Bou-Saïd, B.

Leister, T.

Seemann, W.

English

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German-French-Moroccan Summer School 2018 Marrakesh, MoroccoTRIBOLOGY TODAY – From Research Labs to Industry April 8–14, 2018Modeling and Simulation of Refrigerant-Lubricated Gas Foil BearingsTim Leister, Wolfgang Seemann, Benyebka Bou-SaïdKIT – The Research University in the Helmholtz Association www.kit.eduSelf-Acting Gas Foil Bearings (GFBs)Bump foilTop foilBearing sleeveRotor journalGasdynamiclubrication wedgeSource: rotorlab.tamu.edu/tribgroup/Proj_GasFoilB.htm (adapted) Source: archive.org/details/C-2007-1879High-speed rotor supported by gasdynamic lubrication wedgeOil-free machinery offers high energy efficiency and low wearApplication: Vapor-Compression RefrigerationCondenserEvaporatorMotorTurb. Comp.GFBHeat absorpedfrom cabinHeat expelledto environmentHotvaporWarmvaporLiquidColdmixture(vapor–liquid)GFBR-245faSystem optimized by using refrigerant as lubricating fluidChallenge: Self-Excited VibrationsStationary operating pointstend to become unstable atelevated rotational speedsOccurrence of self-excitedrotor vibrations with largeamplitudes (fluid whirl)ωc > ω2 (Loss of Stability)ω1ω2 > ω1Vibrations calmed down by deliberately introduced friction?Stationary operation:Static bump deflectionModerate excitation:Stiction predominantStrong excitation:Dissipative slidingRefrigerant R-245fa C C C FHFHHFFF(1,1,1,3,3-pentafluoropropane)Vapor pressure 1.23 bar at 20 ◦C50 100 150 2002500.51.01.52.0102030405060PressureP=p /p 0P, D, T SurfaceSat. Vapor LineSat. Liquid LineReg. PR EoSCub. PR EoSCritical PointDensity D = ρ/ρ0Temp. T=ϑ/ϑ0PressureP=p /p 0T=1.37T=1.46T=1.55Cubic Peng–Robinson equation of state (PR EoS)PPR(D, T)=1p0 Rmϑ0T(Mmρ0D)− b− a(ϑ0T)(Mmρ0D)2+ 2b(Mmρ0D)− b2Equilibrium vapor pressure (coexistence curve)by fitting simplified Clausius–Clapeyron solutionPsat(T) = exp(C0− C1T−1− C2 ln T)Regularization of PR EoS by algebraic solutionof cubic equation PPR(D, T) = Psat(T)with roots Dv(T) < Dm(T) < Dl(T)Mass fraction of liquidWl(D, T) =D− Dv(T)Dl(T)− Dv(T)Fluid film thicknessH(ϕ, τ) =Clearance1Structure deformation−Q(ϕ, τ)Rotor journal displacement− ε(τ) cos[ϕ− γ(τ)]Reynolds equation for compressible fluids (density D, pressure P, viscosity V)Fluidcompression∂D∂τH + DStructuresqueeze− ∂Q∂τRotor journal squeeze− ε′(τ) cos[ϕ− γ(τ)]− ε(τ)γ′(τ) sin[ϕ− γ(τ)]=∂∂τ[DH]=12∂∂ϕ[DH3V∂P∂ϕ]+ κ2∂∂Z[DH3V∂P∂Z]Poiseuille flow−Λ ∂∂ϕ[DH]Couette flowFluidDynamicsP(ϕ, Z, τ)StructureDynamicsQ(ϕ, τ),Q′(ϕ, τ)RotorDynamicsε(τ), ε′(τ),γ(τ), γ′(τ)FluidDynamicsP(ϕ, Z, τ)exeyezmBu0(t)w0(t)z0(t)u1(t) u2(t) uNB−2(t) uNB−1(t)lSkBσz, σż, σu̇kWF0(t)lHF1(t) F2(t) FNB−2(t) FNB−1(t)kTlS− 2lBTriangular spring–mass–rod arrangement with superposed elastic beam modelElasto-plastic bristle friction Z′n(τ) = U′n(τ)− αn(τ)βn(τ)∣∣U′n(τ)∣∣ σZZn(τ)µn(τ) f⊥,n(τ)Elastic horizontal rotor symmetrically mounted on two GFBsSmall proportion α/2 of total mass shifted to each journalLgω0 eξeηeζFluidα2mb(1− α)mku0 FoilstructureRotordiskConstant vertical loadand unbalanced diskRotational speed(bearing number)Λ =6µ0ω0p0(RC)2Coordinate transformationε(τ) =√X(τ)2 + Y(τ)2γ(τ) = atan2{−X(τ),−Y(τ)}Computational AnalysisFinite difference discretization on computational grid Nϕ× NZ = 469× 15Simultaneous subproblem solution by means of collective state vectorNonlinear ODE system s′(τ) = k{s(τ), Λ}with k : Rn×R→ Rns(τ) =[· · · Di,j(τ) · · · · · · Un(τ) U′n(τ) Zn(τ) · · · X(τ) X′(τ) · · · XD(τ) X′D(τ) · · ·]>∈ RnResults and ConclusionsDensity DPressure PMass Fraction of Liquid Wl in %−π −π/2 0 +π/2 +πϕ−0.50+0.5Z0.250.500.751.001.251.501.752.001 Fluid pressure build-up limited by local vapor–liquid phase transitionsFrictionForceFrictionForceFrictionForceTime τµ = 0.4−30−101030µ = 0.8−30−101030Pure Stiction−30−1010300 5 10 15 20 25 30 35 40 45 502 Characteristic stick–slip transitions in the contacts−1.50−1.25−1.00−0.75−0.500.6 0.7 0.8 0.9 1.0RotorDiskPositionY D(τ)Rotational Speed ΛPure Stictionµ = 0.8µ = 0.6µ = 0.4No Friction3 Vibrations calmed down by deliberately introduced frictionΓ(t)e(t)eξeηrRq(ϕ, t)h(ϕ, t)ω0eζOPh̃(ϕ,t)eyRϕexezQFluidModelforNon-IdealGasesFoilStructureFrictionModel Jeffcott–LavalRotorModel

Modeling and Simulation of Refrigerant-Lubricated Gas Foil Bearings

https://publikationen.bibliothek.kit.edu/1000092801/24836460

Modeling and Simulation of Refrigerant-Lubricated Gas Foil Bearings

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