Design of Magnetic Circuit and Finite Element Analysis of Stepped Ferrofluid Seal with Small Sealing Gap

For the sake of study the pressure capability of ferrofluid seals with small sealing gap, a diverging stepped ferrofluid structure with small sealing gap was designed according to the theory of stepped ferrofluid and magnetic path design, and the length and cross-sectional area of the permanent magnet (NeFeB) are obtained. On the basis of these parameters, the ferrofluid distribution of seal gap is calculated by using the magnetic ferrofluid element method, and the pressure capability is obtained, then the reliability of the magnetic path design is obtained according to the capability of withstand voltage. There are some findings as following after analysis and comparison. It is the magnetic flux leakage at the junction of the pole boot and NeFeB that cause pressure capability calculated by the magnetic path method is lower than that calculated by the finite element method.


Introduction
Ferrofluid seals is a kind of technology that ferrofluid seals generates sealing ring to the response characteristics of external magnetic field to achieve the function of resisting pressure difference on both sides of the sealing structure [1]. It is widely used in fields of aerospace, defense, machinery, chemical, petroleum, environment, instruments due to its advantages of zero leakage, long life and low friction [2][3]. At present, there are many researches on common ferrofluid seals in small gap. Research on vacuum, dustproof and small diameter magnetic liquid static sealing with low pressure capability and dynamic sealing technology with small sealing gap from 0.1mm to 0.3mm has been relatively mature, but the research on high pressure, high temperature, low temperature and high vacuum is few, which limits its application [4]. Wang Ruijin [5] et al summarized the general law of ferrofluid seals, and focused on the pressure capability of seal gap under different speeds and parts processing quality, then concluded that the seal gap is too small, which can not improve the pressure capability, but will decline. Gu Jianming [6] et al drew a conclusion that the sealing capability of the ferrofluid increases with the increase of seals, and decreases with rise the seal gap through experiments. Besides, the effect is better when the seal gap is 0. 05-0.20 mm, at the same time, the sealing series has an optimum value. Li Decai [7] et al numerically studied the pressure capability of small gap and concluded that sealing pressure capability is greatly decreases in rise sealing gap, but the sealing gap should not exceed 0.3 mm. These studies were only completed on the base of a non-stepped, but the study of stepped ferrofluid seals with small gap has not been carried out. In 2014, Yang Xiaolong and Li Decai et al began to carry out numerical analysis and experimental research on stepped ferrofluid seals with large sealing gap, and compared it with the pressure capability of ordinary ferrofluid seal, then came to a conclusion that the stepped ferrofluid seal is an effective method to improve the sealing performance  [8] demonstrated that the pressure capability of the diverging stepped ferrofluid seal with large sealing gap decreases with the increase of the radial seal gap width based on the theory and experiment of sealing. Yang Xiaolong et al drew a conclusion that the pressure capability of the diverging stepped ferrofluid seal decreases with the increase of the radial seal gap using the magnetic path design method and the finite element analysis method. However, there are no systematic studies about the sealing property of stepped ferrofluid seals with small sealing gap.
In order to improve the pressure capability of the stepped ferrofluid seals with small sealing gap, this paper designs a structure of the diverging stepped magnetic path with small sealing gap based on the theory of magnetic path design, and studies the effects of parameters such as the axial clearance, radial clearance, the number of axial teeth, the number of radial tooth on its sealing capability through finite element method. The research results provide important theoretical guidance for designing a high-reliability stepped ferrofluid seal structure with small gap. Figure 1 is a diverging stepped ferrofluid seal structure, and its equivalent magnetic circuit is shown in Figure 2. It can be seen from the Figure 1 that the magnetic circuit is mainly composed of permanent magnet, pole piece, magnetic fluid and stepped shaft. The permanent magnet generates magnetic field which binds the magnetic fluid in the radial and axial sealing gap formed between the stepped shaft and the pole piece. The magnetic force on the magnetic fluid is to resist the pressure difference between the two sides of the seal structure and to achieve the purpose of blocking the leakage channel. It is assumed that the gravity of the ferrofluid is much smaller than the magnetic force, and the ferrofluid line can be replaced by an arc; the isomagnet line is considered to coincide with the ferrofluid line, and the surface tension of the ferrofluid can be ignored. Then the pressure capability of ferrofluid static seals can be approximately expressed by following formula

Design of magnetic path
where and and and are, respectively, the maximum and minimum magnetic field strengths and the maximum and minimum magnetic flux densities under the th is pole tooth, while N is the total number of sealed poles.
The total sealing capability of the converging stepped ferrofluid as: and can be calculated by formula (1). Two assumptions are contained in magnetic path design of parallel ferrofluid seals in this paper: first, ignore the magnetic flux leakage; second, ignore the edge effect. According to Kirchhoff's first law, That is, at any point in the magnetic path, the sum of the magnetic flux algebra entering the region equals to the sum of the magnetic flux algebras leaving the region. According to the symmetry of the magnetic path structure,    In order to reduce the volume and weight of the NeFeB and improve the utilization rate of the NeFeB, the NeFeB should be operated at its maximum magnetic energy product, and therefore, Since the magnetic permeability of the material of the pole shoe and rotating shaft is much larger than that of the air and the NeFeB, the obstruction of the pole shoe and the rotating shaft can be neglected when calculating the length of the NeFeB.

Finite element analysis of the ferrofluid
The dimensions of each part of magnetic path are shown in table 1. If the sealing pressure capability is required to be not less than 2×10 5 Pa , we know that the total static magnetic induction difference in the sealing gap is 6.5T according to the general static sealing withstand voltage formula. The length of NeFeB is 5.6 mm, the cross-sectional area is 243 mm 2 according to the symmetry of magnetic path and the design of the parallel ferrofluid seals. Create a physical environment of ferrofluid-tight in the preprocessor of ANSYS finite element analysis software. Due to the symmetry of the structure, the three-dimensional axisymmetric problem of the ferrofluid seals can be simplified to a two-dimensional one. NdFeB magnet is chosen as the NeFeB of diverging stepped ferrofluid seals structure with small gap, the coercive force Hc is 1.356×106 A/m, the magnetic permeability is 1.05. The material of the pole shoe and the stepped shaft are 2Cr13, and oil-based ferrofluid, with saturation magnetization of 30.7 KA/m is selected as ferrofluid. Since the ferrofluid strength generated by the NeFeB in the gap under each pole piece is greater than the saturation magnetization of the ferrofluid, the ferrofluid is saturated and magnetized. The saturation magnetization of the ferrofluid is almost the same as that of the air, so the ferrofluid can be treated as air. Each part is endowed with corresponding material properties, and Smart Meshing, with the accuracy of 4, is selected; A boundary condition that parallels with lines of force is applied to the model boundary. Therefore, it is calculated that the pressure capability of the diverging ferrofluid seals structure is less than that required by the magnetic path design, 2×10 5 Pa, through the finite element method. Magnetic flux leakage，which is caused by the direct contact of magnet, pole piece and shell, is the main reason for this phenomenon. In order to improve the accuracy of the magnetic path design and the pressure capability of the ferrofluid seals, it is possible to select a non-magnetically conductive housing and increase the axial length of the intermediate pole piece to reduce the leakage magnetic energy.

Conclusion
• The finite element analysis results verify the reliability of the magnetic path design method.
• The pressure resistance of divergent stepped ferrofluid seal calculated by magnetic path design method is slightly larger than that of finite element analysis, mainly due to magnetic leakage at the direct contact between NeFeB and pole shoe and shell. • We can reduce the leakage of magnetic energy by choosing non-permeable shells.