Effects of Operating Parameters on Mixing Performance and Multi-Objective Optimization of Twin-Blade Planetary Mixer in Viscous Systems

Abstract

The twin-blade planetary mixer is critical for processing highly viscous materials in the chemical and polymer industries, yet optimizing its mixing characteristics alongside energy efficiency remains challenging. This study investigates the twin-blade planetary mixer, using computational fluid dynamics simulation methods to analyze the operating parameters and multi-objective optimization of performance in viscous systems. First, the multi-axis stirring process was simulated numerically based on the Planetary Motion Method, revealing the working process at the cross-section and of the blades, thereby unveiling a mixing mechanism driven by cyclic transitions between local shear-intensive kneading and global convective circulation. Then, through orthogonal experiments and ANOVA, the dominant role of the hollow blade’s self-rotation speed on performance was clarified. Furthermore, based on Kriging and NSGA-II, with LINMAP employed for decision making, an optimal parameter combination, specifically a hollow blade self-rotation speed of 94.86 rpm, a speed ratio of 0.063, and a blade-to-bottom height of 2.79 mm, successfully achieved an 8.15% reduction in power consumption, a 20.03% increase in global axial flow, and a 5.01% enhancement in maximum kneading pressure.

1. Introduction

Mixing is an essential process in industries such as chemicals, food, and pharmaceuticals. Through mixing, materials are homogenized to undergo specific physical or chemical changes to meet the target process parameters or product performance requirements. While various blade configurations have been reported as effective options for laminar mixing, they exhibit distinct limitations when processing high viscosity materials, such as solid propellants. Single-shaft impellers, particularly helical ribbons and anchor impellers, are prone to cavern formation when operating in pseudo-plastic fluids. Under such conditions, fluid motion becomes confined to the region immediately surrounding the rotating blades, while a stagnant core remains in the central zone of the vessel [1]. Similarly, although the Maxblend impeller is widely regarded for its efficiency across a broad viscosity range, it exhibits limitations when handling rheologically complex fluids. Fontaine et al. [2] found that, in highly shear-thinning systems, the flow field can bifurcate into segregated regions, with a stagnant, high-viscosity zone forming in the upper portion of the vessel, thereby hindering overall homogenization.

In contrast, the twin-blade planetary mixer addresses these hydrodynamic deficits through its planetary motion [3]. The intermeshing blades actively sweep the entire vessel volume, effectively preventing the formation of stagnant zones and providing a shear effect by its unique twin-blade structure. However, despite the widespread application of the twin-blade planetary mixer, limited experimental studies and performance parameter research are available due to its structural complexities [4].

With the advancement of computational fluid dynamics (CFD), research on mixers has become increasingly comprehensive. As a crucial approach for analyzing multi-shaft mixing systems, dimensionless analysis has attracted significant attention, with its definitions and calculation methods becoming prominent research focuses. Delaplace et al. [5,6] studied modeling methods for planetary mixers when mixing highly viscous Newtonian fluids and proposed the characteristic velocity $u_{ch}$, modified Reynolds number $R_{eM}$, and modified power coefficient $N_{pM}$, successfully obtaining a unique power curve independent of the speed ratio. André et al. [7] developed modified Froude, power, and mixing time numbers suitable for planetary mixing systems, thereby extending the applicability of traditional dimensionless parameters for mixers. Auger et al. [8] used $u_{ch}$, $R_{eM}$ and $N_{pM}$ to analyze the power characteristics of hook-type planetary mixers, obtaining the master curve of power number versus Reynolds number and predicting the mixer’s power consumption, enabling process optimization by controlling power input.

For vertical twin-blade planetary mixers, researchers have conducted studies from multiple perspectives. In terms of performance analysis, Liu et al. [9] provided new ideas for optimizing blade cross-sectional shapes through theoretical analysis combined with finite element analysis. Liang et al. [10,11] successively investigated the effects of parameters such as blade helix angle, blade clearance, blade eccentricity, and blade arrangement on blade torque and power in vertical twin-blade planetary mixers, and improved the dimensionless numbers proposed by Auger, plotting power curves to describe the performance of the twin-blade planetary mixer. Long et al. [12,13] employed kneading pressure to characterize the extrusion and shear effects of the blades on the fluid within the kneading region, and established, within a certain range, the correlations among power consumption, kneading pressure, and both rheological and kinematic parameters under Newtonian and non-Newtonian fluid conditions. Regarding the mechanism analysis of vertical twin-blade planetary mixers, Coesnon et al. [14] used the sliding mesh method to study a planetary mixer and found that the power consumption was highest when the blades were kneading against each other. Zhang et al. [15] combined CFD with visualization experiments to observe the flow field characteristics of vertical twin-blade planetary mixers and investigated the reasons for the efficiency differences between forward and reverse planetary stirring motions. Guo et al. [16] identified localized pressure and velocity concentrations at the blade–wall interface during propellant mixing. Long et al. [17] employed the Discrete Element Method (DEM) to confirm that convection is the dominant mixing mechanism of the twin-blade planetary mixer. Collectively, these studies have substantially advanced the understanding of blade structure and power prediction. However, a closer examination reveals that geometric parameters, including helix angle, clearance, and eccentricity, are the near-exclusive focus, while the role of adjustable operating parameters has received little systematic attention. Since geometric features are fixed post-manufacturing, this emphasis offers limited guidance for process control in practice.

The combination of numerical simulation and algorithms for the analysis and optimization of fluid machinery has achieved significant research progress, offering the advantage of reducing the computational time required for numerical simulations. Tsugeno et al. [18] conducted a sensitivity analysis of DEM results for different parameters of a ribbon mixer and found that blade width is an important factor in achieving better mixing. Mihailova et al. [19] fitted CFD output data using the least squares method, analyzed the correlation between mixer performance and design parameters, and applied the Hierarchical Evolutionary Engineering Design System (HEEDS) optimization package to perform multi-objective optimization of the fitted functions, obtaining an optimal trade-off solution. Yao et al. [20] employed the Kriging model’s excellent nonlinear approximation capability to construct the objective function and used the Non-dominated Sorting Genetic Algorithm (NSGA) for the multi-objective optimization design of a stirred tank, achieving increased suspension uniformity while reducing power consumption. Bostan et al. [21] developed a comprehensive methodology, using datasets from RANS simulations to construct a Kriging-based response surface model (RSM), and subsequently applying an Evolutionary Algorithm (EA) to optimize the geometric parameters of a hermetic centrifugal pump impeller.

While substantial advancements have been made in analyzing blade structures and predicting power for vertical twin-blade planetary mixers, existing studies have focused predominantly on static geometric parameters, leaving the effects of adjustable operating parameters on mixer performance poorly characterized. Unlike geometric parameters that are fixed after manufacture, operating parameters, including the hollow blade self-rotation speed, speed ratio, and blade-to-bottom height, can be actively regulated during production, making their optimization directly relevant to industrial practice. Moreover, no surrogate-based multi-objective optimization framework has been applied to these operating variables for the twin-blade planetary mixer. Consequently, this study utilizes CFD to examine how these parameters influence power consumption, kneading pressure, and axial flow, and implements a surrogate-based optimization framework to refine these operating conditions, thereby elevating the overall performance of the mixer.

2. Materials and Methods

2.1. CFD Simulation

2.1.1. Mathematical Model and Assumptions

The assumptions for the numerical simulation are as follows: the walls are adiabatic with a no-slip condition; the mixing material is incompressible; the effect of gravity is considered; and the material completely fills the computational domain during the mixing process, neglecting the influence of liquid level variations.

Following standard computational fluid dynamics practices [22], the fluid flow is governed by the principles of mass and momentum conservation. In the Cartesian coordinate system, these governing equations are expressed as follows:

Mass conservation equation (continuity equation):

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overrightarrow{u}\right)=0$$

(1)

Momentum conservation equation (N-S equation):

$${\displaystyle \frac{\partial \left(\rho \overrightarrow{u}\right)}{\partial t}}+\nabla ·\left(\rho \overrightarrow{u}\overrightarrow{u}\right)=-\nabla p+\nabla ·\left(\mu \nabla \overrightarrow{u}\right)+\overrightarrow{S}$$

(2)

where $\overrightarrow{u}$ = (u, v, w) is the velocity vector, with u, v, w denoting the velocity components in the x, y, and z directions, respectively; p is the static pressure; μ is the dynamic viscosity; $\rho$ is the fluid density; and $\overrightarrow{S}$ = ($S_u$, $S_v$, $S_w$) is the generalized momentum source term vector.

For this scenario, the velocity at the tip of the hollow blade was used as the characteristic velocity to calculate the Reynolds number [8], as follows:

$$R_{eM}={\displaystyle \frac{\rho u_{ch}{d}_{H,k}}{\mu }}$$

(3)

$$u_{ch}={\displaystyle \frac{u_{\mathit{\text{impeller tip}}}}{\pi }}=N_{k}d+N_{H,k}{d}_{H,k}$$

(4)

Despite the prevalence of non-Newtonian behaviors in industrial mixing, a highly viscous Newtonian fluid was selected for this study. The primary objective is to address the flow complexity caused by the transient intermeshing of the twin-blade configuration. Using a Newtonian fluid allows for the decoupling of rheological and kinematic effects, thereby clarifying the impact of operating parameters on the fundamental mixing mechanism.

The fluid in the mixing vessel is polybutene with a density of 888.9 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and a viscosity of 10, 20 and 40 $\mathrm{P}\mathrm{a}·\mathrm{s}$. As a Newtonian fluid, its shear stress is linearly proportional to the shear rate. The blade rotational speed is 30~120 rpm, and the speed ratio is 0.0615~0.246. The calculated Reynolds number is 0.003917~0.72827. Therefore, the fluid flow inside the mixing vessel was in the laminar regime, and the laminar model was applied.

2.1.2. Geometry and Mesh

A 1 L twin-blade planetary mixer model was established, its geometric structure is shown in Figure 1, and its geometric dimensions are shown in

Table 1. Geometric parameters of the mixer.

Parameter	Value
$H$	90 mm
$D$	120 mm
$d$	60 mm
$h$	62.2 mm
$a_s$	13.9 mm
$a_k$	27.8 mm
$C_L$	41.7 mm
$\sigma$	1.4~2.2 mm
$d_b$	19 mm
$h_k$	70 mm
$D_k$	25 mm
$i_{s,k}$	2

. The twin-blade planetary mixer consists of two vertically mounted interlocking helical blades, both offset from the center of the mixing vessel. The model incorporates two distinct blade types to optimize mixing: a solid blade and a hollow blade. The mathematical formulations describing the trajectories of the hollow and solid blades are presented in Equations (5) and (6).

$$\left\{\begin{array}{c}{x}_k=C_L\cos {\theta }_k-({\displaystyle \frac{d}{2}}+\sigma )\cos ({\theta }_k-{\displaystyle \frac{{\theta }_k}{i_{s,k}}}+\gamma )\\ y_k={-C}_L\sin {\theta }_k+({\displaystyle \frac{d}{2}}+\sigma )\sin ({\theta }_k-{\displaystyle \frac{{\theta }_k}{i_{s,k}}}+\gamma )\\ 0\le {\theta }_k\le \gamma \end{array}\right.$$

(5)

$$\left\{\begin{array}{c}{x}_s=C_L\cos {\theta }_s-({\displaystyle \frac{d}{2}}+\sigma )\cos ({\theta }_s+{i_{s,k}\theta }_s-\gamma )\\ y_s={-C}_L\sin {\theta }_s+({\displaystyle \frac{d}{2}}+\sigma )\sin ({\theta }_s+{i_{s,k}\theta }_s-\gamma )\\ 0\le {\theta }_s\le \gamma /2\end{array}\right.$$

(6)

The solid blade relies on specially designed surface geometries to induce bulk convection and generate high shear stress. In contrast, the hollow blade features a hollow structure that allows fluid to pass through the blade body, promoting fluid transport, enhancing distributive mixing, and reducing the formation of isolated flow regions. During operation, both blades undergo planetary motion around the mixing vessel’s center, with each blade performing its own rotation, creating a unique shear and folding effect.

The twin-blade planetary mixer features complex blade structures, so dynamic mesh method was used for numerical simulation, with tetrahedral elements for discretization. As shown in Figure 2a, the blade-to-blade, blade-to-bottom, and blade-to-wall distances are extremely small, requiring additional grid refinement to achieve more accurate simulation results. Meshing results are illustrated in Figure 2b.

2.1.3. Discretization and Solution Algorithm

Pressure–velocity coupling was solved using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) Consistent algorithm. The gradient was evaluated with the Green–Gauss node method, while pressure and momentum were discretized using a second-order upwind scheme. The simulation was carried out with a time step size of 0.001 s for 2000 time steps.

To evaluate the performance of the twin-blade planetary mixer, $Q_Z$, $T_k$, $p_{max}$, and $P_k$ were employed as assessment indices. $Q_Z$ and $T_k$ are defined as follows:

$$Q_Z=\underset{S}{\iint }{v}_Z\hspace{0.17em}dS$$

(7)

$$T_k=\underset{A_I}{\iint }{\left[\overrightarrow{r}\times \overrightarrow{F}\right]}_z\hspace{0.17em}dA$$

(8)

$P_k$, as an important factor to be considered in energy-saving optimization, can be obtained from $T_k$ [23]:

$$P_k=T_k\cdot {\omega }_k$$

(9)

2.1.4. Grid Independence

To eliminate the influence of grid density, a grid independence study was conducted. Five mesh configurations were generated, and $T_k$ was monitored as the sensitivity indicator. As illustrated in Figure 3, $T_k$ initially increases with the grid number and exhibits an asymptotic behavior as the cell count exceeds 709,352. As shown in

Table 2. Grid independence verification results.

Mesh ID	Cell Count	${\mathit{T}}_{\mathit{k}}$/N·m	Relative Error
Mesh 1	308,462	0.194	4.89%
Mesh 2	477,497	0.198	2.69%
Mesh 3	709,352	0.202	0.64%
Mesh 4	1,039,756	0.203	0.25%
Mesh 5	1,256,779	0.204	/

, the relative deviation in torque between Mesh 3 and Mesh 5 is less than 1%.

For further analysis, the Velocity Magnitude distribution along the central axis of the vessel was extracted and compared, as shown in Figure 4. It was observed that, apart from Mesh 2, which failed to accurately capture the velocity profile characteristics in the height range of 0.02 to 0.04 m, the results from the other grid densities showed good agreement with negligible differences. Therefore, considering both computational accuracy and efficiency, the grid configuration of Mesh 3 was adopted for all subsequent simulations.

Under dynamic mesh conditions, the planetary stirring motion continuously compresses and stretches the fluid domain, causing variations in mesh quality and count. Five dynamic mesh parameter settings were evaluated, with mesh count monitored over time as shown in Figure 5.

As shown in

Table 3. Numerical results under different dynamic mesh settings.

Group	${\mathit{Q}}_{\mathit{Z}}$$/\mathbf{L}·{\mathbf{s}}^{-1}$	${\mathit{T}}_{\mathit{k}}$$/\mathbf{N}·\mathbf{m}$	${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$$/\mathbf{k}\mathbf{P}\mathbf{a}$
A	5.63 × 10−5	0.200	13.8
B	5.77 × 10−5	0.202	13.9
C	5.58 × 10−5	0.202	13.9
D	5.48 × 10−5	0.202	13.9
E	4.89 × 10−5	0.202	13.9

, the relative errors for $T_k$ and $p_{max}$ across all parameter settings (A–E) are within 1.3%. However, for $Q_Z$, setting E exhibits a significant deviation of 10.7% compared to setting D, whereas settings A through D remain consistent with relative errors within 3%. This indicates that the grid resolution in case E is insufficient, while settings A through D have successfully achieved mesh independence. To balance computational efficiency and accuracy, parameter setting D was selected.

2.1.5. Blade Motion Implementation Scheme

To study the planetary stirring behavior, transient analysis is required. Therefore, dynamic mesh methods were used for numerical simulation. There are two methods currently available for numerical simulation.

The first method is the Rotating Wall Method [24,25], in which the revolution speed is applied to the inner wall of the mixing vessel with its rotational axis at the vessel center and in the direction opposite to the self-rotation of the hollow blade, although this simplified approach requires further investigation to resolve deviations from actual operation. For any position point on the inner wall surface in this method, its velocity is defined as:

$$\left\{\begin{array}{c}{v}_x={\displaystyle \frac{D}{2}}{\omega }_H\cos ({\omega }_H\mathrm{t}+\mathsf{\theta })\\ v_y={\displaystyle \frac{D}{2}}{\omega }_H\sin ({\omega }_H\mathrm{t}+\mathsf{\theta })\\ v_z=0\end{array}\right.$$

(10)

The method proposed in this paper, is the Planetary Motion Method, in which the movement of both the hollow and solid blades is controlled by UDF according to actual conditions. This method can more accurately simulate actual conditions, but due to the superposition of self-rotation and planetary rotation speeds near the blades, the dynamic mesh distorts more significantly, leading to higher computational costs. As an example, the motion equation of the hollow blade in this method is defined as follows:

$$\left\{\begin{array}{c}{v}_x=a_k{\omega }_H\cos {(\omega }_H\mathrm{t}+\mathsf{\theta })\\ v_y=a_k{\omega }_H\sin {(\omega }_H\mathrm{t}+\mathsf{\theta })\\ v_z=0\\ {\omega }_z={\omega }_k+{\omega }_H\end{array}\right.$$

(11)

To ensure the reliability of the numerical simulation, differences between the two methods were compared by monitoring $T_k$, $p_{max}$, $Q_Z$, and time consumption for the same number of iterations.

For $T_k$ and $p_{max}$, as shown in Figure 6a,b, the differences between the two methods are small, with average relative errors of 2.88% and 2.10%, respectively. For $Q_Z$, as shown in Figure 6c, the difference is significant. This occurs because the Rotating Wall Method imposes the revolution speed of the two blades onto the wall surface, which leads to lower fluid velocity near the blades compared with the Planetary Motion Method, thereby modifying the local flow field. At the same time, due to the motion of the vessel wall, the flow field in the regions of the mixing vessel far from the blades is also affected. Based on the monitoring data, this simplification method is not reliable.

Regarding computational speed, as shown in

Table 4. Computational speed of the two methods.

Dynamic Mesh Strategy	Iteration Steps	Time Consumption/s
Rotating Wall Method	2000	453
Planetary Motion Method	2000	472

, after 2000 iterations, the Rotating Wall Method demonstrates an advantage in efficiency. This is because the mesh quality is better and the mesh updating process is faster, leading to less computational time consumption compared with the Planetary Motion Method.

In summary, due to the inherent limitations of the Rotating Wall Method, it may introduce significant errors. To more accurately describe the complex multi-axis planetary stirring motion, the Planetary Motion Method will be used for subsequent numerical simulations.

2.2. Orthogonal Experimental Design

The orthogonal experimental design method is a scientific experimental design method based on orthogonal tables, used for efficiently analyzing multi-factor, multi-level experimental issues [26]. To reduce computational effort, an $L_9\left(3^4\right)$ orthogonal table was used to design experiments for material systems with different viscosities ($\mu$), examining the effect of the self-rotation speed of the hollow blade ($A$), the speed ratio between the self-rotation and revolution speeds ($B$), and the blade-to-bottom height ($C$) on $T_k$, $Q_Z$, $p_{max}$, and $P_k$ during the kneading process, as shown in

Table 5. Factors and levels of the orthogonal experiment.

Level	Factor
	A/rpm	B	C/mm
1	30	0.0615	1.4
2	60	0.123	1.8
3	90	0.1845	2.2

. The selected parameter ranges in Table 5 ensure physical realizability and represent typical industrial operations. Factor A spans the optimal low-to-medium speed regime for viscous systems, covering the laminar flow range established in Section 2.1.1. Factor B is determined by the standard gear configurations available for twin-blade planetary mixers [10,12]. Finally, for Factor C, the range of 1.4 to 2.2 mm is bounded by the minimum clearance necessary to prevent mechanical contact.

Viscosity $\mu$, as an uncontrollable external factor, was considered separately at 10, 20, and 40 for experimental design. The three viscosity levels (10, 20, and 40 Pa·s) span the range of commercially available high-viscosity polybutene grades characterized by varying molecular weights, with the upper bound (40 Pa·s) corresponding to the Polybutene N15000 grade used in prior validation studies [8,10]. Therefore, a total of 27 orthogonal experiments were conducted to study the effects of variations in speed ratio, rotation speed, and blade-to-bottom height at different viscosities on $T_k$, $Q_Z$, $p_{max}$ and $P_k$. The orthogonal experimental scheme for each viscosity is shown in

Table 6. Orthogonal experimental scheme.

No.	Factor
	A	B	C
1	1	1	1
2	1	2	2
3	1	3	3
4	2	1	2
5	2	2	3
6	2	3	1
7	3	1	3
8	3	2	1
9	3	3	2

.

2.3. Surrogate Model Construction

2.3.1. Optimized Latin Hypercube Design

Optimized Latin Hypercube Sampling (OLHS) is a random sampling method based on stratified sampling proposed by Morris et al. [27]. Compared to the orthogonal experimental method, it can efficiently fill the space and accurately describe the mapping relationship between design variables and response objectives through a small number of representative sample points. Compared to Latin Hypercube Sampling (LHS), as shown in Figure 7, it introduces the principle of $\max \left(\underset{i\ne j}{\min }|x_i-x_j|\right)$, which makes the spatial distribution of data points more even.

The study uses 24 sets of CFD numerical simulation data obtained from OLHS and 11 sets of multi-factor level combination CFD numerical simulation data as the training set, as shown in

Table 7. Training set.

Group	A/rpm	B	C/mm	Pk	Qz	${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$
1	70.70	0.1457	2.6273	0.2812	0.4401	0.4653
2	114.71	0.2154	1.8232	1.0000	0.9367	0.9529
3	93.74	0.1270	2.2139	0.5412	0.5664	0.7257
4	103.19	0.0958	1.6775	0.6657	0.9988	0.8408
5	60.77	0.0707	2.7947	0.1649	0.3465	0.3564
6	58.47	0.2380	2.4210	0.2140	0.3413	0.3143
7	85.94	0.1661	1.4098	0.5211	0.7328	0.6435
8	46.88	0.1091	1.9815	0.0875	0.1139	0.1964
9	33.06	0.1880	2.0892	0.0255	0.2752	0.0313
10	95.63	0.1807	2.6542	0.6018	0.7762	0.7557
11	76.88	0.2268	2.3625	0.4040	0.4658	0.5232
12	69.38	0.2114	1.8375	0.3232	0.6096	0.4472
13	106.88	0.2191	2.7708	0.8063	1.0000	0.8661
14	110.63	0.1730	2.1875	0.8474	0.8679	0.9064
15	118.13	0.0884	2.2458	0.8457	0.9181	1.0000
16	73.13	0.2037	1.5458	0.3676	0.6938	0.4878
17	80.63	0.2345	1.6042	0.4932	0.7652	0.5688
18	39.38	0.1653	2.7125	0.0482	0.1104	0.1027
19	91.88	0.1345	1.8958	0.5398	0.5889	0.7032
20	99.38	0.0807	2.0125	0.5762	0.6345	0.7968
21	46.88	0.1422	1.7208	0.0986	0.1838	0.1956
22	61.88	0.2422	1.9542	0.2575	0.3502	0.3542
23	43.13	0.1115	1.6625	0.0672	0.2143	0.1507
24	114.38	0.1038	2.1292	0.8200	0.8594	0.9723
25	30.00	0.0615	1.4000	0.0000	0.0000	0.0018
26	30.00	0.1230	1.8000	0.0047	0.0233	0.0037
27	30.00	0.1845	2.2000	0.0090	0.0615	0.0000
28	30.00	0.1230	2.2000	0.0020	0.0380	0.0013
29	60.00	0.0615	1.8000	0.1720	0.2603	0.3501
30	60.00	0.1230	2.2000	0.1852	0.3181	0.3448
31	60.00	0.1845	1.4000	0.2319	0.2735	0.3512
32	90.00	0.0615	2.2000	0.4548	0.5764	0.6917
33	90.00	0.1230	1.4000	0.5218	0.4839	0.7019
34	90.00	0.1230	2.2000	0.4973	0.5791	0.6967
35	90.00	0.1845	1.8000	0.5666	0.8344	0.6921

. The model was constructed with the input variables: self-rotation speed of the hollow blade ($A$), the speed ratio between the self-rotation and revolution speeds ($B$), and the blade-to-bottom height ($C$). The outputs are the surrogate model for $P_k$, $Q_Z$, and $p_{max}$. Additionally, 5 sets of data obtained from OLHS were used as the testing set, as shown in

Table 8. Test set.

Group	$\mathbf{A}/\mathbf{r}\mathbf{p}\mathbf{m}$	B	C/mm	${\mathit{P}}_{\mathit{k}}$	${\mathit{Q}}_{\mathit{Z}}$	${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$
1	45.00	0.1360	2.0000	0.0803	0.1251	0.1716
2	112.80	0.1080	1.7000	0.8327	0.8706	0.9537
3	57.60	0.2230	1.4200	0.2231	0.3485	0.3136
4	70.80	0.0670	2.6800	0.2411	0.3878	0.4601
5	86.40	0.1950	2.5000	0.5011	0.5489	0.6504

. All data were normalized using the max-min normalization method as shown in Equation (12):

$$X_{\mathrm{normalized}}={\displaystyle \frac{X-X_{min}}{X_{max}-X_{min}}}$$

(12)

2.3.2. Kriging Model

A surrogate model is a simplified model used to replace a complex original model. It approximates the input–output relationship of the original model based on sampling data, significantly improving computational efficiency while maintaining a certain level of accuracy. The Kriging model is an interpolation model, which can be expressed as [28]:

$$Y\left(x\right)=\sum _{j=1}^k{\beta }_j{f}_j\left(x\right)+Z\left(x\right)$$

(13)

The regression function $\sum _{j=1}^k{\beta }_j{f}_j\left(x\right)$ can take constant, linear, or quadratic forms to capture global trends. The stochastic process function $Z\left(x\right)$ characterizes local deviations and provides a quantitative estimate of prediction uncertainty, satisfying the following relationship:

$$E\left[Z\left(x\right)\right]=0$$

(14)

$$Var\left[Z\left(x\right)\right]={\sigma }^2$$

(15)

$$Cov\left(Z{(x}_i\right),Z{(x}_j\left)\right)={\sigma }^2{R}_{ij}(\theta ,x_i,x_j)$$

(16)

In this paper, the Kriging model was constructed with parameters set to quadratic functions and Gaussian kernels.

3. Results and Discussion

3.1. Model Validation

To ensure the reliability of the numerical simulation results, a model of the same size operating in the laminar flow regime was established based on the experiment by Long et al. [12], as shown in Figure 8, using the same material parameters as listed in

Table 9. Rheological and physical properties of the material.

Material	$\mathit{\rho }$$/\mathbf{k}\mathbf{g}·{\mathbf{m}}^{-3}$	$\mathit{\mu }$$/\mathbf{P}\mathbf{a}·\mathbf{s}$	T/°C
Corn syrup	1340	4	22

, and pressure was measured to validate the simulation model. Since both corn syrup and the polybutene act as typical Newtonian fluids, this comparison provides a valid methodological validation.

According to the experiment, the self-rotation speeds of the hollow blade were set to 60 and 80 rpm, with a speed ratio of 3. The maximum kneading pressure and peak-to-peak pressure were measured in the kneading region at a height of 11.5 mm above the vessel bottom, based on the spherical average over a diameter of 9.53 mm. Figure 9 shows the kneading pressure–time curves at the same rotation angle under different speeds. It can be observed that the experimental and CFD kneading pressure curves exhibit good overall agreement. The discrepancies observed in the local peak regions may be attributed to the measurement limitations of the thin-film pressure sensor used in the experiments. As noted in CFD validation studies of intermeshing kneading elements, pressure transducers measure averaged pressure over their face area rather than point values, producing signal dampening at locations where the pressure gradient is steepest [29]. Additionally, the sensor’s physical intrusion alters local boundary conditions and induces flow disturbances, further deviating the measured pressure from the ideal CFD flow field [30]. As shown in

Table 10. Comparison between CFD and experimental results.

${\mathit{N}}_{\mathit{k}}$$/\mathbf{r}\mathbf{p}\mathbf{m}$	${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}-\mathit{E}\mathit{X}\mathit{P}}$/Pa	${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}-\mathit{C}\mathit{F}\mathit{D}}$/Pa	$\mathbf{Error} \mathbf{of} {\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{p}}_{\mathit{p}-\mathit{p}-\mathit{E}\mathit{X}\mathit{P}}$/Pa	${\mathit{p}}_{\mathit{p}-\mathit{p}-\mathit{C}\mathit{F}\mathit{D}}$/Pa	$\mathbf{Error} \mathbf{of} {\mathit{p}}_{\mathit{p}-\mathit{p}}$
60.00	1227.80	1265.37	3.06%	1573.75	1504.12	4.42%
80.00	1662.85	1686.06	1.40%	2088.11	2009.72	3.75%

, the average relative errors of pressure are 3.74% for $N_k$ = 60 rpm and 2.58% for $N_k$ = 80 rpm, with an overall relative error of 3.16%.

Although discrepancies exist between the experimental and CFD kneading pressure curves in some regions, the average relative errors of $p_{max}$ and $p_{p-p}$ are low. Therefore, the CFD model demonstrates good overall consistency with the experimental results, confirming its reliability for subsequent analysis.

3.2. Planetary Stirring Behavior Analysis

To quantitatively characterize the local kneading intensity, the maximum kneading pressure $p_{max}$ at the cross-section is adopted as the key indicator. The cross-sectional working processes of the mixer at liquid levels of 21.1 mm, 31.1 mm, and 41.1 mm are depicted in Figure 10. It can be observed that $p_{max}$ at the cross-section exhibits periodic variations, with a period corresponding to the self-rotation cycle of the hollow blade T = 2 s. Regarding the influence of observation height, due to the helical geometry of the blades, variations in the axial position primarily induce a phase shift in the curves, while the amplitude and waveform remain consistent. Therefore, the pressure variation at the cross-section is independent of the selection of the cross-sectional height. This is consistent with the finding of Liang et al. [10] that the helical angle governs the timing of blade interactions in the kneading region, thereby affecting the periodic characteristics of the pressure curves without changing their fundamental waveform.

Next, the analysis focuses on the specific case where $h_{b-b}$ = 31.1 mm. The process can be roughly divided into four stages: 1, 2, 3, and 4. The corresponding contour plots are shown in Figure 11, Figure 12, Figure 13 and Figure 14. In Stage 1, the kneading pressure curve exhibits a local peak. The concentration of maximum pressure and strain rate within the clearance between the hollow blade and the vessel wall identifies this phase as the blade-to-wall kneading stage. Furthermore, the velocity gradient at the blade tip induces flow separation and recirculation in the blade’s wake, generating vortices as depicted in the red dashed circle in Figure 12a. This phenomenon enhances the local shear effect and mixing intensity to a certain extent. Concurrently, the high-speed rotation of the hollow blade drives local internal circulation. By channeling fluid between the leeward and windward sides through the hollow cavity, this mechanism facilitates a large-scale axial circulation as illustrated in Figure 14a.

In Stage 2, the curve exhibits a relatively stable kneading pressure at a lower magnitude. At this stage, the tip of the hollow blade has moved away from the vessel wall and is gradually approaching the kneading surface of the solid blade as illustrated in Figure 11b. Due to the absence of direct kneading interaction during this phase, the strain rate in the vicinity of the blade is lower compared to Stage 1, whereas the spatial range of action is comparatively larger. Furthermore, the global flow field retains the characteristics of large-scale, high axial circulation. Consequently, the primary significance of this stage lies in ensuring the homogeneity of the mixing process by leveraging broad flow field coverage.

In Stage 3, the kneading pressure curve rises to a global peak, representing the maximum kneading capacity of the equipment. At this stage, intense kneading interaction occurs between the hollow blade and the kneading surface of the solid blade. Driven by blade rotation and surface geometry, fluid is drawn into the narrow inter-blade gap. Subjected to the opposing local velocity vectors of the hollow and solid blades, the fluid undergoes significantly higher pressure and stronger squeezing effects compared to the blade-to-wall region, as illustrated in Figure 11c. The maximum pressure zone is concentrated within the blade-to-blade kneading region. In contrast, the maximum strain rate is observed in the region between the blade and the wall. This distinction is attributed to the fact that pressure magnitude is primarily governed by the relative normal velocity, whereas the strain rate is determined by the relative tangential velocity. As evidenced by Figure 12c, the relative normal velocity in the blade-to-blade region is significantly higher than that in the blade-to-wall gap. Conversely, the relative tangential velocity reaches its maximum in the blade–to-wall kneading region. However, the axial transport zone is relatively confined, resulting in weaker circulation capacity during this phase. Consequently, the primary significance of this stage lies in leveraging locally extreme pressure and shear to compress the fluid, marking it as the most critical phase in the kneading process.

In Stage 4, the max kneading pressure curve exhibits a stable trend at a lower magnitude. As the hollow and solid blades gradually separate, significant negative pressure is generated near the blade tips. Driven by this pressure differential, the fluid is forced through the gap between the blade tips, where it is subjected to intense stretching and separation effects. Simultaneously, induced by the significant difference in relative tangential velocity, the fluid within this region undergoes strong shear deformation, as illustrated in Figure 12d and Figure 13d. Notably, the scope of global axial circulation expands relative to Stage 3, indicating sustained axial convection during the separation process. Consequently, the primary significance of this stage lies in exerting strong stretching and shearing effects to facilitate material dispersion.

To investigate the macroscopic operational behavior and energy consumption characteristics, the instantaneous torque of the hollow blade is analyzed. The behavior of the hollow blade during the operation of the twin-blade planetary mixer can be roughly divided into two stages, I and II, with two peaks in the overall torque variation within one cycle, as shown in Figure 15. Due to the limitations in blade height and fluid surface height, the hollow blade and solid blade cannot always be in a kneading state. Therefore, the hollow blade torque exhibits a periodic variation characteristic corresponding to its self-rotation. In Stage I, both blades gradually knead, as shown in Figure 16a. When the area of engagement between the two blades is maximized, the torque of the hollow blade reaches its peak. In Stage II, the hollow blade kneads with some of the non-kneading surfaces of the solid blade and the mixing vessel wall, as shown in Figure 16b, resulting in a local peak in the hollow blade torque. Compared to the large shear effect on the upper part of the hollow blade in Stage I, as shown in Figure 17a, the shear effect in Stage II is limited to the area near the tip of the blade, as shown in Figure 17b. Therefore, the kneading action in Stage II is weaker, and the peak torque consumed by the hollow blade is also relatively smaller.

3.3. Effect of Operating Parameters on Kneading Performance

3.3.1. Orthogonal Experimental Analysis

Based on the data obtained from the orthogonal experiment, the mean value $k_{ij}$ is defined to evaluate the influence of each factor level on performance. The definition is as follows:

$$k_{ij}={\displaystyle \frac{{\sum }_{level=ij}Index}{3}}$$

(17)

The curves for each factor level are plotted in Figure 18. It can be observed that, for the axial flow rate $Q_Z$ and power consumption $P_k$, the influence of factors on performance does not follow a monotonic trend with the change in levels. Relying solely on trend curves makes it difficult to distinguish between genuine physical effects and random fluctuations. Therefore, to rigorously analyze the influence of factor levels, Analysis of Variance (ANOVA) is introduced [31]. The percentage contribution ($C_j$) of each factor reflects its relative importance to the total variation and is calculated as:

$$C_j={\displaystyle \frac{S_j}{S_T}}$$

(18)

The p-value evaluation criteria for ANOVA are presented in

Table 11. Significance codes of variables based on p-value.

p-Value	Significance Level
>0.1	Insignificant
0.05–0.1	Significant
0.01–0.05	Significant
0.001–0.01	Significant
0–0.001	Significant

. The p-value results for each factor under different viscosities obtained in this study are listed in

Table 12. Analysis of variance results.

$\mathit{\mu }$	Index	p-ValueA	p-ValueB	p-ValueC
40 Pa·s	$Q_Z$	0.039	0.470	0.509
	$p_{max}$	4.10 × 10−5	0.74	0.340
	$P_k$	0.002	0.111	0.236
20 Pa·s	$Q_Z$	0.022	0.504	0.797
	$p_{max}$	5.68 × 10−5	0.756	0.412
	$P_k$	0.002	0.128	0.216
10 Pa·s	$Q_Z$	0.021	0.417	0.718
	$p_{max}$	5.58 × 10−5	0.221	0.481
	$P_k$	0.002	0.133	0.239

, and the contribution of each factor at $\mu$ = 40 $\mathrm{P}\mathrm{a}·\mathrm{s}$ is illustrated in Figure 19. According to the p-value analysis, Factor A is the primary influencing factor for all viscosities and performance indicators.

For $p_{max}$, the influence of $A$ is extremely significant (p < 0.001) under any viscosity, occupying an almost completely dominant position. This characteristic is consistent with the twin-blade kneading principle analyzed above. The kneading pressure between the twin blades is governed by the velocity difference in the kneading surfaces. In the planetary motion reference frame, since the speed ratio of the blades is constant, this velocity difference is determined solely by $A$. Conversely, $B$ and $C$ have a negligible effect on the pressure at the cross-section, leading to the conclusion that A dominates $p_{max}$.

For $P_k$, $A$ is the primary influencing factor (p < 0.01), while $B$ and $C$ are secondary factors. Compared to $C$, the significance level of $B$ is relatively higher. This is attributed to the fact that during the operation of the hollow blade, a portion of the power consumption is directly accounted for by the revolution motion. Moreover, as the viscosity rises and the overall torque increases, the p-value of $B$ decreases, indicating a more significant influence. The influence of $C$ on power consumption lies in inducing changes in the blade torque; however, this change is limited solely to the bottom region of the blade, as shown in Figure 20. The mechanism of $C$ involves altering the drag effect of the vessel bottom on the lower surface of the blade, which induces variations in the shear force at the blade bottom. Specifically, a reduction in bottom height intensifies the wall shear between the blade and the vessel floor, thereby increasing the torque. However, while this leads to local torque changes, the impact on the overall blade torque is limited.

For $Q_Z$, $A$ remains the primary influencing factor (p < 0.05), while $B$ and $C$ are secondary factors. However, compared to $p_{max}$ and $P_k$, the contribution of $B$ and $C$ to $Q_Z$ is relatively higher, with a total contribution of approximately 8%. This indicates that although the rotation speed determines the magnitude of the flow rate, $B$ and $C$ play roles in regulating the flow field structure. $B$ modifies the motion trajectory and sweeping area of the blades, influencing the flow field in distal regions and thereby affecting axial circulation efficiency. $C$ affects flow by regulating bottom fluid passage as illustrated in Figure 21. Rotation drives axial circulation, which intensifies near the blades. While both the hollow section and the blade-to-bottom height function as circulation conduits, they exhibit distinct flow dynamics. The hollow section is characterized mainly by axial flow, whereas the bottom channel is governed by radial motion, which redirects fluid toward the vessel wall for subsequent axial conversion. As the blade-to-bottom height increases, the blade pumping region shifts upwards, enhancing the intensity of the bulk circulation in the upper part of the vessel and thereby increasing the axial flow. However, due to low mass flux and a localized effective range, its regulation of global axial flow is limited. Notably, its influence exhibits a non-monotonic trend, initially decreasing and then increasing with rising viscosity.

Since the $L_9\left(3^4\right)$ orthogonal array provides only two degrees of freedom for error estimation, the statistical power of the ANOVA is inherently limited, which may contribute to the non-significance observed for factors $B$ and $C$. However, their combined contribution ratio of approximately 8% to $Q_Z$ nonetheless indicates a physically meaningful influence. Moreover, as discrete orthogonal levels limit precision, advanced optimization is required. Factor $A$ presents a trade-off between performance and power, while factors $B$ and $C$ exhibit non-monotonic effects on $Q_Z$ with minimal energy cost. To address these conflicts and identify local optima, a Kriging model coupled with the NSGA-II is employed.

3.3.2. Effect of Viscosity on Kneading Performance

Material viscosity $\mu$, as an uncontrollable external factor, is a key factor affecting the performance of the twin-blade planetary mixer in industrial production. Its variation will have a certain impact on the mixer’s performance. To further study the effect of material viscosity on the performance of the twin-blade planetary mixer, the data obtained from orthogonal experiments at viscosities of $\mu$ = 10 $\mathrm{P}\mathrm{a}·\mathrm{s}$, 20 $\mathrm{P}\mathrm{a}·\mathrm{s}$, and 40 $\mathrm{P}\mathrm{a}·\mathrm{s}$ are analyzed.

As material viscosity increases, the power load and maximum kneading pressure of the hollow blade generally increase linearly, as shown in Figure 22a,b. With the blade speed kept constant, as the material viscosity increases, according to the shear law for Newtonian fluids, the resistance of the fluid to the blade also increases, requiring more torque to maintain the speed, resulting in an increase in the hollow blade’s power load. For viscous flow in narrow gaps, the pressure drop is approximately linearly related to viscosity, and as the material viscosity increases, the maximum kneading pressure also increases linearly. Additionally, it can be observed that the graph displays three overlapping curve characteristics. Based on the conclusions drawn by ANOVA, it can be concluded that the maximum kneading pressure is almost unaffected by the revolution speed and blade-to-bottom height, and the three overlapping curves are distinguished by the self-rotation speed of the hollow blade.

For the variation in axial flow with viscosity, as shown in Figure 22c, it can be observed that, as viscosity increases, there is no clear pattern in the change in axial flow. Similarly, the curves are roughly divided into three groups according to the rotation speed of the hollow blades, with the fluctuations in axial flow induced by viscosity changes becoming more pronounced as the self-rotation speed increases.

3.4. Optimization of Twin-Blade Planetary Mixer

3.4.1. Accuracy of Kriging Model

The accuracy of the model is quantified using Mean Squared Error (MSE) and R-Squared ($R^2$), where smaller MSE and $R^2$ values closer to 1 indicate better model prediction performance. The specific expressions are as follows:

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2$$

(19)

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}}=1-{\displaystyle \frac{MSE}{Var\left(y\right)}}$$

(20)

The accuracy of the trained Kriging model is calculated using Equations (19) and (20), and the results are shown in

Table 13. Accuracy evaluation of Kriging model.

Parameter	$\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathit{R}}^2$
$P_k$	1.23 × 10−5	0.999
$Q_Z$	3.39 × 10−3	0.944
$p_{max}$	5.02 × 10−5	0.999

. It can be observed that MSE is small, and $R^2$ is greater than 94%, indicating that the model’s prediction accuracy is high and meets engineering requirements. A comparison of the predicted values with the CFD simulation values is shown in Figure 23.

3.4.2. Three-Objective Optimization

Multi-objective optimization of the twin-blade planetary mixer performance requires the use of relevant algorithms. In this paper, NSGA-II is used to obtain the Pareto front, from which suitable optimal solutions are selected. Compared to other optimization algorithms, NSGA-II ensures the diversity of the solution set through crowding distance sorting to prevent it from falling into local optima, and it guarantees the retention and updating of excellent solutions through the elite strategy in the GA [32]. The specific process is shown in Figure 24.

Based on the previous Kriging model, the following expression is used as the multi-objective optimization target for the NSGA-II:

$$\left\{\begin{array}{c}\begin{array}{l}\min P_k(N_k,\phi ,h_{b-b})\\ \max Q_Z(N_k,\phi ,h_{b-b})\\ \max p_{max}(N_k,\phi ,h_{b-b})\end{array}\\ \begin{array}{l}30\le N_k\le 120\\ 0.0615\le \phi \le 0.246\\ 1.4\le h_{b-b}\le 2.8\end{array}\end{array}\right.$$

(21)

The three-objective optimization of the operating parameters of the twin-blade planetary mixer was carried out based on NSGA-II, and the Pareto front was obtained, as shown in Figure 25. The parameter settings for the NSGA-II are shown in

Table 14. NSGA-II parameter settings.

Parameter	Value
Population size	100
Maximum generations	500
Elite fraction	0.8
Crossover rate	0.8
Mutation intensity	0.3

. In general, the Pareto front represents a set of optimal solutions under different trade-offs among conflicting objectives. The conflicts among the three objectives are rooted in their shared dependence on $N_k$. Since $N_k$ dominates all three responses, reducing $N_k$ and $\phi$ to save power inevitably suppresses $Q_Z$ and $p_{max}$. The partial decoupling is enabled by $h_{b-b}$, which preferentially reduces $P_k$ and enhances $Q_Z$ through bottom wall shear reduction, while leaving pmax largely unaffected, as it is governed solely by the blade-to-blade velocity difference. This mechanism allows the optimizer to navigate the three-way conflict without uniformly sacrificing any single objective.

Although all points on the Pareto front are non-dominated solutions, practical applications still require identifying a single most suitable design. To achieve this, the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) method was employed. This approach normalizes all objective functions into a dimensionless space and quantifies the Euclidean distance of each solution from a hypothetical ideal point, at which all performance metrics attain their optimal values. The solution with the smallest geometric distance to this ideal point is identified as the best compromise solution.

The solutions satisfying these thresholds (highlighted in blue in Figure 25) are regarded as the optimal combinations of operating parameters, and the corresponding results are summarized in

Table 15. Comparison of parameters between optimal and initial solutions.

Group	${\mathit{N}}_{\mathit{k}}/\mathbf{r}\mathbf{p}\mathbf{m}$	$\mathit{\phi }$	${\mathit{h}}_{\mathit{b}-\mathit{b}}/\mathbf{m}\mathbf{m}$	$\mathit{\mu }/\mathbf{P}\mathbf{a}·\mathbf{s}$
Optimal	94.86	0.063	2.79	40
Initial	89.08	0.124	1.43	40

.

To verify the accuracy of the solution, geometric modeling and CFD numerical simulation were conducted on the optimization scheme, and the comparison between the results and predicted values is shown in

Table 16. Verification results of Pareto optimal solutions.

Parameter	Predicted Value	CFD Solution	Relative Error
$P_k$/W	6.20	6.24	0.60%
$Q_Z$/$\mathrm{L}·{\mathrm{s}}^{-1}$	8.20 × 10−5	8.76 × 10−5	6.35%
$p_{max}$/$\mathrm{k}\mathrm{P}\mathrm{a}$	43.5	43.3	0.51%

. It can be observed that the errors for $P_k$ and $p_{max}$ do not exceed 0.6%, and the error for $Q_Z$ is around 6%. To demonstrate the improvement potential of the proposed optimization framework, a worst-case baseline was selected from the design space—specifically, the dominated solution exhibiting the largest aggregate deviation from the optimal solution across all three objectives. This selection is intended to illustrate the maximum performance gap that the Kriging–NSGA-II framework can bridge. As shown in

Table 17. Comparison of performance between optimal and initial solutions.

Parameter	Optimal	Initial	Improvement
$P_k$/W	6.24	6.75	8.15%
$Q_Z$/$\mathrm{L}·{\mathrm{s}}^{-1}$	8.76 × 10−5	7.00 × 10−5	20.03%
$p_{max}$/$\mathrm{k}\mathrm{P}\mathrm{a}$	43.3	41.1	5.01%

, compared to this baseline, the optimal solution achieves an 8.15% reduction in power consumption, a 20.03% increase in axial flow, and a 5.01% enhancement in maximum kneading pressure.

The optimal configuration is characterized by a significant increase in $h_{b-b}$ from 1.43 mm to 2.79 mm. This adjustment reduces the intense wall shear stress at the vessel bottom, directly driving the 8.15% reduction in $P_k$. Furthermore, the enlarged bottom clearance facilitates radial fluid flow, shifting the pumping region upwards and enhancing $Q_Z$ by 20.03%. To maintain kneading intensity despite reduced bottom shear, the algorithm selected a slightly higher rotation speed and a lower speed ratio. This kinematic combination optimizes the blade-to-blade interaction, achieving a 5.01% increase in $p_{max}$ while minimizing energy costs.

Based on the results of multi-objective optimization, the number of optimal solutions in different parameter ranges can be analyzed, as shown in Figure 26. For $N_k$, the distribution of the number of optimal solutions across different ranges is relatively uniform, with the highest proportion of 40% in the 90~120 range. In contrast, the distributions for $\phi$ and $h_{b-b}$ are highly uneven. For $\phi$, the highest proportion is 68.75% in the 0.0615~0.123 range, while the lowest proportion is only 11.25% in the 0.1845~0.246 range. For $h_{b-b}$, the highest proportion is 78.75% in the 2.33~2.88 range, while the lowest proportion is only 2.5% in the 1.4~1.87 range.

These characteristics also confirm the results of the operating parameter analysis: the probability of obtaining non-dominated solutions for speed, as the main influencing factor, is relatively even across the range, while the non-dominated solutions for the secondary factors are concentrated in certain ranges.

From this, general recommendations suggest flexible selection of hollow blade speed due to the existence of multiple optima. Lower speed ratios are generally preferred, with higher values requiring validation, while larger blade-to-bottom heights are recommended over smaller ones.

4. Conclusions

Through numerical simulations, the planetary stirring behavior of the twin-blade planetary mixer was studied, and the influence analysis and multi-objective optimization of its operating parameters were carried out. The conclusions are as follows:

(1)

A comprehensive analysis of the planetary stirring behavior was performed. For the operation process at a fixed cross-section, it can be divided into four stages. The global peak of the kneading pressure ($p_{max}$ = 13.9 kPa) occurs at the blade-to-blade engagement stage, while the local peak occurs at the blade-to-wall engagement stage. Stage 2 is characterized as the key stage for distributive mixing, in contrast to Stage 3, which is the primary phase for kneading. For the operation process of the hollow blade, it can be divided into two stages: the kneading face engagement stage and the non-kneading face engagement stage. The global torque peak occurs at the kneading face engagement stage, while the local peak occurs at the non-kneading face engagement stage.

(2)

The self-rotation speed of the hollow blade is the main factor affecting the performance of the twin-blade planetary mixer, while the speed ratio and blade-to-bottom height are secondary factors. The self-rotation speed of the hollow blade (Factor A) is the dominant factor affecting all three performance indicators, with p-values below 0.001 for $p_{max}$ and below 0.01 for $P_k$ across all viscosity conditions. The speed ratio (Factor B) and blade-to-bottom height (Factor C) are secondary factors, with a combined contribution of approximately 8% to $Q_Z$. This finding implies that $N_k$ should be the primary control variable in practice, while $\phi$ and $h_{b-b}$ can be used to fine-tune axial circulation with minimal energy cost.

(3)

By designing the optimal Latin Hypercube experiment, a Kriging model with high prediction accuracy was constructed, and multi-objective optimization of the operating parameter combinations was achieved through the NSGA-II. Compared to the conventional scheme, the solution reduced power consumption by 8.15%, increased axial flow by 20.03%, and improved the maximum kneading pressure by 5.01%.

(4)

Based on the Pareto front analysis, the distribution of optimal solutions across the operating parameter space was characterized. Optimal solutions are relatively uniformly distributed across the speed range, reflecting its role as the dominant but trade-off-prone variable. In contrast, 68.75% of optimal solutions are concentrated in the lowest speed ratio range (0.0615 to 0.123), and 78.75% in the largest blade-to-bottom height range (2.33 to 2.88 mm). This strongly suggests that lower speed ratios and larger blade-to-bottom heights are preferred operating strategies, providing practical guidance for process engineers to narrow the parameter search space in real applications.