Magnetic field finite element analysis of diverging stepped ferrofluid seal with a large gap and two magnetic sources

Abstract

In order to improve the pressure capability of the ferrofluid seal under the condition of large sealing gap, a diverging stepped ferrofluid seal with a large gap and two magnetic sources was designed. The magnetic field distributions in the gap of the ferrofluid seal were calculated by finite element method. The effects of the pole teeth number, the height of the pole teeth and the sealing gap on the pressure capabilities of the diverging stepped ferrofluid seal are studied and compared with ordinary ferrofluid seal. The results show that the change of the radial pole teeth number has little effect on the pressure capability of the stepped ferrofluid seal. The pressure capability of stepped ferrofluid seal decreases with the increase of the axial pole teeth number. The pressure capability of the stepped ferrofluid seals reach maximum when the radial and axial pole teeth height is 0.6 mm. The pressure capability of stepped ferrofluid seal decreases as the radial sealing gap increases. The pressure capability of stepped ferrofluid seal decreases firstly and then increases with the increase of the axial sealing gap. Comparing with ordinary ferrofluid seal, the stepped ferrofluid seal has better pressure capability.

Keywords

Large gap diverging ferrofluid seal numerical simulation

1. Introduction

Ferrofluid is a new type of functional material [1] which is a stable colloid liquid, composed of magnetic particles, base carrier fluid and surfactant [2,3]. Ferrofluid seal technology is a kind of technology that makes use of the ferrofluid subjected to applied magnetic field to form a sealing ring and achieve sealing effect. Compared to traditional seal, ferrofluid seal has the advantages of long life, zero leakage and pollution-free, and is widely used in national defense, machinery, chemical industry and petroleum and other fields [4–6]. In the field of ferrofluids seal, the gap less than 0.3 mm is usually called small gap [7], and the gap greater than 0.3 mm is called large gap [8]. In the application field requiring large gap, the pressure capability of ordinary ferrofluids seal will decrease significantly with the increase of the sealing gap, so improving the pressure capability of ordinary ferrofluids seal is of great significance to broaden the application field of ferrofluids seal.

Zhao [9] and Radionov [10] carried out numerical analysis of ferrofluid seal with small gap. Szczech [11] and Wang and Li [12] carried out experimental research on ferrofluid seal with small gap. Chen and Yang [13] studied the effects of structural parameters of ferrofluid seal with small gap on sealing capabilities. The above research is focused on non-stepped ferrofluids seal with small gap. Parmar S. et al. [14] conducted an experimental study on ordinary ferrofluid seal with a large sealing gap of 0.7 mm. To improve the pressure capabilities, Yang and Sun [15] have carried out numerical and experimental research the influence of the sealing gap and leakage path on converging stepped ferrofluids seal with the large gap. Yang and Li [16] carried out experimental research the capabilities of ordinary and stepped ferrofluids seal with large gap. However, the stepped ferrofluid seal of above studies has the defects that the numbers of radial and axial sealing gaps are not equal and the influences of the key parameters such as the number of pole teeth and the height of pole teeth on the sealing capabilities are not investigated.

In this paper, a diverging stepped ferrofluid seal with a large gap and two magnetic sources is designed to make the numbers of radial and axial sealing gaps consistent. The effects of the key parameters such as the number of pole teeth, the radial and axial sealing gaps, and the height of pole teeth on its sealing capability are analyzed by finite element method, and its pressure capabilities are compared with that of traditional ferrofluids seal. It provides an important theoretical guiding significance for designing a diverging stepped ferrofluid seal with a large gap and two magnetic sources.

2. Diverging stepped ferrofluid seal formula

Normally, the Bernoulli equation of a magnetic fluid can be represented by the following equation [17] $\begin{eqnarray}\displaystyle P+\frac{1}{2}{\rho}_{f}V^{2}+{\rho}_{f}gh-{\mu}_{0}\int _{0}^{H}MdH=C & & \displaystyle\end{eqnarray}$ (1) Where P is the pressure of magnetic fluid at a certain position; 𝜌_f is the density of magnetic fluid; V is the velocity of magnetic fluid at a certain position; g is gravitational acceleration; h is the height of the magnetic fluid at a certain position; μ₀ is the vacuum permeability; M is magnetization of ferrofluid; H is external magnetic field strength; and c is the constant. For the static pressure in ferrofluid seals, the effect of velocity on ferrofluid seals can be neglected and the gravity effect in the sealing gap should also be ignored. So, the total sealing capability of a ferrofluid seal is simplified into the following expression: $\begin{eqnarray}\displaystyle {\rm\Delta}P={\mu}_{0}M_{S}\mathop{\sum }_{i=1}^{N}(H_{max}^{i}-H_{min}^{i})=M_{S}\mathop{\sum }_{i=1}^{N}(B_{max}^{i}-B_{min}^{i}) & & \displaystyle\end{eqnarray}$ (2) where $H_{max}^{i},H_{min}^{i},B_{max}^{i}$ and $B_{min}^{i}$ are the maximum and minimum magnetic field strengths and the maximum and minimum magnetic flux densities under the i-th pole piece respectively. N is the total number of sealed pole pieces. We express the total sealing capability of the diverging ferrofluid as [15] $\begin{eqnarray}\displaystyle {\rm\Delta}P_{c}=\mathop{\sum }_{i}^{N}(P_{jr}+{\lambda}P_{ja}). & & \displaystyle\end{eqnarray}$ (3)

In (3), P _jr and P _ja are the pressure capabilities of the ferrofluid seal in the radial and in the axial sealing gaps formed by the j-th pole piece and the stepped shaft. We assume that when P _jr < P _ja then 𝜆 = 1 and, otherwise 𝜆 = 0.

3. Structure design of diverging stepped ferrofluid seal with a large gap and two magnetic sources

In order to study the influence of the key parameters of the stepped ferrofluid seal on the sealing capabilities, this paper designs a diverging stepped ferrofluid seal with a large gap and two magnetic sources as shown in Fig. 1. Its structural parameters are shown in Table 1. The permanent magnet material in the diverging stepped ferrofluid seal is NdFeB and its coercive force and relative magnetic permeability are 1.356 × 10⁶ A∕m and 1.05 respectively. The material of the pole piece and the stepped shaft are both 2Cr13. The magnetic fluid uses an oil-based magnetic fluid having a saturation magnetization of 40 KA/m.

Fig. 1.

Magnetic field distributions for different radial sealing gaps. (i), 0.4 mm; (ii), 0.5 mm; (iii), 0.6 mm; (iv), 0.7 mm.

Table 1

Diverging stepped ferrofluid seal with a large gap and two magnetic sources structure parameters

Item	Value
Inner radius of the 1/2/3 (mm)	14.9/18.5/22.1
Outer radius of the 1/2/3 (mm)	30
Length of the pole piece (mm)	5/8/5
Permanent magnet height (mm)	5
Inner radius of each permanent magnet (mm)	24
Outer radius of each permanent magnet (mm)	30
Axial width of pole teeth (mm)	5/8/5
Axial sealing gap width (mm)	0.4/0.5/0.6/0.7
Radial sealing gap height (mm)	0.4/0.5/0.6/0.7

Introduce a two-dimensional physical model of the diverging stepped ferrofluid seal with a large gap and two magnetic sources built in the Auto CAD software into ANSYS finite element software to define the material properties of each component. The smart meshing is selected and the accuracy of the grid is 1. Boundary conditions are applied to make sure that magnetic field lines are parallel to the model boundary. The magnetic field strength can be solved by a solver.

4. Results and discussion

4.1. Influence of the pole teeth number

The theoretical pressure capability of the diverging stepped ferrofluid seal with a large gap and two magnetic sources is related to the number of pole teeth. Study on the influence of pole teeth number of the diverging stepped ferrofluid on the sealing capability, provides an important theoretical basis for the design of the diverging stepped ferrofluid seal with a large gap and two magnetic sources.

4.1.1. Influence of the axial pole teeth number

When the axial and radial sealing gap are 0.4 mm, the axial and radial pole teeth height are 0.7 mm and the radial pole teeth number is 4, the magnetic flux density in the sealing gap under the different axial pole teeth number is shown in Fig. 2 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different axial pole teeth number is shown in Fig. 3.

Fig. 2.

Magnetic field distributions for different axial pole teeth number. (i), 1; (ii), 2; (iii), 3.

Fig. 3.

Critical leakage pressure ΔP as a function of the number of axial pole teeth.

It can be seen from Fig. 2 that as the number of axial pole teeth increases, the magnetic flux density in the axial seal gap gradually decreases. This is because the magnetic resistance in the axial seal gap decreases as the number of axial pole teeth increases and the magnetic flux density in the magnetic circuit is decrease. It is not difficult to find that the magnetic flux density in the radial seal gap increases as the number of axial pole teeth increases. This is because the total magnetic potential in the magnetic circuit is constant. So the magnetic flux density in the axial seal gap is decreasing. At the same time, the magnetic flux density in the radial sealing gap increases slowly.

Figure 3 shows that as the number of axial pole teeth increases, the theoretical pressure capability of the stepped ferrofluid gradually decreases. This is because as the number of axial pole teeth increases, the magnetic flux density difference in the axial sealing gap gradually decreases and the magnetic flux density difference in the radial sealing gap gradually increases. Therefore, the magnetic flux density difference in the axial sealing gap is smaller than the magnetic flux density difference in the radial sealing gap. According to the pressure capability theory of the diverging stepped ferrofluid seal, its pressure capability depends on pressure capability in the radial sealing gap and the pressure capability in radial sealing gap decrease as the number of axial pole teeth increase. Therefore, as the number of axial pole teeth increases, the pressure capability of the stepped ferrofluid seal gradually decreases. It can also be seen from Fig. 3 that the pressure capability of the stepped ferrofluid seal far exceeds the pressure capability of the ordinary ferrofluid seal, and the maximum pressure capability can exceed 10 atmospheres of ordinary ferrofluid seal.

4.1.2. Influence of the radial pole teeth number

When the axial and radial sealing gaps are 0.4 mm, the axial and radial pole teeth height are 0.7 mm and the axial pole teeth number is 1, the magnetic flux density in the sealing gap under the different radial pole teeth number is shown in Fig. 4 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different radial pole teeth number is shown in Fig. 5.

Fig. 4.

Magnetic field distributions for different radial pole teeth number. (i), 1; (ii), 2; (iii), 3; (iv), 4.

Fig. 5.

Critical leakage pressure ΔP as a function of the number of radial pole teeth.

It can be seen from Fig. 4 that as the number of radial pole teeth increases, the magnetic flux density in the radial sealing gap gradually decreases and the magnetic flux density in the axial sealing gap increases. This is because as the number of radial pole teeth increases, the magnetic resistance in the magnetic circuit will decrease and the magnetic flux density in the magnetic circuit increase. According to the magnetic circuit theorem, the magnetic flux density in the axial sealing gap decreases while the magnetic flux density in the radial sealing gap increases.

From Fig. 5, it can be seen that the theoretical pressure capability of the stepped ferrofluid seal increases firstly and then decreases with the increase of the radial pole teeth number. The pressure capability increases firstly is because the magnetic flux leakage in the radial sealing gap decreases as the number of radial pole teeth increases. The theoretical pressure capability decreases later because the theoretical pressure capability reaches maximum value when the number of radial pole teeth is 4. Part of the radial pole teeth will not work when the number of radial pole teeth continues to increase. It is not difficult to find that the theoretical pressure capability of the stepped ferrofluid seal changes with the increase of the radial pole teeth number by no more than one atmosphere. Therefore, the number of radial pole teeth has no effect on the theoretical pressure capability of the stepped ferrofluid seal. It can also be seen from Fig. 5 that the theoretical pressure capability of ordinary ferrofluid seal increases with the increase of the radial pole teeth number. The reason is that the magnetic flux leakage of ordinary ferrofluid seal will be more serious when the number of radial pole teeth is small. Comparing the theoretical pressure capability between the ordinary ferrofluid seal and the stepped ferrofluid seal, it can be found that the stepped ferrofluid seal pressure capability is superior to the ordinary ferrofluid seal when the number of radial pole teeth is small. This is because ordinary ferrofluid seal only has a radial sealing gap, while stepped ferrofluid seal has both axial and radial sealing gaps.

4.2. Influence of pole teeth height

The theoretical pressure capability of diverging stepped ferrofluid seal with a large gap and two magnetic sources is related to its pole teeth height. Studying the influence of pole teeth height of the diverging stepped ferrofluid seal on the sealing capability will provide an important theoretical basis for the design of the diverging stepped ferrofluid seal with a large gap and two magnetic sources.

4.2.1. Influence of radial pole teeth height

When the axial and radial sealing gap are 0.4 mm, the axial pole teeth height is 0.7 and the numbers of axial and radial pole teeth are 1 and 4 respectively, the magnetic flux density in the sealing gap under the different radial pole teeth height is shown in Fig. 6 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different radial pole teeth height is shown in Fig. 7.

Fig. 6.

Magnetic field distributions for different radial pole teeth height. (i), 0.4 mm; (ii), 0.5 mm; (iii), 0.6 mm; (iv), 0.7 mm.

Fig. 7.

Critical leakage pressure ΔP as a function of the radial pole teeth height.

It can be seen from Fig. 6 that the magnetic flux density in the axial sealing gap of the diverging stepped ferrofluid seal increases as the height of the radial pole teeth increases. This is because the increase of magnetic potential in the radial sealing gap with the height of the radial pole teeth increases and the increase of magnetic potential in the axial sealing gap with increase of magnetic potential in the radial sealing gap. So the magnetic flux density in the axial sealing gap will increase. It can also be seen from Fig. 8 that the magnetic flux density in the radial sealing gap of the diverging stepped ferrofluid increases as the height of the radial pole teeth decreases. This is because the magnetic resistance of the radial sealing gap increase as the height of the radial pole teeth increases and the magnetic resistance of whole magnetic circuit will decrease. Therefore, the magnetic flux density in the whole magnetic circuit will decrease and the decrease of magnetic flux density in the radial sealing gap with the increase of magnetic flux density in the axial sealing gap.

Fig. 8.

Magnetic field distributions for different axial pole teeth height. (i), 0.4 mm; (ii), 0.5 mm; (iii), 0.6 mm; (iv), 0.7 mm.

It can be seen from Fig. 7 that the theoretical pressure capability of the diverging stepped ferrofluid seal increases firstly and then decreases with the increase of the radial pole teeth height. The pressure capability increases firstly because the magnetic flux density difference in the radial sealing gap does not change too much as the radial pole teeth height increases and the magnetic flux density difference in the axial sealing gap increase slowly. The theoretical pressure capability of the diverging stepped ferrofluid seal decreases later because the theoretical pressure capability reaches maximum value when the height of radial pole teeth is 0.6 mm. The magnetic flux leakage occurs in the radial sealing gap when the height of radial pole teeth continues to increases. It can be seen from Fig. 7 that the theoretical pressure capability of the ordinary ferrofluid seal decreases with the increase of the radial teeth height. The reason is that the ordinary ferrofluid seal only has radial sealing gap and its magnetic flux density in radial sealing gap decreases as radial pole teeth height increases. Comparing the theoretical pressure capability between the ordinary ferrofluid seal and the stepped ferrofluid seal, it can be found that the theoretical pressure capability of stepped ferrofluid seal is superior to the ordinary ferrofluid seal when the height of radial pole teeth is 0.4 mm. This is because ordinary ferrofluid seal only has a radial sealing gap, while stepped ferrofluid seal have both axial and radial sealing gaps.

4.2.2. Influence of axial pole teeth height

When the axial and radial sealing gap are 0.4 mm, the radial pole teeth height is 0.7 and the numbers of axial and the radial pole teeth are 1 and 4 respectively, the magnetic flux density in the sealing gap under the different axial pole teeth height is shown in Fig. 8 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different axial pole teeth height is shown in Fig. 9.

Fig. 9.

Critical leakage pressure ΔP as a function of the axial pole teeth height.

It can be seen from Fig. 8 that the magnetic flux density in the radial sealing gap of the diverging stepped ferrofluid seal increases as the height of the axial pole teeth increases. This is because the increase of magnetic potential in the axial sealing gap with the height of the axial pole teeth increases and the increase of magnetic potential in the radial sealing gap with increase of magnetic potential in the axial sealing gap. So the magnetic flux density in the radial sealing gap will increase. It can also be seen from Fig. 8 that the magnetic flux density in the axial sealing gap of the diverging stepped ferrofluid increases as the height of the radial pole teeth decreases. This is because the magnetic resistance of the axial sealing gap increase as the height of the radial pole teeth increases and the magnetic resistance of whole magnetic circuit will decrease. Therefore, the magnetic flux density in the whole magnetic circuit will decrease and the decrease of magnetic flux density in the radial sealing gap with the increase of magnetic flux density in the radial sealing gap.

It can be seen from Fig. 9 that the theoretical pressure capability of the diverging stepped ferrofluid seal increases firstly and then decreases with the increase of the axial pole teeth height. The pressure capability increases firstly because as the increases of axial pole teeth height, the magnetic flux density difference in the radial sealing gap is gradually increased and part of the magnetic flux density difference in the axial sealing gap is smaller than the magnetic flux density difference in the radial sealing gap. According to the pressure capability theory of the diverging stepped ferrofluid seal, the axial seal gap will invalid. The theoretical pressure capability decreases later because the theoretical pressure capability reaches maximum value when the height of axial pole teeth is 0.6 mm. The magnetic flux leakage in the axial sealing gap when the height of radial pole teeth continues to increases. It is difficult to find that the theoretical pressure capability of stepped ferrofluid seal is always superior to the ordinary ferrofluid seal. This is because ordinary ferrofluid seal only have a radial sealing gap, while stepped ferrofluid seal have both axial and radial sealing gaps.

4.3. Influence of sealing gap

The theoretical pressure capability of the diverging stepped ferrofluid seal is related to its sealing gap. Studying the influence of radial sealing gap height and axial sealing gap width on the sealing capability of the diverging stepped ferrofluid seal will provide an important theoretical basis for the design of the diverging stepped ferrofluid seal with a large gap and two magnetic sources.

4.3.1. Influence of radial sealing gap

When the numbers of axial and the radial pole teeth are 1 and 4 respectively, the axial and radial pole teeth height are 0.7 mm and the axial sealing gap width is 0.4 mm, the magnetic flux density in different radial sealing gap is shown in Fig. 10 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different radial sealing gap height is shown in Fig. 11.

Fig. 10.

Magnetic field distributions for different radial sealing gaps. (i), 0.4 mm; (ii), 0.5 mm; (iii), 0.6 mm; (iv), 0.7 mm.

Fig. 11.

Critical leakage pressure ΔP as a function of the height of the radial sealing gap.

It can be seen from Fig. 10 that the magnetic flux density in the axial sealing gap of the diverging stepped ferrofluid seal increases as the radial sealing gap height increases. This is because the increase magnetic potential in the radial sealing gap with the radial sealing gap height increases and the increase of magnetic potential in the axial sealing gap with increase of magnetic potential in the radial sealing gap. So the magnetic flux density in the axial sealing gap will decrease. It can also be seen from Fig. 10 that the magnetic flux density in the radial sealing gap of the diverging stepped ferrofluid increases as the radial sealing gap height decreases. This is because the magnetic resistance of the radial sealing gap increase as the radial sealing gap height increases and the magnetic resistance of whole magnetic circuit will decrease. Therefore, the magnetic flux density in the whole magnetic circuit will decrease and the decrease of magnetic flux density in the radial sealing gap with the increase of magnetic flux density in the axial sealing gap.

It can be seen from Fig. 11 that the theoretical pressure capability of the diverging stepped ferrofluid seal decreases with the increase of the radial sealing gap height. This is because with the increase of the radial sealing gap height, the increase of magnetic flux density difference in the axial sealing gap are smaller than decrease values of magnetic flux density difference in the radial sealing gap. It can be seen from Fig. 11 that with the increase of the radial sealing gap height of the diverging stepped ferrofluid seal, the decreasing trend of the pressure capability of ordinary ferrofluid seals is greater than that of diverging stepped ferrofluid seals. This is because ordinary ferrofluid seal only have a radial sealing gap, while stepped ferrofluid seal have both axial and radial sealing gaps.

4.3.2. Influence of axial sealing gap

When the numbers of axial and the radial pole teeth are 1 and 4 respectively, the axial and radial pole teeth height are 0.7 mm and the radial sealing gap height is 0.4 mm, the magnetic flux density of different axial sealing gap width is shown in Fig. 12 and the capabilities of ordinary ferrofluid seal and stepped ferrofluid seal with different axial sealing gap height is shown in Fig. 13.

Fig. 12.

Magnetic field distributions for different axial sealing gaps. (i), 0.4 mm; (ii), 0.5 mm; (iii), 0.6 mm; (iv), 0.7 mm.

Fig. 13.

Critical leakage pressure ΔP as a function of the width S of the axial sealing gap.

It can be seen from Fig. 12 that the magnetic flux density in the radial sealing gap of the diverging stepped ferrofluid seal increases as the axial sealing gap width increases. This is because the increases of magnetic potential in the axial sealing gap with the axial sealing gap width increases and the increase of magnetic potential in the radial sealing gap with increase of magnetic potential in the axial sealing gap. So the magnetic flux density in the radial sealing gap will increase. It can also be seen from Fig. 12 that the magnetic flux density in the axial sealing gap of the diverging stepped ferrofluid increases as the axial sealing gap width decreases. This is because the magnetic resistance of the axial sealing gap increase as the axial sealing gap width increases and the magnetic resistance of whole magnetic circuit will decrease. Therefore, the magnetic flux density in the whole magnetic circuit will decrease and the decrease of magnetic flux density in the radial sealing gap with the increase of magnetic flux density in the radial sealing gap.

It can be seen from Fig. 13 that the theoretical pressure capability of the diverging stepped ferrofluid seal decreases firstly and then increases with the increase of the axial sealing gap. The pressure capability decreases firstly because the magnetic flux density difference in the axial sealing gap is smaller than the magnetic flux density difference in the radial sealing when the axial sealing gap is 0.5 mm. According to the pressure capability theory of the diverging stepped ferrofluid seal, the axial seal gap will invalid. The theoretical pressure capability increases later because the magnetic flux density difference in radial sealing gap increases as increases of axial sealing gap. It can be found that the diverging stepped ferrofluid seal pressure capability is superior to the ordinary ferrofluid seal when the axial sealing gap width is 0.5 mm. This is because ordinary ferrofluid seal only have a radial sealing gap, while stepped ferrofluid seal have both axial and radial sealing gaps.

5. Conclusion

In this paper, the diverging stepped ferrofluid seal with a large gap and two magnetic sources is designed. The finite element method is used to numerically calculate the magnetic field distribution in the sealing gap of the stepped ferrofluid seal and the ordinary ferrofluid seal. The theoretical pressure capability of the diverging stepped ferrofluid seal and the ordinary ferrofluid seal is obtained. The influences of key parameters such as the numbers of axial and radial pole teeth, axial and radial pole teeth heights, radial sealing gap height and axial sealing gap width on the sealing capability of the diverging stepped ferrofluid seal are studied. The results show that the radial pole teeth number of the diverging stepped ferrofluid seal have little influence on its pressure capability which is maximum when the axial pole teeth number is 1. The pressure capability of diverging stepped ferrofluid seal reached maximum when the radial and axial pole teeth heights are 0.6 mm. The pressure capability of the diverging stepped ferrofluid seal decreases with the increase of the radial seal gap height and its pressure capability decreases firstly and then increases with the increase of the axial sealing gap width. The pressure capability of diverging stepped ferrofluid seal is significantly improved compared to that of the ordinary ferrofluid seal.

Footnotes

Acknowledgements

This work was supported in part by the National Nature Science Foundation of China under Grant 51905114, in part by the Science and Technology Project of Guangxi Province under Grant 2016GXNSFBA380213, in part by Innovation Project of Guangxi University of Science and Technology Graduate Education GKYC201901, and in part by the Science and Technology Project of Liuzhou under Grant 2017BC20204.

References

Zhao

Sheng

Lin

Zhang

and Jiao

, The model of seal mechanism for magnetic fluid and related experimental study, Mechanika 22(4) (2016), 260–264.

, Magnetic Fluid Sealing Theory and Application, Science Publications, 2010.

Huang

Rui

Jiang

and Rui

, Research on sealing technology of magnetic fluid in medicinal high-speed grinder gearbox, Advanced Materials Rsesarch 433-440 (2012), 600–605.

Wang

, , Theoretical and experimental study on magnetic fluid rotary seal for sealing fluids, Ph.D. Dissertation, Beijing Jiaotong University, 2018.

Zhang

Chen

Yang

and Yang

, Experimental validation and numerical simulation of static pressure in multi-stage ferrofluid seals, IEEE Transaction on Magnetics 55(3) (2019).

Chen

and Zhang

, Numerical analysis on boundary and flow regime of magnetic fluid in the sealing clearance with a rotation shaft, IEEE Transaction on Magnetics 55(2) (2019).

Yang

Hao

Gao

Chen

and Sun

, Leakage path study of a diverging stepped magnetic fluid seal with large gap, International Journal of Applied Electromagnetics and Mechanics 60(3) (2019), 327–335.

Wang

and Li

, Influence of gravity on the anti-pressure ability of magnetic fluid seal, International Journal of Applied Electromagnetics and Mechanics 60(1) (2019), 21–32.

Zhao

Zou

and Hu

, An analysis on the magnetic fluid seal capacity, Journal of Magnetism and Magnetic material 303(2) (2006), 428–431.

10.

Radionov

Podoltsev

and Zahorulko

, Finite-element analysis of magnetic field and the flow of magnetic fluid in the core of magnetic-fluid seal for rotational shaft, Prcedia Engineering 39(11) (2012), 327–338.

11.

Szczech

, Experimental study on the pressure distribution mechanism among stages of the magnetic fluid seal, IEEE on Transactions on Magnetics 54(6) (2018).

12.

Wang

and Li

, Theoretical analysis and experimental study on loading process among stages of magnetic fluid seal, International Journal of Applied Electromagnetics and Mechanics 48(1) (2015), 101–110.

13.

Chen

Yang

Gao

and Chen

, Magnetic field finite element analysis of concentric biaxial stepped ferrofluid seals, International Journal of Applied Electromagnetics and Mechanics 58(4) (2018), 471–481.

14.

Saurabh

Venkat

Upadhyay

and Kinnari

, Design and development of large radial gap static and dynamic magnetic fluid seal, Vacuum 156 (2018), 325–333.

15.

Yang

Sun

Chen

Hao

and Thomas

, Numerical and experimental studies of a novel converging stepped ferrofluid seal, IEEE Transaction on Magnetics 55(3) (2019).

16.

Yang

and Li

, Experimental investigation of diverging stepped magnetic fluid seals with large sealing gap, International Journal of Applied Electromagnetics and Mechanics 50(3) (2016), 407–415.

17.

Rosensweig

R.E.

, Ferrohydrodynamics, Dover Publications, 2002.