The magnetic circuit in this sealing mechanism can be considered in static condi- tion so that the ampere enclosed circuit and H–B curve are able to be used to analyze the functionality of this rare-earth magnet steel. The magnetic circuit in this magnetic sealing system can be assumed as a series magnetic circuit mainly consisting of magnet steel and sealing gap. The following equations can be derived based on Fig.8.3(Ekren et al. 2011):
HLỵHgLgẳ0 ð8:1ị HLẳ LgΦ
U0Ag ð8:2ị The magnetic curve in magnetic circuit is shown in Fig.8.4.
If Fm(Φ)ẳHL, the intersection point between function {Fm(Φ)} and line of {[Lg/(U0Ag)]Φ} at vertical coordinate system is the magnetic flux in sealing gap that needs to be determined. The sealing gap reduces fromLgtoLg0 if more magnetic particles are put into the gap of magnetic circuit. As the thickness of magnetic particle layer in the gap between contact surfaces of magnet steel and shaft to be sealed varies from 0 to b, the functioning point of magnet steel varies
Fig. 8.3 2D cross-section view of new high-pressure gas compressive system
8.2 Computer-Aided Simulation on Magnetic Sealing System 131
along the line QK and magnetic flux in sealing gap can be solved. The coefficient of magnetic efficiencyηeff can be applied to verify if the magnetic field in sealing mechanism is correctly designed (Evans et al. 2006):
ηeffẳ Bg2Vg
BH
ð ịmaxV ð8:3ị A higherηeffvalue shows more reliable magnetic circuit design. The normalηeff
value is around 40 % in standard spec. The computational simulation indicates that theηeffvalue in this new magnetic sealing mechanism is 48.8 % which confirms the proper magnetic circuit design in this newly developed magnetic sealing system.
Figure8.5shows the cross-section view of magnetic steel in this sealing system.
The mathematical equation of calculating sealing capacityΔPcan be derived from energy balance law (Feil-Seifer et al. 2007).
Based on diagram in Fig. 8.5, R1ẳ sin2ð ịbα, R2ẳ sin2ð ịbβ, S1ẳR1α, and S2ẳR2β.
So,
ΔSẳS2S1ẳhsinβð ịβ sinαð ịαi
2band OO0 ẳ2βẵctgð ị α ctgð ịβ . Since the work done by each magnetic force line equals to {TΔS}, total work done by all magnetic force lines in magnetic circuit is
WMLẳBDT2b β
sinð ịβ α sinð ịα
ð8:4ị
And the work done by gaseous media pressure exerted to the body of magnetic particles is
WGP ẳ4b2ΔP sinð ị ỵα πẵcosð ị α 22 8sinð ị α sin4ð ịα
" #
ð8:5ị Fig. 8.4 Magnetic curve of
circuit
132 8 Magnetic Sealing System
Based on the energy balance law, the work done by magnetic force lines applied in magnetic circuit should equal to the work done by gaseous media pressure applied to the body of magnetic particles. So,
ΔPẳ
1
2 BDTẵsinð ị α α
2bðsinαị ỵα8ẵcos2ð ị α 4cosð ị ỵα 3
ð8:6ị
The above equation can be analyzed using computer-aided modeling and numer- ical simulation. The stress and deflection profiles of major components in this new magnetic sealing system are shown in Figs.8.6,8.7,8.8,8.9,8.10,8.11,8.12,8.13, 8.14,8.15,8.16,8.17,8.18,8.19, and8.20.
The computer-aided simulation and analysis in Figs. 8.7 and 8.8display the stress and deflection of aluminum adaptor in this new magnetic sealing system. The analytic results show that the maximum stress of 2,923.21 psi in this aluminum adaptor is less than the material yield strength and maximum deflection of 0.000016 in. is within material allowable deflection limit.
The computer-aided simulation and analysis in Figs.8.10and8.11tell the stress and deflection of Armco iron ring in this new magnetic sealing system. The analytic results present that the maximum stress of 2,428.89 psi in this Armco iron ring is less than the material yield strength and maximum deflection of 0.00001 in. is within material allowable deflection limit.
The computer-aided simulation and analysis in Figs.8.13and8.14demonstrate the stress and deflection of compressive unit in this new magnetic sealing system.
The analytic results indicate that the maximum stress of 19,376.24 psi in this compressive unit is less than the material yield strength of 36,000 psi and maximum deflection of 0.00105 in. is within material allowable deflection limit.
Fig. 8.5 Cross-section view of magnetic steel
8.2 Computer-Aided Simulation on Magnetic Sealing System 133
The computer-aided simulation and analysis in Figs. 8.16 and 8.17 state the stress and deflection of piston shaft in this new magnetic sealing system. The analytic results display that the maximum stress of 19,484.07 psi in this piston shaft is less than the material yield strength of 36,000 psi and maximum deflection of 0.00021 in. is within material allowable deflection limit.
Fig. 8.6 Aluminum adaptor
Fig. 8.7 Stress profile in aluminum adaptor
134 8 Magnetic Sealing System
Fig. 8.8 Deflection profile in aluminum adaptor Fig. 8.9 Armco iron ring
8.2 Computer-Aided Simulation on Magnetic Sealing System 135
Fig. 8.10 Stress profile in Armco iron ring
Fig. 8.11 Deflection profile in Armco iron ring
Fig. 8.12 Compressive unit
Fig. 8.13 Stress profile in compressive unit
8.2 Computer-Aided Simulation on Magnetic Sealing System 137
Fig. 8.14 Deflection profile in compressive unit
Fig. 8.15 Piston shaft
138 8 Magnetic Sealing System
Fig. 8.16 Stress profile in piston shaft
Fig. 8.17 Deflection profile in piston shaft
Fig. 8.18 Rare-earth magnetic steel
Fig. 8.19 Stress profile in rare-earth magnetic steel
140 8 Magnetic Sealing System
The computer-aided simulation and analysis in Figs.8.19 and 8.20 show the stress and deflection of rare-earth magnetic steel in this new magnetic sealing system. The analytic results tell that the maximum stress of 2,963.27 psi in this rare earth magnetic steel is less than the material yield strength and maximum deflection of 0.000005 in. is within material allowable deflection limit.
These above computational simulation results demonstrate that the maximum stresses on these major components are all less than the material yield stress and maximum material deflections are all within material allowable deformation limits. The seal capacity of this new magnetic sealing system can keep the oil leakage from chamber of crankshaft into the gaseous chamber of cooling system and reciprocating machineries. The computational solutions verify that this newly developed magnetic sealing system can work properly for better sealing functions.
Fig. 8.20 Deflection profile in rare-earth magnetic steel
8.2 Computer-Aided Simulation on Magnetic Sealing System 141