institute of engineering mechanics (itm) simulation of
TRANSCRIPT
Institute of Engineering Mechanics (ITM)
Contact and Structure Mechanics Laboratory(LaMCoS UMR 5259)
Simulation of Refrigerant-Lubricated Gas Foil BearingsTim 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
MotorTurb. Comp.
GFB
Heat absorpedfrom cabin
Heat expelledto environment
Hotvapor
Warmvapor
Liquid
Coldmixture(vapor–liquid)
GFB
System optimized by using refrigerant as the lubricating fluid
Challenge: Self-Excited Vibrations
Stationary operating pointstend to become unstable atelevated rotational speed
Occurrence of self-excitedrotor vibrations with largeamplitudes (fluid whirl)
ωc > ω2 (Loss of Stability)
ω1
ω2 > ω1
Key requirements for simulation of GFB rotor systems
1 Realistic fluid model accounting for phase transitions2 Dissipative foil structure model considering dry friction3 Rotor model mutually coupled to nonlinear GFB forces
Refrigerant R-245fa C C C FH
F
H
H
FF
F
(1,1,1,3,3-pentafluoropropane)
Normal boiling point at 15.3 ◦C
50 100 150 200 2500.51.0
1.52.0
10
20
30
40
50
60
Pre
ssur
eP=
p/p 0
P, D, T SurfaceSat. Vapor LineSat. Liquid Line
Reg. PR EoSCub. PR EoSCritical Point
Density D = ρ/ρ0
Temp. T=
ϑ/ϑ0
Pre
ssur
eP=
p/p 0
T=
1.37
T=
1.46
T=
1.55
Cubic Peng–Robinson equation of state (PR EoS)
PPR(D, T)
=1p0
Rϑ0T(M
ρ0D
)− b− a(ϑ0T)(
Mρ0D
)2+ 2b
(M
ρ0D
)− b2
Equilibrium vapor pressure (coexistence curve)by fitting simplified Clausius–Clapeyron solution
Psat(T) = exp(
C0− C1T−1− C2 ln T)
Regularization of PR EoS requires algebraic solutionof cubic equation PPR(D, T) = Psat(T)for roots Dv(T) < Dm(T) < Dl(T)
Mass fraction of liquid
Wl(D, T) =D− Dv(T)
Dl(T)− Dv(T)
Fluid film thickness
H =
Clearance
1
Rotor position
− ε(τ) cos[
ϕ− γ(τ)] Structure deformation
−Q
Reynolds equation for compressible fluids (density D, pressure P, viscosity V)Fluid
compression
∂D∂τ
H + D
Rotor squeeze
− ε′(τ) cos[
ϕ− γ(τ)]− ε(τ)γ′(τ) sin
[ϕ− γ(τ)
]Structuresqueeze
− ∂Q∂τ
=
12
Poiseuille flow
∂
∂ϕ
[DH3
V∂P∂ϕ
]+ κ2 ∂
∂Z
[DH3
V∂P∂Z
] Couette flow
−Λ∂
∂ϕ
[DH
]
FluidDynamics
P(ϕ, Z, τ)
StructureDynamicsQ (ϕ, Z, τ),
Q′(ϕ, Z, τ)
RotorDynamics
ε(τ), ε′(τ),
γ(τ), γ′(τ) FluidDynamicsP(ϕ, Z, τ)
Deformation field Q(ϕ, Z, τ) calculated with beam model (top foil)coupled to deflections of NB bumps per foil strip (free and fixed ends)
Axially averaged pressure load P(ϕ, τ) =∫ +0.5
−0.5P(ϕ, Z, τ)dZ
u1(t)u0(t) uNB−2(t) uNB−1(t)
kB
mF
µCθ0(t) θ1(t) θNB−2(t) θNB−1(t)
. . . FixedEnd
FreeEnd
lR
Coulomb friction forces fn(τ) = µC
[f⊥, n(τ) + f⊥, preload
]sgn U′n(τ)
Continuous signum regularization sgn U′n(τ) ≈ tanh[ΞU′n(τ)
]with Ξ� 1
eξ
eη
eζ
L
Fluid
Structure
Rotor
2mg
ω0
Symmetric horizontalrotor (rigid or elastic)
Constant vertical loadand static unbalance
Rotational speed(bearing number)
Λ =6µ0ω0
p0
(RC
)2
Bearing forces bypressure integration
f(τ) =[
fξ(τ)fη(τ)
]{eξ,eη}
=∫ +0.5
−0.5
∫ 2π
0P(ϕ, Z, τ)
[sin ϕcos ϕ
]{eξ,eη}
dϕ dZ
Computational AnalysisFinite difference discretization on computational grid Nϕ× NZ = 469× 15
Simultaneous subproblem solution by means of collective state vector
Nonlinear ODE system s′(τ) = k{
s(τ), Λ}
with k : Rn×R→ Rn
s(τ) =
[D1,1(τ) · · · DNϕ−2,NZ−2(τ) Θ0(τ) Θ′0(τ) · · · ΘNB−1(τ) Θ′NB−1(τ) ε(τ) ε′(τ) γ(τ) γ′(τ)
]>∈ Rn
Results and Conclusions
Density D
Pressure P
Mass Fraction of Liquid Wl in %
−π −π/2 0 +π/2 +πϕ
−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
Tang
entia
lBum
pF
oilD
ispl
acem
enti
nµm
Time in ms
0.0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8 9
Pro
port
ion
ofD
issi
pate
dE
nerg
y
Free End← Contact #→ Fixed End
2 Important effect of bump–bump interaction mechanisms
0.50.60.70.80.9
0 10000 20000 30000 40000 50000Rot
.Spe
edΛ
Time τ
Run-Up Coast-Down
0.0
−1.0
+1.0
0 10000 20000 30000 40000 50000
Rot
orP
os.ξ
Time τ
0.0
−1.0
+1.0
0 10000 20000 30000 40000 50000
Rot
orP
os.η
Time τ
Dissipative Structure ModelRigid Structure Model
Dissipative Structure ModelRigid Structure Model
3 Undesirable self-excitation vibrations reduced by friction
Γ(t)
e(t)eξ
eη
r
R
q(ϕ, z, t)
h(ϕ, z, t)
ω0
eζ
O
P
h̃(ϕ,t)
ey
Rϕ
exezQ
Flu
idM
od
elS
tru
ctu
reM
od
el Ro
tor
Mo
del