Leeds–Lyon Symposium on TribologySeptember 2–4, 2019, Lyon, France

A Thermo-Gas-Dynamic Model for the Bifurcation Analysisof Refrigerant-Lubricated Gas Foil Bearing Rotor Systems

Tim Leister (KIT/INSA), Benyebka Bou-Saïd (INSA), Wolfgang Seemann (KIT)

KIT – The Research University in the Helmholtz Association www.kit.edu

Self-Acting Gas Foil Bearings (GFBs)Bump foilTop foil

Bearing sleeveRotor journal

Gasdynamiclubrication wedge

Source: rotorlab.tamu.edu/tribgroup/Proj_GasFoilB.htm (adapted) Source: archive.org/details/C-2007-1879

High-speed rotor supported by gasdynamic lubrication wedge

Oil-free machinery offers high energy efficiency and low wear

Application: Vapor-Compression Refrigeration

Condenser

Evaporator

MotorTurb. Comp.

GFB

Heat absorbedfrom cabin

Heat expelledto environment

Hotvapor

Warmvapor

Liquid

Coldmixture(vapor–liquid)

GFB

R-245fa

System optimized by using refrigerant as lubricating fluid

Challenge: Self-Excited Vibrations

Stationary operating pointstend to become unstable atelevated rotational speeds

Occurrence of self-excitedrotor vibrations with largeamplitudes (fluid whirl)

ωc > ω2 (Loss of Stability)

ω1

ω2 > ω1

Vibrations calmed down by deliberately introduced friction

Stationary operation:Static bump deflection

Moderate excitation:Stiction predominant

Strong excitation:Dissipative sliding

Refrigerant R-245fa C C C FH

F

H

H

FF

F

(1,1,1,3,3-pentafluoropropane)

Vapor pressure 1.23 bar at 20 ◦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

Rmϑ0T(Mmρ0D

)− b− a(ϑ0T)(Mm

ρ0D

)2+ 2b

(Mmρ0D

)− 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 by algebraic solutionof cubic equation PPR(D, T) = Psat(T)with 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(ϕ, τ) =

Clearance

1

Structure deformation

−Q(ϕ, τ)

Rotor journal displacement

− ε(τ) cos[

ϕ− γ(τ)]

Reynolds equation

∂

∂τ(DH) +

Couette flow

Λ2

∂

∂ϕ(DH) =

Poiseuille flow

∂

∂ϕ

(DH3

2V∂P∂ϕ

)+ κ2 ∂

∂Z

(DH3

2V∂P∂Z

)Energy equation

ΘcpDH2

[∂T∂τ

+∂T∂ϕ

(Λ2− H2

2V∂P∂ϕ

)+ κ2∂T

∂Z

(−H2

2V∂P∂Z

)]

= ΘαTH2

[∂P∂τ

+∂P∂ϕ

(Λ2− H2

2V∂P∂ϕ

)+ κ2∂P

∂Z

(−H2

2V∂P∂Z

)]

+ 2V

[Λ2

12+

H4

4V2

(∂P∂ϕ

)2+ κ2 H4

4V2

(∂P∂Z

)2]

Viscous dissipation

+ 2ΘkH(Ta− T)

Heat transfer

FluidDynamics

P(ϕ, Z, τ)

StructureDynamicsQ(ϕ, τ),

Q′(ϕ, τ)

RotorDynamics

ε(τ), ε′(τ),

γ(τ), γ′(τ)

FluidDynamicsP(ϕ, Z, τ)

exey

ezmB

u0(t)

w0(t)z0(t)

u1(t) u2(t) uNB−2(t) uNB−1(t)lS

kB

σz, σz, σu

kW

F0(t)

lH

F1(t) F2(t) FNB−2(t) FNB−1(t)kT

lS− 2lB

Triangular spring–mass–rod arrangement with superposed elastic beam model

Elasto-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 GFBs

Small proportion α/2 of total mass shifted to each journal

L

g

ω0 eξ

eη

eζ

Fluid

α2m

b

(1− α)m

k

u0 Foilstructure

Rotordisk

Constant vertical loadand unbalanced disk

Rotational speed(bearing number)

Λ =6µ0ω0

p0

(RC

)2

Unbalance transformation

s′Λ(τ) = +ΛcΛ(τ)− sΛ(τ)[sΛ(τ)

2 + cΛ(τ)2− 1

]c′Λ(τ) = −ΛsΛ(τ)− cΛ(τ)

[sΛ(τ)

2 + cΛ(τ)2− 1

]

Computational Analysis

Finite difference discretization on computational grid Nϕ× NZ = 469× 15Simultaneous subproblem solution by means of collective state vector

Nonlinear ODE system s′(τ) = k{

s(τ), Λ}

with k : Rn×R→ Rns(τ) =

[· · · Di,j(τ) · · · Ti,j(τ) · · · · · · Un(τ) U′n(τ) Zn(τ) · · · X(τ) X′(τ) · · · XD(τ) X′D(τ) · · ·

]>∈ Rn

Results and Conclusions

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

−π −π2 0 +π

2 +π

0.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

Non

dim

.Pre

ssur

ean

dD

ensi

ty

Nondim

.Temp.and

Vapor

Frac.

Circumferential Coordinate ϕ

Pressure PDensity D

Temperature TVapor Fraction Wv

1 Stable stationary operation under two-phase flow conditions

Fric

tion

For

ce

−30−10

1030 Low Friction Coefficient

Fric

tion

For

ce

−30−10

1030 Moderate Friction Coefficient

Fric

tion

For

ce

Time τ

−30−10

1030

0 5 10 15 20 25 30 35 40 45 50

High Friction Coefficient

2 Detailed investigation of stick–slip transitions in foil contacts

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14

Ver

tical

Rot

orJo

urna

lPos

ition

Y

Rotational Speed Λ

HBHB

LP

LP

LP

LP

Friction Coeff.µ→ 0µ = 0.4µ→ ∞

(Limit Point of Cycles)

(Subcrit. Hopf Bifurcation)

(Supercrit. Hopf Bifurcation)

3 Bifurcation diagram giving insights into coexisting solutions

Γ(t)

e(t)eξ

eη

r

R

q(ϕ, t)

h(ϕ, t)

ω0

eζ

O

P

h(ϕ,t)

ey

Rϕ

exezQ

Flu

idM

od

elfo

rN

on

-Id

ealG

ases

Fo

ilS

tru

ctu

reF

rict

ion

Mo

del Jeffco

tt–LavalR

oto

rM

od

el