Hydrodynamic fluid film bearings and their effect on the stability - PowerPoint PPT Presentation

Notes 5 – Modern Lubrication Hydrodynamic fluid film bearings and their effect on the stability of rotating machinery Dr. Luis San Andres Mast-Childs Professor Lsanandres@tamu.edu http://rotorlab.tamu.edu/me626 1 September 2010

Lubricated Journal Bearings Radial and axial load support of rotating machinery – low friction and long life Advantages Disadvantages Do not require external source of Thermal effects affect performance if pressure. film thickness is too small or available flow rate is too low. Support heavy loads. The load support is a function of the lubricant viscosity, surface speed, Potential to induce hydrodynamic surface area, film thickness and instability , i.e. loss of effective geometry of the bearing. damping for operation well above critical speed of rotor-bearing system Long life (infinite in theory) without wear of surfaces. Typically use MINERAL OIL as lubricant. Modern trend is to Provide stiffness and damping replace with working fluid (water) coefficients of large magnitude. 2

Fundamentals of Thin Film Lubrication (U,V) surface velocities • Film thickness << other dimensions • No curvature effects V V y • Laminar flow, inertialess V z V x U h(x,z,t) z TYP ( c/L *) = 0.001 x Lx Lz h << Lx,Lz ρ U * c Re = y SMALL Couette flow Reynolds # V μ V y U V x V z h(x,z,t) Flow equations: continuity + momentum (x,y) x Lx ( ) ( ) ( ) ∂ ∂ ∂ v v v + + = y x z 0 ∂ ∂ ∂ x y z D B =2 R B D J =2 R J ∂ ∂ ∂ ∂ 2 2 P v P v = − + μ = − + μ x x 0 ; 0 ∂ ∂ ∂ ∂ 2 2 x y z y Quasi-static (pressure forces = viscous forces) Cylindrical bearing 3 Figures 1 & 2 Geometry of flow region in a thin fluid film bearing (h << Lx, Lz)

Importance of fluid inertia in thin film flows Reynolds numbers Absolute Kinematic viscosity ( ν ) fluid viscosity ( µ ) Re at 1,000 rpm Re at 10,000 rpm lbm.ft.s x 10 -5 centistoke Air 1.23 15.4 9.9 99 Thick oil 1,682 30.0 5.1 51 Light oil 120 2.14 71 711 Water 64 1.00 159 1,588 Liquid hydrogen 1.075 0.216 705 7,052 Liquid oxygen 10.47 0.191 794 7,942 Liquid nitrogen 13.93 0.179 848 8,477 R134 refrigerant 13.30 0.163 930 9,296 Fluid inertia is important for operation at high speeds and with process fluids. These are prevalent conditions in HP turbomachinery 4 Importance of fluid inertia effects on several fluid film bearing Table 1 applications. ( c/R J ) =0.001, R J = 38.1 mm (1.5 inch)

Fluid inertia effects at inlet & edges U U U U P P P P Δ P ~ ½ ρ U 2 Δ P ~ ½ ρ U 2 Fluid inertia (Bernoulli’s effect) causes sudden pressure drop (or raise) at sharp inlets (exits). Most important effect on annular pressure seals and hydrostatic bearings with process fluids 5 Pressure drop & rise at sudden changes in film thickness Figure 3

Thin Film Lubrication: Reynolds Equation Elliptical PDE in film region Θ ⎧ ⎫ ⎧ ⎫ ∂ Ω ∂ ∂ ρ ∂ ∂ ρ ∂ 3 3 { } { } 1 h P h P ρ + ρ = + ⎨ ⎬ ⎨ ⎬ h h Bearing center ∂ ∂ Θ ∂ Θ μ ∂ Θ ∂ μ ∂ θ 2 ⎩ ⎭ ⎩ ⎭ t 2 R 12 z 12 z Y Film thickness = + Θ + Θ = θ h c e cos e sin e sin Ω X Y e journal Pressure = ambient on sides Pressure > P cavitation = + θ + θ h C e cos e sin X Y X e Y Bearing Kinematics of journal motion: Y center e X = e cos ( φ ); e Y = e sin ( φ ) e X e φ journal X 6 Cylindrical journal bearing & coordinates Figure 4

Kinematics of journal motion x=R Θ e X = e cos ( φ ); e Y = e sin ( φ ) y t Θ t e Y Y O B φ − φ ⎡ � ⎤ ⎡ ⎤ ⎡ � ⎤ e cos sin e = θ Ω X e ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V t � φ φ φ � ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ e sin cos e O B Y Y h e X O J e A O J V r Journal φ r φ Set: incompressible fluid (oil) X Bearing r Θ = θ + φ Reynolds Eqn. in fixed coordinates (X,Y) ⎧ ⎫ ⎧ ⎫ ∂ ∂ ∂ ∂ Ω Ω ⎧ ⎫ ⎧ ⎫ 3 3 1 h P h P + = + Θ + − Θ � � ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ e e cos e e sin ∂ Θ μ ∂ Θ ∂ μ ∂ X Y Y X ⎩ ⎭ ⎩ ⎭ 2 ⎩ ⎭ ⎩ ⎭ 12 12 2 2 R z z Reynolds Eqn. in moving coordinates) ⎧ ⎫ ⎧ ⎫ ∂ ∂ ∂ ∂ Ω ⎧ ⎫ 3 3 1 h P h P � + = θ + φ − θ � ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ e cos e sin ∂ θ μ ∂ θ ∂ μ ∂ ⎩ ⎭ 2 ⎩ ⎭ ⎩ ⎭ 12 12 2 R z z � φ = Ω For circular centered orbits:: radius ( e ) and / 2 Hydrodynamic pressure P=0 7 Loss of load capacity

Journal bearing reaction force Force = integration of pressure t P.sin θ Θ field on journal surface P.cos θ θ P Y θ π L 2 ∫ ∫ P θ ⎡ ⎤ ⎡ ⎤ F cos ( ) = θ ⋅ θ r journal ⎢ ⎥ , , ⎢ ⎥ r P z t R d dz θ F t ⎣ ⎦ ⎣ ⎦ F sin t 0 0 X F r φ − φ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F F cos sin = X r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ φ φ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ F sin cos F Y t ⎛ Ω ⎞ ⎡ ⎤ Dynamic forces = fn. of ( ) � = Ω = ⎜ φ − ⎟ � � � F F , e , e F e , e ⎜ ⎟ ⎢ ⎥ α α α journal position and X Y ⎣ ⎦ ⎝ ⎠ 2 velocities, rotational speed ( Ω ), viscosity (μ) and geometry ( L, D, c ) 8 Figure 5 Fluid film force acting on journal surface

LONG journal bearing (limit geometry) L L/D >>> 1 bearing Ω ⎧ ⎫ ∂ Ω ∂ ∂ ∂ D 3 { } { } h P journal + = ⎨ ⎬ h h ∂ ∂ Θ ∂ μ ∂ ⎩ ⎭ 2 12 t z z Axial pressure field LONG BEARING MODEL Pressure does not vary axially. Not applicable for most practical L/D >> 1 cases, except sealed squeeze film dampers dP/dz → 0 9 Figure 6

SHORT journal bearing (limit geometry ) L L/D < 0.50 bearing L/D << 1 ⎧ ∂ ⎫ ∂ ∂ Ω ∂ Ω 3 1 h P { } { } D dP/d θ → 0 = + journal ⎨ ⎬ h h ∂ θ μ ∂ θ ∂ ∂ Θ 2 ⎩ ⎭ R 12 t 2 Applicable to actual rotating machinery Axial pressure field SHORT JOURNAL BEARING MODEL ⎡ Ω ⎤ ⎛ ⎞ Hydrodynamic pressure is � μ θ + φ − θ � ⎜ ⎟ 6 ⎢ e cos e sin ⎥ proportional to viscosity ( μ), speed ⎧ ⎫ 2 ⎝ ⎠ ⎪ ⎛ ⎞ ⎪ ⎣ ⎦ 2 L θ − = − ⎜ ⎟ ⎨ 2 ⎬ ( Ω) , and most important to: P ( , z , t ) P z a ⎪ ⎝ ⎠ ⎪ 3 3 3 2 C H ⎩ ⎭ 1/C Control of tolerances in machined clearance is critical for reliable performance 10 Figure 7

STATIC LOAD PERFORMANCE Bearing reaction force = applied Static load static load (% of rotor weight) W t Journal bearing Rotation Ω Y μ Ω ε μ Ω π ⋅ ε 3 2 3 R L R L = − = + Y F ; F ( ) ( ) r t -F r 3 − ε 2 2 − ε 3 / 2 c c 2 2 1 4 1 W e F t Rotor r fluid Static Forces for short length bearing (journal) 1 . 105 film φ X φ X Radial and Tangential forces [N] 1 . 104 Force Balance for Static Load F radial Radial and tangential forces for L/D =0.25 bearing. μ =0.019 Pa.s, L=0.05 1 . 103 m, c=0.1 mm, 3, 000 rpm, F tangential 100 0 0.2 0.4 0.6 0.8 1 Journal bearing can generate large reaction forces. Highly nonlinear journal eccentricity (e/C) * -Fr functions of journal eccentricity Ft 11 Figures 8 & 9

DESIGN PARAMETER: STATIC LOAD PERFORMANCE μ 2 Sommerfeld number ⎛ ⎞ N L D R N rotational speed (rev/s) = ⎜ ⎟ S W static load ⎝ ⎠ W c L, D=2R, c : clearance & μ viscosity Given S, iterative solution to find ( ) 2 operating journal eccentricity ( ε = e/c) μ Ω 2 − ε ⎛ ⎞ 2 ( ) 1 L R L σ = π = = 2 ⎜ ⎟ S L D { ( ) } and attitude angle (φ): ⎝ ⎠ ε ε + π − ε 4 W c 2 2 2 16 1 ( ) π − ε 2 1 F φ = − = t tang ey/c Attitude angle ε F 4 r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 speed increases, Low load, high speed, 0.1 load loads, large viscosity Low load, high speed, large viscosity high viscosity 0.2 Journal locus 0.3 W e/c load Clearance circle Locus of journal center for short 0.4 ex/c length bearing spin attitude 0.5 direction angle 0.6 clearance 0.7 circle load increases, low speed, low 0.8 High load, low speed, small viscosity viscosity 0.9 12 Figure 12 1

DESIGN PARAMETER: STATIC LOAD PERFORMANCE Sommerfeld number N rotational speed (rev/s) μ Ω 2 ⎛ ⎞ ( ) L R L σ = π = 2 ⎜ ⎟ S L D W static load ⎝ ⎠ 4 W c σ L, D=2R, c : clearance & μ viscosity 10 * Low load, high speed, large viscosity Sommerfeld number 1 High load, low speed, small viscosity 0.1 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Large e journal eccentricity (e/c) Centered journal 13 Sommerfeld # vs journal eccentricity Figure 10

DESIGN PARAMETER: STATIC LOAD PERFORMANCE μ Ω 2 ⎛ ⎞ Sommerfeld number N rotational speed (rev/s) ( ) L R L σ = π = 2 ⎜ ⎟ S L D W static load ⎝ ⎠ 4 W c L, D=2R, c : clearance & μ viscosity φ 90 * Low load, high speed, large viscosity 80 70 Attitude angle 60 50 40 High load, low speed, 30 small viscosity 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 journal eccentricity (e/c) Large e Centered journal 14 Attitude angle # vs journal eccentricity Figure 11