A novel twisted rotor blade suitable for high-speed turbomolecular pumps: Numerical modelling and structural optimisation

Abstract

To meet the urgent need for a new design concept and solve the inaccuracy of existing performance prediction algorithms for high-speed turbomolecular pumps (TMPs), a new algorithm based on a novel twisted rotor blade is proposed. In this algorithm, the blade angle of the turbine rotor row progressively decreases from the root to the tip of the blade tooth. The feasibility and accuracy of the simulation algorithm were verified through experiments. The dependence of the simulation results on the number of simulated molecules was discussed. Both theoretical analysis and simulations confirmed the necessity of setting a twisted rotor blade in the turbine combined blade row. A comparative analysis on the performance of conventional straight-blade and twisted-blade structures based on the first-four stages of turbine combined blade rows of the F-63/55 TMP was conducted. The results indicated that the maximum pumping speed coefficient and maximum compression ratio of the optimised twisted-blade structure increased by 4.59% and 22.26%, respectively. This novel blade structure overcomes the limitations of the conventional straight-blade structure. Progressively decreasing the rotor blade angle from the root to the tip of the blade tooth is beneficial for improving the performance of TMPs. This study provides a new design concept and performance prediction algorithm for the structural optimisation of high-speed TMPs.

1. Introduction

Turbomolecular pumps (TMPs) are mainstream equipment that rely on the high-speed rotation of the rotor blade row to achieve momentum transfer of gas molecules. The momentum transfer causes directional transmission of rarefied gas, thus obtaining and maintaining a clean and high vacuum environment. TMPs have stable and reliable operation, achieving high pumping gas efficiency under a molecular flow state. Thus, they have become essential vacuum components in high-end applications such as scientific instruments, aerospace engineering, large scientific devices, integrated circuits, and coating equipment [[1], [2], [3]].

In recent years, with the rapid development of magnetic levitation and motor technology, magnetic levitation bearings have been successfully applied to TMPs [[4], [5], [6]]. Moreover, the rotational speed of TMPs has been significantly improved from thousands of revolutions per minute in the past to tens of thousands of revolutions per minute, which also brings opportunities and challenges to their development.

Owing to the high rotational speed, the present design concept of TMPs is significantly different from that in the past [7], and the existing performance prediction algorithms need to be more accurate and adapt to the current development trend [8]. In terms of geometry, the blades of the turbine blade row in the past used a straight-blade structure with a constant inclination angle [[9], [10], [11], [12]]. Although this structure has the advantages of easy processing and manufacturing, it does not match the motion law of gas molecules in a free molecular flow state, limiting improvements in the pumping speed coefficient. In addition, this structure negatively affects the gas pumping efficiency in high-speed TMPs. In the aspect of simulation algorithms, almost all calculation models established in the past were based on simplified two-dimensional [9,13,14] or three-dimensional [[15], [16], [17]] geometric models. Most performance prediction studies were conducted on single-stage [[18], [19], [20]] or two-stage blade rows [[21], [22], [23]]. Under low speeds at that time, the simulation algorithms were reasonable; however, for the current development trend of high-speed TMPs, these performance prediction algorithms are no longer suitable.

Considering the shortcomings of the conventional straight-blade structure with a constant inclination angle and the limitations of simulation algorithms, the emerging trend of high rotational speeds provides the possibility for improving the performance of TMPs. Therefore, this study aims to demonstrate the radial variation in the rotor blade angle through theoretical analysis. Using numerical modelling, a new performance prediction algorithm for TMPs is proposed, and the structural parameters of conventional straight blades are optimised through simulations. Based on a previously established calculation model with a constant blade inclination angle [12], the variation coefficient $\eta$ of the blade inclination angle along the radial direction of the rotor blades is introduced for the first time. A true twisted three-dimensional multi-stage combined blade row structure model is then established. Using this twisted rotor blade structure, a prediction algorithm for the performance of TMPs is developed and verified through experiments. Introducing a twisted combined blade row structure with improved pumping performance to replace the conventional straight-blade structure paves the way for the structural optimisation of high-speed TMPs. The limitation caused by the mismatch between the conventional straight blades and high rotational speeds is overcome.

2. Theoretical analysis

TMPs operate in a free molecular flow state [24]. As the vacuum degree increases, the gas being pumped gradually becomes rarefied. Additionally, the intermolecular forces become increasingly smaller. The conventional fluid mechanics theory is no longer applicable for analysing this free molecular flow state; instead, molecular gas dynamics principles are required to study the motion laws of gas molecules [25]. To clarify the symbols used in the following theoretical analysis and numerical modelling, Table 1 provides their definitions.

Table 1.

Definitions of symbols used in this study.

Symbol	Meaning	Symbol	Meaning
$a\_r_r$	rotor blade root radius	$b\_r_r$	stator blade root radius
$a\_r_t$	rotor blade tip radius	$b\_r_t$	stator blade tip radius
$a\_\alpha$	rotor blade angle	$b\_\alpha$	stator blade angle
$a\_h$	rotor blade height	$b\_h$	stator blade height
$a\_b$	rotor blade thickness	$b\_b$	stator blade thickness
$a\_l$	rotor blade chord length	$b\_l$	stator blade chord length
$a\_s$	rotor blade slot width	$b\_s$	stator blade slot width
$a\_s_r$	rotor blade spacing root	$b\_s_r$	stator blade spacing root
$a\_s_t$	rotor blade spacing tip	$b\_s_t$	stator blade spacing tip
$a\_m$	m-th blade of rotor blades	$v$	resultant velocity of gas molecules
$a\_n$	number of rotor blade teeth	$v_{mp}$	most probable velocity
$a\_{\alpha }_r$	rotor blade root angle	$v_r$	rotor blade linear velocity
$a\_{\alpha }_t$	rotor blade tip angle	$\omega$	rotor blade angular velocity
$R_g$	universal gas constant	$r$	rotor blade radius
$T$	gas temperature	$t$	flight time of gas molecules
$M$	gas molecular mass

Fig. 1 shows a photograph of a conventional straight-blade structure and a cross-sectional schematic of the rotor-stator blade rows for a TMP. It is evident that the angle between the tooth profile of each rotor blade and the horizontal plane (i.e. the rotor blade angle $a\_\alpha$, as depicted in Fig. 1 (b)) is fixed and unchanged along the radial direction. From the two-dimensional cross-sectional diagram of the rotor-stator blade rows, the turbine blade row can be simplified into a parallel plate model, and the slot width of the rotor blade rows can be calculated by averaging, that is, $a\_s=\left(a\_s_r+a\_s_t\right)/2$ [19]. However, this installation method with a fixed blade inclination angle conflicts with the changing airflow direction, thereby limiting the pumping gas capacity of TMPs. The specific analysis is as follows.

Fig. 1.

(a) Photograph of a conventional turbine rotor with a straight-blade structure. (b) Cross-sectional schematic of rotor-stator blade rows.

The gas molecules in a free molecular flow state move constantly, and their velocities follow the Maxwell velocity distribution [26]. Based on a large number of gas molecule motion laws and probability statistics, the self-thermal motion velocity of gas molecules is the most probable velocity $v_{mp}$, as shown in Eq. (1).

$$v_{mp}=\sqrt{\frac{2R_{g}T}{M}}$$ (1)

After colliding with the high rotational speed turbine blades, the gas molecules exhibit a certain tangential velocity $v_r$, which can be calculated using Eq. (2).

$$v_r=\omega r$$ (2)

Therefore, combined with the velocity decomposition diagram of the gas molecules in the rotor blade rows, as displayed in Fig. 2, the direction angle $\alpha$ of the airflow can be obtained using Eq. (3).

$$\alpha =\arctan \left(\frac{v_{mp}}{v_r}\right)=\arctan \left(\frac{\sqrt{2R_{g}T/M}}{\omega r}\right)$$ (3)

Fig. 2.

Velocity decomposition diagram of gas molecules in the rotor blade rows: (a) airflow not parallel to the blade inclination angle; (b) airflow parallel to the blade inclination angle.

From Eq. (3), it is evident that for the same type of gas, assuming that the gas temperature is invariable, the most probable velocity $v_{mp}$ of gas molecules also remains constant. However, owing to the influence of the rotational motion of the rotor blade rows, the tangential velocity $v_r$ increases with an increase in blade radius. Therefore, the moving direction of the airflow is also constantly changing, and the direction angle $\alpha$ between the radial and horizontal planes is gradually decreasing. To make the airflow pass unobstructed through the turbine rotor blade row, the rotor blade angle should progressively decrease along the blade radius direction in the turbine rotor blade row. In other words, the rotor needs to have a twisted-blade structure.

3. Establishing the calculation model.

Based on a previously established two-stage combined blade row calculation model [12], a true three-dimensional multi-stage combined blade row geometric structure model is established. This model considers the gaps between the rotor and stator blade rows and between the turbine blade rows and pump chamber. Moreover, a variation coefficient $\eta$ of the blade inclination angle along the radial direction of the rotor blade rows is introduced. In Fig. 3, only the cross-sectional view of the four-stage combined blade rows is indicated, where the rotor blade tip angle $a\_{\alpha }_t$ differs from the rotor blade root angle $a\_{\alpha }_r$. Actually, the geometric model established in this study is based on a multi-stage combined blade model with more than four stages.

Fig. 3.

Cross-sectional view of turbine blade rows: (a) rotor-stator-rotor-stator blade rows; (b) rotor blade tip; (c) rotor blade root.

According to Eq. (3), the tangent value of the gas molecule motion direction angle $\alpha$ of the rotor blade rows varies inversely along the radial direction. To match this trend, the variation equation of the blade inclination angle $a\_\alpha$ along the radial direction of the rotor blade rows is given by Eq. (4).

$$a\_\alpha =\frac{a\_{\alpha }_t-a\_{\alpha }_r}{a\_r_t-a\_r_r}\eta +\frac{a\_r_t\cdot a\_{\alpha }_r-a\_r_r\cdot a\_{\alpha }_t}{a\_r_t-a\_r_r}$$ (4)

The variation coefficient $\eta$ of the blade inclination angle along the radial direction of the rotor blade rows can be solved using Eq. (5). It is directly related to the physical state parameters (position coordinates, flight time, etc.) of the gas molecules in the turbine blade rows, making the rotor blade rows more consistent with the motion direction of the airflow.

$$\eta =y\cos \left(\frac{2\pi a\_m}{a\_n}+\omega t\right)+x\sin \left(\frac{2\pi a\_m}{a\_n}+\omega t\right)$$ (5)

A calculation algorithm for predicting the performance of multi-stage combined blade rows was developed using Visual Basic 6.0, as illustrated in Fig. 4. The blade equations for the rotor and stator blade rows, solution equation for the flight time of gas molecules, initial position equation, and velocity equation of gas molecules involved in this study can be referred to in a previous study [12]. Using the test particle Monte Carlo method [27] to simulate the flight trajectory of gas molecules in the turbine blade rows, the number of molecules that reach the outlet [28] was counted, and the performance of TMPs was accurately predicted (i.e. maximum pumping speed coefficient $Q_{\max }$ and maximum compression ratio $K_{\max }$).

Fig. 4.

Calculation algorithm for predicting the performance of multi-stage combined blade rows.

Equations (6), (7) provide the solutions for the maximum pumping speed coefficient $Q_{\max }$ and maximum compression ratio $K_{\max }$, respectively. In these equations, ${\sum }_{12}$ is the positive transmission probability, which represents the probability of gas molecules passing through from region 1 to region 2 (see Fig. 3). In contrast, ${\sum }_{21}$ is the reverse transmission probability, which represents the probability of gas molecules passing through from region 2 to 1. $A_1$ and $A_2$ are the effective pumping areas of the inlet and outlet stages of the turbine blade rows, respectively, which make the performance prediction of TMPs more accurate. Their formulae are provided in Ref. [23]. The values of $A_1$ and $A_2$ were not considered in Ref. [12]. The maximum pumping speed coefficient $Q_{\max }$ and compression ratio $K_{\max }$ are two important performance parameters of TMPs [29].

$$Q_{\max }={\sum }_{12}-\frac{A_2}{A_1}{\sum }_{21}$$ (6)

$$K_{\max }=\frac{A_1\cdot {\sum }_{12}}{A_2\cdot {\sum }_{21}}$$ (7)

3. Experimental verification

To verify the feasibility and accuracy of the multi-stage combined blade row performance prediction algorithm for TMPs based on the three-dimensional geometric model of twisted rotor blade rows, experiments and simulations were conducted using the FF-100/150 TMP. Considering the issue of intellectual property protection for this TMP, only the structural parameters of the first eight stages of turbine combination blade rows are presented in Table 2.

Table 2.

Structural parameters of the first eight stages of combined blade rows of the FF-100/150 TMP.

Stage no.	Blade root diameter (mm)	Blade tip diameter (mm)	Blade height (mm)	Blade angle (°)	Blade thickness (mm)	No. of blades
1- rotor	48	100	10	40	1.4	14
2-stator	58	105	5	35	1.2	24
3-rotor	62	100	5	34	1.2	26
4-stator	68	105	3	20	0.8	24
5- rotor	70	100	3	18	0.6	24
6-stator	77	105	3	11	0.8	20
7-rotor	79	100	3	12	0.6	20
8-stator	77	105	3	11	0.8	20

A performance test platform for the TMP was established in the laboratory, and a simplified schematic of the experimental measurement device is depicted in Fig. 5. The pumping performance experiments were conducted using conventional straight turbine blades of the FF-100/150 TMP, with the following working conditions: rated speed - 51000 rpm, ambient temperature - 19 °C, air humidity - 31%, and type of gas - nitrogen. The measurement devices, methods, and steps of this experimental test are primarily the same as those in Ref. [30] and will not be detailed here.

Fig. 5.

Simplified schematic of experimental measurement device: 1. nitrogen gas bag; 2. adjustable gas inlet valve; 3. test cover; 4. vacuum gauge for measuring the pressure on the upper end of the test cover; 5. vacuum gauge for measuring the TMP inlet pressure; 6. TMP; 7. vacuum gauge for measuring the TMP outlet pressure; 8. backup pump.

Fig. 6 displays the experimental and simulation results for the FF-100/150 TMP. It is evident that the simulation values obtained from the pumping performance prediction algorithm for multi-stage combined blade rows are almost consistent with the experimental values. Therefore, the feasibility and accuracy of the established numerical calculation model are verified, laying the foundation for the subsequent investigation of the performance of the four-stage combined blade rows.

Fig. 6.

Experimental and simulation values for the FF-100/150 TMP.

4. Results and discussion

4.1. Determination of the number of simulated molecules

The test particle Monte Carlo method [27] has been used to study the motion laws of gas molecules by tracking the trajectory of test particles This method can accurately reproduce the pumping process of TMPs. It uses particles to replace gas molecules; thus, the number of test particles (simulated molecules) affects the error in the simulation results. To make the results of the simulations more reliable and reduce sample statistical errors, this study analysed the independence of the number of simulated molecules to determine the appropriate value.

Taking the first four stages of combined blade rows, which have significant impacts on the pumping speed, as an example, the F-63/55 TMP [31] was selected to study the variation in its performance with the simulated number of molecules. The structural parameters are listed in Table 3. In the simulation process, the TMP rotational speed was 60000 rpm, the simulated number of stages was 4, the relative molecular mass of gas molecules was 29, and the ambient temperature was 25 °C.

Table 3.

Structural parameters of the first four stages of combined blade rows of F-63/55 TMP [30].

Stage no.	Blade root diameter (mm)	Blade tip diameter (mm)	Blade height (mm)	Blades angle (°)	Blade thickness (mm)	No. of blades
First-stage rotor	44	67	6	40	0.8	16
Second-stage stator	44	67	3	30	0.3	26
Third-stage rotor	52	67	2	30	0.8	24
Fourth-stage stator	52	67	3	26	0.3	30

Fig. 7 depicts the variation in the performance of the F-63/55 TMP with the simulated number of molecules. Initially, the errors of the maximum pumping speed coefficient and compression ratio fluctuate significantly; however, as the simulated number of molecules increases, the errors become increasingly smaller. When the number of simulated molecules is $1.1\times {10}^6$, the results of the simulation gradually stabilise. Theoretically, the larger the number of molecules selected for the simulation, the more accurate the results. However, considering that increasing the number of molecules increases the time required for the calculations, only $1.1\times {10}^6$ was selected to perform the subsequent analysis. With this number of molecules, the maximum pumping speed coefficient $Q_{\max }$ and compression ratio $K_{\max }$ were 0.3263 and 26.8003, respectively.

Fig. 7.

Variation in the F-63/55 TMP performance with the simulated number of molecules.

4.2. Investigation of pumping performance

The maximum pumping speed coefficient and compression ratio are two important performance indicators for TMPs; thus, it is crucial to investigate their variation with the blade velocity ratio ($c=2\pi \omega \left(a\_r_r+a\_r_t\right)/v_{mp}/2$) [19]. Taking the first four stages of combined blade rows, which have significant impacts on the pumping speed, as an example, the structural parameters of the F-63/55 TMP were selected [31]. The inclination angles of the first- and third-stage rotor blade rows were changed to obtain the variations in the maximum pumping speed coefficient and compression ratio for the four-stage combined blade rows with the blade velocity ratio.

Fig. 8 illustrates the variations in the maximum pumping speed coefficient and compression ratio for the four-stage combined blade rows under different inclination angles in the first-stage rotor blade rows with the blade velocity ratio. It can be observed that as the blade velocity ratio increases, almost all pumping performances of the turbine blade rows increase.

Fig. 8.

Variation in performance with blade velocity ratio at different angles for the first-stage rotor blade rows: (a) maximum pumping speed coefficient; (b) maximum compression ratio.

The variation in the maximum pumping speed coefficient with the blade velocity ratio under different inclination angles was examined. It was found that, during the process of increasing the blade velocity ratio (i.e. the variation in the linear velocity of the rotor blade row from the blade root to the blade tip), the blade inclination angle corresponding to the maximum value of the maximum pumping speed coefficient decreases from 55° to 35°, as shown in Fig. 8 (a). Similarly, by observing the variation in the maximum compression ratio in Fig. 8 (b), when the blade velocity ratio is larger, the blade inclination angle corresponding to the values from the minimum to the maximum of the maximum compression ratio decreases from 55° to 35°. This indicates that the blade inclination angle should be smaller at the blade tip in the rotor blade row.

Fig. 9 shows the variations in the maximum pumping speed coefficient and compression ratio for the four-stage combined blade rows under different inclination angles in the third-stage rotor blade rows with the blade velocity ratio. It can be observed that as the blade velocity ratio increases, the performance of the combined blade row also increases; however, the growth trend gradually decreases. Theoretically, the performance of TMPs continuously increases with an increase in the rotational speed of the turbine rotor blade row, and the maximum compression ratio and blade velocity ratio have an exponential relationship [22].

Fig. 9.

Variation in performance with blade velocity ratio at different angles for the third-stage rotor blade row: (a) maximum pumping speed coefficient; (b) maximum compression ratio.

By examining the maximum pumping speed coefficient alone, it seems that the influence of the blade inclination angle of the third-stage rotor blade row is relatively small, as depicted in Fig. 9 (a). This is mainly because the maximum pumping speed coefficient of the combined blade row depends on the magnitude of the blade inclination angle of the first-stage rotor blade row. Fig. 9 (b) displays the variation in the maximum compression ratio for the four-stage combined blade row under different inclination angles in the third-stage rotor blade row with the blade velocity ratio. It is evident that the inclination angle of the third-stage rotor blade row significantly affects the maximum compression ratio of the combined blade row. As the blade velocity ratio increases, the influence of the blade inclination angle on the maximum compression ratio increases. In addition, the smaller the blade inclination angle, the larger the maximum compression ratio of the combined blade row.

From the analysis of Fig. 8, Fig. 9, as the blade velocity ratio increases, the blade angle required for the turbine rotor blade row becomes increasingly smaller. In other words, under high rotational speeds, the blade inclination angle of the turbine rotor blade row should be progressively decreasing from the blade root to the blade tip. This is consistent with the theoretical analysis described in Section 2. Therefore, the blade structure of the turbine combined blade row must be twisted, and the blade inclination angle should be progressively decreasing from the blade root to the blade tip. This configuration is consistent with the direction of the airflow movement.

4.3. Optimisation of the conventional straight-blade structure

According to the above findings, using a twisted structure for the turbine rotor blade row is more conducive for improving the performance of TMPs. Therefore, in this investigation, the structural parameters of the first four stages of turbine combined blade rows of the F-63/55 TMP [31] were selected, as listed in Table 3. This TMP model was available in the laboratory, and the structural parameters of the first four stages were chosen because they had the greatest impact on the performance of the TMP. For the structural optimisation of the rotor blade inclination angle, the previous operating and environmental parameters applied in the TMP simulation were used.

Fig. 10 illustrates the variation in the performance of the four-stage combined blade row with the inclination angle of the first-stage rotor blades. Because of the involvement of the rotor blade root angle and rotor blade tip angle in this analysis, a control variable method was used to study the variation in the performance of the four-stage combined blade row with the rotor blade root angle and tip angle. From Fig. 10, it is evident that as the first-stage rotor blade root angle and tip angle increase, both the maximum pumping speed coefficient and compression ratio of the combined blade row first increase and then decrease. The optimal values can be found by determining the angle that matches the airflow direction. When the first-stage rotor blade root angle and tip angle are 46° and 43°, the maximum pumping speed coefficient and compression ratio of the four-stage combined blade row are 0.3339 and 31.1723, respectively.

Fig. 10.

Variation in performance of the four-stage combined blade row with the first-stage rotor blade (a) root angle and (b) tip angle.

Using the same method, optimisation was conducted for the third-stage rotor blade root angle and tip angle, as shown in Fig. 11. It is clear that as the third-stage rotor blade root angle and tip angle increase, the maximum pumping speed coefficient and compression ratio of the four-stage combined blade row fluctuate; however, maximum values can be observed. This is because the third-stage rotor blade row belongs to the middle-stage of the combined blade row; therefore, its impact on the performance differs from that of the first-stage rotor blade row. When the third-stage rotor blade root angle is 31° and the tip angle is 19°, the maximum pumping speed coefficient and compression ratio of the four-stage combined blade row are 0.3413 and 32.7672, respectively, as depicted in Fig. 11.

Fig. 11.

Variation in performance of the four-stage combined blade row with the third-stage rotor blade (a) root angle and (b) tip angle.

By analysing Fig. 10, Fig. 11, the optimised blade inclination parameters and simulation results for the conventional straight-blade structure were obtained, as displayed in Table 4. Fig. 7 indicates that the maximum pumping speed coefficient and compression ratio obtained from the conventional straight-blade structure parameters before optimisation are 0.3263 and 26.8003, respectively. Comparative analysis revealed that with continuous optimisation of the blade root angle and tip angle of the first- and third-stage rotor rows, the growth rates of the maximum pumping speed coefficient and compression ratio also continued to increase. The maximum pumping speed coefficient and compression ratio for the final optimised structural parameters increased by 4.59% and 22.26%, respectively. Experimental testing and platform construction of the twisted-blade row are currently ongoing, and this will be the focus of future work.

Table 4.

Optimised blade inclination parameters and simulation results for the conventional straight-blade structure.

Stage no.	Blade angle (°)		$Q_{\max }$	$Q_{\max }$ growth rate (%)	$K_{\max }$	$K_{\max }$ growth rate (%)
	Root angle	Tip angle
First-stage rotor row	46	40	0.3314	1.58	30.5569	14.02
	46	43	0.3339	2.35	31.1723	16.31
Second-stage stator row	30	30	–	–	–	–
Third-stage rotor row	31	30	0.3354	2.79	31.8672	18.91
	31	19	0.3413	4.59	32.7672	22.26
Fourth-stage stator row	26	26	–	–	–	–

5. Conclusions

A new algorithm for predicting the performance of high-speed TMPs was proposed based on a multi-stage combined blade row geometric structure model of a twisted rotor blade. The results of the simulations and experiments were consistent, verifying the feasibility and accuracy of the prediction algorithm. Taking the first four stages of combined blade rows, which have significant impacts on the pumping speed, as an example, this study found that.

• 1.As the blade velocity ratio increased, the required angle of the turbine rotor blade decreased.
• 2.The concept of adopting a twisted rotor blade design for high-speed TMPs, which was validated by theoretical analysis, was proposed.
• 3.The maximum pumping speed coefficient and compression ratio of the optimised blade structure were increased by 4.59% and 22.26%, respectively.
• 4.This study provides a new performance prediction algorithm and design concept for enhancing the structural design of TMPs, overcoming the limitations of the performance of conventional TMPs with a straight-blade structure.

In subsequent work, experimental testing of the twisted rotor blade will be conducted.

Ethical approval

Not applicable.

Data availability statement

No data were used for the research described in this article.

CRediT authorship contribution statement

Kun Sun: Writing – original draft, Validation, Methodology, Funding acquisition. Haishun Deng: Funding acquisition. Shiwei Zhang: Conceptualization. Xin Hu: Investigation. Cheng Wang: Funding acquisition. Senhui Wang: Software, Data curation. Kun Li: Supervision. Long Wang: Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.