy in comparison. Therefore, the RFR model is preferred in this study as the best model for predicting cut size and classifying sharpness index of the classifying device.

3.3. MLA Model Analysis and Optimization

3.3.1. Model Analysis

To analyze the influence of individual factors on the classification performance of the classifying device, the cut size and classifying sharpness index are predicted using the RFR prediction model for different combinations of factors. On this basis, the effects of the number of rotor blades, length of rotor blade, and rotor blade installed angle on the cut size and classifying sharpness index of the classifying device are analyzed.

Correlation analysis is a statistical method used to measure the degree of correlation between two variables, and the correlation coefficient is commonly used to express this correlation. The covariance between each independent variable can be seen by calculating the Pearson Correlation Coefficient (PCC), which evaluates the linear correlation between two independent variables within a range of −1 to 1. A larger value of PCC indicates a stronger correlation, and a plus or minus sign indicates either a positive or negative correlation. A PCC close to 0 indicates that there is no linear correlation between the two variables. The PCC for each factor and performance indicator is shown in Figure 9.

As seen in Figure 9, the calculated values of PCC between each characteristic parameter are small, indicating no significant correlation, and the effects on the performance indexes are independent of each other. For cut size and classifying sharpness index, the influence of blade installed angle is the largest, followed by the length of rotor blade, and the influence of the number of rotor blades is the smallest. The rotor blade installed angle is negatively correlated with the cut size and classifying sharpness index; i.e., the larger the blade installed angle, the smaller the cut size and classifying sharpness index. There is a positive correlation between the length of blade and cut size and classifying sharpness index; i.e., the greater the length of blade, the larger the cut size and classifying sharpness index. The PCC value of the number of blades on cut size and classifying sharpness index is close to 0, indicating no significant positive or negative correlation. The qualitative conclusions of this section are consistent with the conclusions of the extreme value analysis of variance in Section 3.1.

The SHAP value originates from game theory, the core idea of which is to decompose the importance of each feature into a weighted sum of the Shapley values of different feature values. Shapley values are used in game theory to measure the degree of contribution of each participant to the final payoff of a cooperative game, and Shapley values are used in machine learning to measure the degree of contribution of each feature to the model prediction. As shown in Figure 10, the SHAP summary plot is ranked according to the importance of the selected influencing factors on both the cut size and the classifying sharpness index. Each point in the plot represents the Shapley value of a feature for an instance. The SHAP value taken as zero serves as the middle demarcation line: samples on the left side show a negative effect on the prediction value, samples on the right side show a positive effect. The colors indicate the size of the corresponding feature value: red indicates a higher predicted value, and blue indicates a lower predicted value.

As seen in Figure 10, a larger blade installed angle value results in a larger negative impact on the model output, i.e., a smaller cut size and classifying sharpness index; a smaller value results in a larger positive impact on the model output, i.e., a larger cut size and classifying sharpness index. A larger blade length value results in a larger positive impact on the model output, i.e., a larger cut size and classifying sharpness index; a smaller value results in a larger negative impact on the model output, i.e., a smaller cut size and classifying sharpness index. The number of blades does not significantly affect the cut size output; however, it negatively affects the classifying sharpness index, with more blades resulting in smaller SHAP values and a lower classifying sharpness index.

Figure 11 shows the SHAP feature importance plot, where the global importance of each feature is considered to be the average absolute value of that feature over all given samples. As seen from the figure, the length of rotor blades and rotor blade installed angle have a more significant effect on the predictive model.

The Partial Dependence Plot (PDP) demonstrates the dependency of prediction results on input features. In this study, the PDP is used to analyze the dependence of individual input features on output features. From Figure 12a, it can be seen that the cut size does not change significantly with the number of rotor blades; the cut size first increases and then tends to stabilize with the length of rotor blades; and the cut size decreases with the rotor blade installed angle. From Figure 12b, it can be seen that the classifying sharpness index decreases and then increases with the number of rotor blades; the classifying sharpness index decreases and then increases significantly with the length of rotor blades; and the classifying sharpness index decreases with the rotor blade installed angle. This conclusion is consistent with the results of the orthogonal experimental analysis, further confirming the accuracy of the model predictions.

3.3.2. GA for Solving Optimal Solutions

The preceding analysis qualitatively illustrates the impact of the number of rotor blades, length of rotor blades, and rotor blade installed angle on the classification performance of the classifying device. In this section, a genetic algorithm is used to determine the optimal geometric parameters quantitatively.

The genetic algorithm is an optimization and search technique inspired by biological evolution and molecular genetics, deriving its principles from biological theory. It is known for its simplicity in computational methods, effective optimization outcomes, and robust capacity to address combinatorial optimization problems. Utilizing the genetic algorithm as the focal point, we optimize the classification performance, as depicted in the algorithm flow shown in Figure 4.

3.3.3. Optimization Results

The optimal solution in this paper is to determine the optimal combination of characteristic parameters that minimizes the cut size ($\min y_1(x)$) and maximizes the classifying sharpness index ($\max y_2(x)$) within the characteristic parameter value range. Using $\min (y_1-y_2)$ as the fitness function and each characteristic parameter’s value range as constraints, we set the initial population to 300, the number of iterations to 100, the mutation rate to 0.001, and the precision to 0.01. This prevents the algorithm from converging to a local optimal solution. The optimal feature parameter combination is achieved with 29 rotor blades, a rotor blade length of 232.8 mm, and a rotor blade installed angle of 36.8°.

At this stage, the performance index parameters of the optimal solution from orthogonal experiments and the optimal solution from ML-GA are compared through CFD simulation, as depicted in

Table 7. Comparison of optimization results between orthogonal experimental scheme and ML-GA scheme.

Scheme	Factors of Influence			Performance Parameters
	A	B	C	Cut Size	Classifying Sharpness Index
Orthogonal experiment	24	272	24	52.5	0.56
ML-GA	29	232.8	36.8	47.6	0.62

. Among these, the cut size under the optimal feature parameter combination of ML-GA is 47.6 μm, and the classifying sharpness index is 0.62. Compared with the optimal feature parameter combination of orthogonal experiments, the ML-GA solution reduces the cut size of the classifying device by 9.33%, and the classifying sharpness index is improved by 9.68%.

It is evident that the ML-GA optimized rotor cage structure results in improved classification performance. Compared to the rotor cage structure optimized by orthogonal experiments, the number of blades after ML-GA optimization increased by 5, which is a minor change, indicating that a moderate number of blades is beneficial for enhancing classification performance within the scope of this study. The installed angle of the blades increased by 34.8%, demonstrating that increasing the installed angle can effectively reduce the cutting particle size and improve the accuracy of separation. This is because the installed angle of the blades can mitigate the impact of airflow on the blades [20,38], thus preventing particle rebound.

The length of the blades is reduced by 14.4%. To analyze the intrinsic reason why reducing blade length can improve classification performance, the pressure cloud of the longitudinal section of the classifying device at y = 0 is intercepted, as shown in Figure 13. The pressure distribution around the rotor cage affects the flow path of particles into the outlet inside the classifying chamber. As observed in Figure 13, the rotor cage structures optimized by both the orthogonal experimental scheme and the ML-GA scheme share common features. They exhibit more layers of pressure bands near the rotor cage blades and more uniform variations. However, in the optimized structure after orthogonal experiments, the local positive pressure area at the gap between the rotor cage and the lower port of the discharge port is significantly increased. This can cause large particles to be easily carried into the finished product by the fine particle flow. Additionally, the concentrated negative pressure area inside the rotor cage is significantly larger than that in the ML-GA scheme, hindering the rapid passage of fine particles through the rotor cage. This result explains the improved classification performance due to the ML-GA scheme.

A CFD-ML-GA method for optimizing rotor cage structural parameters is presented in this paper. Conventional CFD methods typically use a control variable approach to optimize individual structural parameters, which may ignore the interactions between multiple feature parameters. In this paper, ML and GA are combined with CFD to enable a more comprehensive optimization process that considers multiple feature parameters simultaneously, thus reducing the overall computational time and resources required. Previous optimization techniques for rotor cage parameters usually use simple parameter comparison or local optimization methods, which may miss the global optimization values due to their limited scope. In contrast, GA can achieve the optimization of the entire system. Combining ML and GA to construct a prediction model enables more efficient global optimization, providing a more accurate and practical solution for engineering applications.

4. Conclusions

By combining CFD, ML, and GA, this study not only enhances the accuracy and efficiency of the optimization process but also improves the global search capability for complex systems. This integration offers innovative solutions for similar engineering challenges. The multidisciplinary approach effectively addresses the limitations of individual methods, demonstrating significant potential for optimizing complex engineering systems. The main research results are as follows:

(1)

The results of the orthogonal experiments indicate that the influence of the studied characteristic parameters on classification performance, from strongest to weakest, is as follows: installed angle of blades, length of blades, and number of blades. The best combination of orthogonal experiments is the number of rotor blades being 24, the length of rotor blades being 272 mm, and the rotor blade installed angle being 24°. Compared with the original structure, this orthogonal experimental scheme reduced the cutting size of the classifying device by 7.57% and increased the classifying sharpness index by 3.57%.

(2)

The RFR model excelled in predicting the classification performance with an R² of 0.9572 and an MSE of 0.4632 between its prediction and CFD simulation results. Correlation, SHAP, and PDP analyses are conducted on it, aligning with orthogonal experiment results.

(3)

The quantitative analysis of the GA shows that when the number of blades is 29, the length of blade is 232.8 mm, and the blade installed angle is 36.8°, the cut size is reduced to 47.6 μm, and the classifying sharpness index is improved to 0.62. Compared with the optimal scheme of orthogonal experiment, the scheme reduces the cut size by 9.33%, and the classifying sharpness index is improved by 9.68%.

(4)

Due to the size of the dataset, the model’s accuracy in this study is not absolutely precise, which somewhat limits the findings. Future researchers could focus on optimizing the computational efficiency of the algorithms and exploring their application to larger and more complex datasets. In addition, the integration of advanced machine learning techniques should be explored to improve model prediction accuracy.