A finite element analysis of the impact of split pole shoes on magnetic liquid sealing performance

Figures 1 and 2 show the radial and axial section diagrams of split magnetic liquid seal structure combing magnetic liquid rotary seal with split pole shoe's plane seal. Figure 3 is the axial section diagram of optimised seal structure based on Figures 1 and 2.

Fig. 1

Fig. 2

Fig. 3

In Figures 2 and 3, the magnetic loop consists of permanent magnet, pole shoe, magnetic liquid and shaft. Permanent magnet, as the magnetic source of the entire magnetic loop, provides magnetic flux to restrain magnetic liquid in the gap between pole shoe and half-shell, and magnetic liquid can resist pressure difference from both sides under the action of magnetic field force, so as to achieve sealing [11]. As the pole shoe is split, the junction surface of the two sections is bonded by ordinary or magnetic sealant, and therefore its relative magnetic permeability is far less than that of pole shoe material, equivalent to a magnetic shielded ring with the same thickness as pole shoe. See equivalent magnetic circuit in Figures 4 and 5. In Figure 4, R_m is reluctance of permanent magnet, R_j1 and R_j2 are reluctance of split pole shoe, and R_g1, R_g2, R_gn, R_g1′, R_g2′ and R_gn′ are reluctance of magnetic liquid at each pole. teeth of seal gap, and F_c is magnetomotive force. In Figure 5, R_j1, R_j2, R_j3 and R_j4 represent the reluctance at split pole shoe, and R_g1, R_g2, R_g3 and R_g4 represent the total reluctance at each pole shoe seal gap, where the number of reluctance depends on the number of pole teeth of main shaft, and reluctance are connected in parallel. The magnetic permeability of permanent magnet is μ_m, B_r is remanence, H_c is coercivity, l_m is magnet thickness, and Rm=lmμmSm=lmHcBrSm {R_m} = {{{l_m}} \over {{\mu_m}{S_m}}} = {{{l_m}{H_c}} \over {{B_r}{S_m}}} .

Fig. 4

Fig. 5

Theoretical research is the basis of experimental study. Without support of theory, all study is meaningless. This section studies the pressure resistance theory of split pole shoe and provides the foundation for follow-up sections, to ensure research correctness and significance.

An approximate calculation is made on the pressure resistance of static seal of magnetic sealant under the action of magnetic field; the formula is [12, 13] (1.1) Δpm=μ0Ms∑i=1N(Hmaxi-Hmini)=Ms∑i=1N(Bmaxi-Bmini) \Delta {p_m} = {\mu_0}{M_s}{\sum\limits_{i = 1}^N}\left({H_{\max}^i - H_{\min}^i} \right) = {M_s}{\sum\limits_{i = 1}^N}\left({B_{\max}^i - B_{\min}^i} \right) in which Δp_m is the theoretical pressure resistance of magnetic sealant under action of magnetic field, μ_i and M are magnetic permeability of vacuum and saturation magnetisation of magnetic sealant, Hmaxi H_{\max}^i and Hmini H_{\min}^i are the maximum and minimum magnetic field intensity of seal gap between pole shoe in the i^th pole teeth and half-shell of magnetic liquid plane structure, Bmaxi B_{\max}^i and Bmini B_{\min}^i are the maximum and minimum magnetic flux density of seal gap between pole shoe in the i^th pole teeth and half-shell of magnetic liquid plane structure, and N is the total seal series. From Eq. (1), if total magnetic flux density difference is known, we can get the pressure resistance difference of magnetic liquid plane seal.

According to Kirchhoff's First Law, (1.2) ∑iϕi=0 \sum\limits_i {\phi_i} = 0 i.e., the magnetic flux of each magnetic circuit section in magnetic loop sums up to zero. (1.3) ∑iUmi=∑kFmk \sum\limits_i {U_{mi}} = \sum\limits_k {F_{mk}} in which U_mi is the magnetic pressure drop of magnetic field in this section of magnetic circuit, and F_mk is the magnetomotive force of magnetic circuit; when U_mi = H_il_i, its positive direction is the same as magnetic field direction, and when F_mk = I_kN_k, its positive direction is the same as current direction.

According to Ohm's Law for magnetic circuit, (1.4) ϕm=NIl/μcs {\phi_m} = {{NI} \over {l/{\mu_c}s}} in which l is the perimeter of pole shoe inwall, μ_c=μ₀ × μ_r is magnetic permeability of pole shoe material, μ_r is relative magnetic permeability and s is the area of pole shoe in wall.

Let R be reluctance of magnetic circuit. (1.5) R=lμcs R = {l \over {{\mu_c}s}}

Simplify magnetic force line around pole tooth as arcs. Let reluctance of middle vertical part be R₁.S₁ is the area of a single pole tooth. It can be inferred from the reluctance Formula (5) that (1.6) R1=lgμ0S1 {R_1} = {{{l_g}} \over {{\mu_0}{S_1}}}

In Figure 6, let the reluctance of arc-shaped magnetic force line on both sides be R₂. L is the length of pole shoe; then we can get (1.7) dR2=πrμcdrL d{R_2} = {{\pi r} \over {{\mu_c}drL}}

Fig. 6

Let t=∫1dR2 t = \int {1 \over {d{R_2}}} ; then (1.8) t=∫1dR2=μcLπ∫1rdr t = \int {1 \over {d{R_2}}} = {{{\mu_c}L} \over \pi}\int {1 \over r}dr

Since R₁ and R₂ are connected in parallel, (1.9) Rg1=11/R1+t R{g_1} = {1 \over {1/{R_1} + t}}

As shown in Figure 4, several reluctance connected in parallel have the same structure and size; therefore, their values are equal, i.e., Rg1=Rg2=...=Rgn=Rg1'=Rg2'=...=Rgn' {R_{g1}} = {R_{g2}} = ... = {R_{gn}} = R_{g1}^{'} = R_{g2}^{'} = ... = R_{gn}^{'} ; then (1.10) Rg=Rg12n {R_g}{\rm{=}}{{{R_{g1}}} \over {2n}} and (1.11) Rj=ljμjsj {R_j} = {{{l_j}} \over {{\mu_j}{s_j}}} in which R_j is the reluctance of junction surface of split pole shoe, l_j is the thickness of split pole shoe, μ_j is the magnetic permeability of sealant, and s_j is the area of junction surface.

The magnetic flux in magnetic circuit is given by (1.12) ϕm=FcRm+Rg+Rj {\phi_m} = {{{F_c}} \over {{R_m} + {R_g} + {R_j}}}

Suppose the magnetic flux value ϕ_pt is equals in each pole tooth, then we can work out (1.13) ϕpt=ϕm2n {\phi_{pt}} = {{{\phi_m}} \over {2n}} so that the magnetic flux density B_pt at each pole tooth is (1.14) Bpt=ϕptS1 {B_{pt}} = {{{\phi_{pt}}} \over {{S_1}}}

Substituting Eqs (6)–(14) into Eq. (1), we can ascertain the theoretical pressure resistance at plane seal of split structural magnetic sealant under the action of magnetic field force.

Ordinary sealant is the most direct way to bond split pole shoe. An analysis on magnetic field number at sealant gap between pole shoe and shaft is made, and the gradient difference of magnetic field intensity in this seal gap is simulated. Substituting into Eq. (1), the maximum simulated pressure resistant property can be obtained.

The distribution of magnetic field lines when ordinary sealant is used to bond the split pole shoes.

In Figure 8, the magnetic field lines distribute in a very thin manner. Beside pole tooth, there are also some closed magnetic field line loops on top of magnet, which indicates a severe magnetic flux leakage. Particularly, there is no magnetic field distribution at pole tooth which is farther from magnet. This suggests that when split pole shoes are bonded by ordinary sealant, the thin sealant is not magnetically permeable, only equivalent to a magnetically shielded ring that is perpendicular to magnetic field direction and is as thick as pole shoe width. Thereby, the magnetic field intensity at the junction gap between sealant and pole tooth is reduced to a large extent.

Fig. 8

Distribution of magnetic flux density B in the seal gap between split pole shoes and shaft bonded by ordinary sealant is exported, as shown in Figure 9.

Fig. 9

In Figure 9, only at two pole teeth close to magnet is the magnetic flux density higher and the corresponding magnetic field gradient difference larger. The situation is just opposite in other positions, i.e., weaker magnetic flux density and smaller magnetic field gradient difference. After exporting B distribution point value in seal gap, we calculate the sum of all magnetic field gradient differences as 3.45 T, and substitute into Eq. (1) to ascertain the theoretical pressure resistance in seal gap between pole shoe and shaft when using ordinary sealant to bond split pole shoe, which comes out to be 0.6 atmospheric pressure.

In order to solve the seal problem of split pole shoe, magnetic sealant that owns not only magnetism but also air tightness of sealant, is adopted to bond split pole shoe. This extends the application field of magnetic sealant. The distribution of magnetic field lines using magnetic sealant to bind split pole shoes is shown in Figure 10.

Fig. 10

Comparing Figures 8 and 10, it is discovered that in Figure 10, the distribution of magnetic field lines is denser, with less closed loops of magnetic field lines appearing on top of magnet. This indicates a decrease of magnetic flux leakage. Especially, the distribution of magnetic field also exists at pole tooth farther from magnet, which indicates that when magnetic sealant is used to bond split pole shoes, the magnetic permeability of the thin magnetic sealant layer is higher than that of air. Compared to nonmagnetic, ordinary sealant, magnetic sealant enhances the magnetic field intensity of junction gap between sealant and pole tooth to a large extent.

Distribution of magnetic flux density B in the seal gap between split pole shoes and shaft in the case when magnetic sealant is used to bond the split pole shoes is exported, as shown in Figure 11.

Fig. 11

Comparing Figures 9 and 11, it is discovered that in Figure 11, the magnetic flux density is increased significantly. Not only is the magnetic flux density close to magnet pole tooth higher, with larger magnetic field gradient difference, but also the magnetic flux density in other positions is somewhat increased. After exporting B distribution point values in seal gap, we work out the sum of all magnetic field gradient difference as 7.88 T, substitute into Eq. (1) and calculate the theoretical pressure resistance in seal gap between pole shoes and main shaft using magnetic sealant to bond split pole shoe, which comes out to be 1.5 atmospheric pressure.

To further enhance the magnetic field intensity in seal gap between junction surface of split pole shoe and pole tooth, magnetic sealant is used to bond split pole shoes. In case the number of pole shoes remains unchanged, the thickness of split pole shoe is reduced and two cylinder magnets are added. Under this situation, the distribution of magnetic field values is studied and the pressure resistance of seal gap is worked out.

The distribution of magnetic force lines in seal gap between junction surface of split pole shoes and pole tooth is shown in Figure 12.

Fig. 12

Comparing Figures 10 and 12, it is observed that in Figure 12, the distribution of magnetic force lines becomes denser. Due to reduced thickness of split pole shoes, magnetic field distribution exits at every pole tooth farther from magnet. On satisfaction of the condition of same number of pole teeth, and when compared to original two-sectional split pole shoe, this suggests that the use of magnetic sealant reduced pole shoe thickness and that the increase of cylinder magnets can largely enhance the magnetic field intensity in seal gap between sealant and pole tooth.

Distribution of magnetic flux density B in the seal gap between split pole shoes and shaft for split shoes with optimised structure is exported, as shown in Figure 13.

Fig. 13

Comparing Figures 11 and 13, it is observed that in Figure 13, there are four groups of regular concave-convex lines corresponding to pole tooth under the four sections of pole shoes. The magnetic flux density is increased significantly, and the position where the magnetic flux density is the highest is at the two split pole shoes in the middle. This is because the seal gap between pole teeth corresponding to the split two pole shoes in the middle is affected by the action of two magnets with opposite N–S poles. After exporting B distribution point values in seal gap, we work out the sum of all magnetic field gradient difference as 16.83 T, substitute into Eq. (1) and calculate the theoretical pressure resistance in seal gap between pole shoes and main shaft using magnetic sealant to bond split pole shoe, which comes out to be 3.3 atmospheric pressure.