Institute of Engineering Mechanics (ITM) Contact and Structure Mechanics Laboratory (LaMCoS UMR 5259) Simulation of Refrigerant-Lubricated Gas Foil Bearings Tim Leister, Benyebka Bou-Saïd, Wolfgang Seemann [email protected] [email protected] [email protected] KIT – The Research University in the Helmholtz Association www.kit.edu Gas Foil Bearings (GFBs) Bump foil Top foil Bearing sleeve Rotor journal http://rotorlab.tamu.edu/tribgroup/Proj_GasFoilB.htm https://archive.org/details/C-2007-1879 High-speed rotor supported by aerodynamic lubrication wedge Oil-free machinery offers high energy efficiency and low wear Application: Vapor-Compression Refrigeration Condenser Evaporator Motor Turb. Comp. GFB Heat absorped from cabin Heat expelled to environment Hot vapor Warm vapor Liquid Cold mixture (vapor–liquid) GFB System optimized by using refrigerant as the lubricating fluid Challenge: Self-Excited Vibrations Stationary operating points tend to become unstable at elevated rotational speed Occurrence of self-excited rotor vibrations with large amplitudes (fluid whirl) ω c > ω 2 (Loss of Stability) ω 1 ω 2 > ω 1 Key requirements for simulation of GFB rotor systems 1 Realistic fluid model accounting for phase transitions 2 Dissipative foil structure model considering dry friction 3 Rotor model mutually coupled to nonlinear GFB forces Refrigerant R-245fa C C C F H F H H F F F (1,1,1,3,3-pentafluoropropane) Normal boiling point at 15.3 ◦ C 50 100 150 200 250 0.5 1.0 1.5 2.0 10 20 30 40 50 60 Pressure P = p/ p 0 P, D, T Surface Sat. Vapor Line Sat. Liquid Line Reg. PR EoS Cub. PR EoS Critical Point Density D = ρ/ρ 0 Temp. T = ϑ/ϑ 0 Pressure P = p/ p 0 T = 1.37 T = 1.46 T = 1.55 Cubic Peng–Robinson equation of state (PR EoS) P PR ( D, T ) = 1 p 0 Rϑ 0 T M ρ 0 D - b - a ( ϑ 0 T ) M ρ 0 D 2 + 2b M ρ 0 D - b 2 Equilibrium vapor pressure (coexistence curve) by fitting simplified Clausius–Clapeyron solution P sat ( T )= exp C 0 - C 1 T -1 - C 2 ln T Regularization of PR EoS requires algebraic solution of cubic equation P PR ( D, T )= P sat ( T ) for roots D v ( T ) < D m ( T ) < D l ( T ) Mass fraction of liquid W l ( D, T )= D - D v ( T ) D l ( T ) - D v ( T ) Fluid film thickness H = Clearance 1 Rotor position - ε ( τ ) cos h ϕ - γ ( τ ) i Structure deformation - Q Reynolds equation for compressible fluids (density D, pressure P, viscosity V ) Fluid compression ∂ D ∂τ H + D Rotor squeeze - ε 0 ( τ ) cos h ϕ - γ ( τ ) i - ε ( τ ) γ 0 ( τ ) sin h ϕ - γ ( τ ) i Structure squeeze - ∂ Q ∂τ = 1 2 Poiseuille flow ∂ ∂ϕ " DH 3 V ∂ P ∂ϕ # + κ 2 ∂ ∂ Z " DH 3 V ∂ P ∂ Z # Couette flow - Λ ∂ ∂ϕ h DH i Fluid Dynamics P ( ϕ, Z, τ ) Structure Dynamics Q ( ϕ, Z, τ ), Q 0 ( ϕ, Z, τ ) Rotor Dynamics ε ( τ ), ε 0 ( τ ), γ ( τ ), γ 0 ( τ ) Fluid Dynamics P ( ϕ, Z, τ ) Deformation field Q( ϕ, Z, τ ) calculated with beam model (top foil) coupled to deflections of N B bumps per foil strip (free and fixed ends) Axially averaged pressure load P ( ϕ, τ )= Z +0.5 -0.5 P ( ϕ, Z, τ ) d Z u 1 ( t ) u 0 ( t ) u N B -2 ( t ) u N B -1 ( t ) k B m F μ C θ 0 ( t ) θ 1 ( t ) θ N B -2 ( t ) θ N B -1 ( t ) ... Fixed End Free End l R Coulomb friction forces f n ( τ )= μ C h f ⊥, n ( τ )+ f ⊥, preload i sgn U 0 n ( τ ) Continuous signum regularization sgn U 0 n ( τ ) ≈ tanh ΞU 0 n ( τ ) with Ξ 1 e ξ e η e ζ L Fluid Structure Rotor 2m g ω 0 Symmetric horizontal rotor (rigid or elastic) Constant vertical load and static unbalance Rotational speed (bearing number) Λ = 6μ 0 ω 0 p 0 R C 2 Bearing forces by pressure integration f ( τ )= f ξ ( τ ) f η ( τ ) { e ξ , e η } = Z +0.5 -0.5 Z 2π 0 P ( ϕ, Z, τ ) sin ϕ cos ϕ { e ξ , e η } d ϕ d Z Computational Analysis Finite difference discretization on computational grid N ϕ × N Z = 469 × 15 Simultaneous subproblem solution by means of collective state vector Nonlinear ODE system s 0 ( τ )= k n s ( τ ), Λ o with k : R n × R → R n s ( τ )= " D 1,1 ( τ ) ··· D N ϕ -2, N Z -2 ( τ ) Θ 0 ( τ ) Θ 0 0 ( τ ) ··· Θ N B -1 ( τ ) Θ 0 N B -1 ( τ ) ε ( τ ) ε 0 ( τ ) γ ( τ ) γ 0 ( τ ) # > ∈ R n Results and Conclusions Density D Pressure P Mass Fraction of Liquid W l in % -π -π /2 0 +π /2 +π ϕ -0.5 0 +0.5 Z 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 1 Fluid pressure build-up limited by local vapor–liquid phase transitions -3 -2 -1 0 +1 +2 +3 0 4 8 12 16 Tangential Bump Foil Displacement in μm Time in ms 0.0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 7 8 9 Proportion of Dissipated Energy Free End ← Contact # → Fixed End 2 Important effect of bump–bump interaction mechanisms 0.5 0.6 0.7 0.8 0.9 0 10000 20000 30000 40000 50000 Rot. Speed Λ Time τ Run-Up Coast-Down 0.0 -1.0 +1.0 0 10000 20000 30000 40000 50000 Rotor Pos. ξ Time τ 0.0 -1.0 +1.0 0 10000 20000 30000 40000 50000 Rotor Pos. η Time τ Dissipative Structure Model Rigid Structure Model Dissipative Structure Model Rigid Structure Model 3 Undesirable self-excitation vibrations reduced by friction Γ ( t ) e ( t ) e ξ e η r R q ( ϕ, z, t ) h ( ϕ, z, t ) ω 0 e ζ O P ˜ h ( ϕ, t ) e y R ϕ e x e z Q Fluid Model Structure Model Rotor Model