Title
Application of fluid-structure interaction technique for the
TEHD lubrication problems of the bidirectional thrust bearings
(
)
0
j
j
U
tx
r
r
¶¶
+=
¶¶
Liming Zhai1, 2, Zhengwei Wang1, 2, *, Yongyao Luo1, 2, Zhongjie
Li1, 2, Xin Liu1, 2
ISROMAC 2016
International Symposium on Transport Phenomena and Dynamics of
Rotating Machinery
Hawaii, Honolulu April 10-15, 2016
Abstract
Thrust bearing lubrication involves fluid-thermal-structural
interactions between the oil film, the pad and the runner collar.
This study used the FSI technique to investigate the lubrication
characteristics of a bidirectional thrust bearing for several
typical operating conditions to analyze the influences of the
operating conditions and the thrust load on the lubrication
characteristics. The results show only a very small part of heat is
dissipated through the pad and collar, while most of the heat is
carried away into the fresh oil by the film flow. The heat out of
the inner radius surface, trailing surface and outer radius surface
account for almost 80 percent of the total heat transferred into
the pad. The heat transfer coefficients on the pad surfaces are
quite uneven with the largest on the leading surface and the least
on the trailing surface. The eddies in the space between the
adjacent two pads result to the larger heat transfer coefficients
on the leading surface and less on the trailing surface.
Keywords
Bidirectional thrust bearing — TEHD — FSI — Heat transfer
1 State Key Laboratory of Hydroscience and Engineering, Tsinghua
University, Beijing, China
2 Department of Thermal Engineering, Tsinghua University,
Beijing, China
*Corresponding author: [email protected]
INTRODUCTION
The thrust bearings are one of the most crucial components in
large hydropower units which greatly affects units’ safe and stable
operation. A thrust bearing includes a thrust collar, a mirror
plate and several pads. The rotor load is transferred to each pad
through the thrust collar and the mirror plate, and then to the
base. A very narrow clearance between the mirror plate and each pad
is filled with the lubricating oil during operation.
The lubrication of large thrust bearings is the
thermal-elastic-hydrodynamic (TEHD) problem. During operation, high
pressure is formed in the oil film between the mirror plate and the
pad which support the thrust load. In addition, the oil film is
heated by viscous friction which leads to the temperature gradient
in the pad and collar (or mirror plate). Thus, the high pressure in
the film will result in mechanical deformation on the pad and
collar, while temperature gradient in the pad and collar will cause
the thermal deformation. In turn, the total deformation will change
the oil film thickness and affect the pressure and temperature
distribution in the film. Luo et al. [1] theoretically and
experimentally analyzed the applications and operating conditions
of thrust bearings with different centrally supporting structures
and various operating conditions. Huang et al. [2] [3] and Wu et
al. [4] conducted experiments on a bidirectional thrust bearing in
a test rig and verified 3D TEHD numerical results. They solved the
Reynolds equation, the energy equation, the film thickness equation
for the film with assumpted inlet temperatures and then the heat
conduction equation and the elastic equilibrium equation for the
pad with assumpted heat transfer coefficients on the pad surfaces.
The results showed that the pressure center is almost at the pad
angular center rather than off the angular center in a
bidirectional thrust bearing. Wang et al. [5] used a CFD model to
do a 3D HD analysis of a bidirectional thrust bearing to analyze
the effects of the pad inclination angle and the rotor speed on the
lubrication. However, only the HD model was used in the analysis.
Recently, TEHD calculations using FSI procedures which combine
computational fluid dynamics (CFD) and finite element method (FEM)
models have become more and more popular for analyzing journal
bearings [6] [7] and thrust bearings [8] [9]. Wodtke et al. [10]
used the FSI technique to analyze the hydrodynamic lubrication
bearing in journal bearings and thrust bearings.. The oil film and
the flow surrounding the bearing pads were modeled with inclusion
of the viscosity shearing heat generation without assuming
temperatures at the film inlet and the heat convection coefficients
at the pad free surfaces which were evaluated in the calculations
and differed in different locations in the FSI model. The
temperatures, displacements, heat fluxes and forces were exchanged
at the FSI interfaces between the oil flow and the pad.
Pajaczkowski et al. [11] used the FSI approach in transient
simulations of hydrodynamic tilting pad thrust bearings with a
ring-disc support system. The static oil pocket and the inlet and
outlet chambers were all modeled. The results showed that the
minimum oil film thickness almost immediately stabilizes, while
stabilization of the deformations requires much more time.
This study applied the FSI technique to analyze the lubrication
of a bidirectional thrust bearing in a pump storage unit. A 3D TEHD
model was used for the thrust bearing without the thermal and
pressure boundary condition assumptions for the pad and film. The
basic lubrication characteristics like oil film pressure,
temperature and thickness, the heat transfer coefficient on the pad
surfaces were analyzed first. Then the mechanism of the heat
dissipation and the wall heat transfer coefficients on the pads are
further discussed.
1. GOVERNING EQUATIONS
The numerical model of the thrust bearing consisted of the fluid
region (the oil film and the surrounding oil) and the solid region
(the pad and the runner collar). This study coupled the
computational hydrodynamics (CFD) model for the fluid domain and
finite elements analysis (FEA) model for the solid domain to
analyze the thermal-elastic-hydrodynamic lubrication of the thrust
bearing.
1.1 Fluid region equations:
The transient three-dimensional turbulent Navier-Stokes
equations were solved for the oil film flow including the effects
of the inertial force and body force terms. The computational fluid
dynamics method was used to solve the momentum, continuity and
energy conservation equations with a temperature-dependent
viscosity. The lubricant was treated as a single-phase,
incompressible, Newtonian fluid.
The Reynolds averaged continuity equation is
(1)
The Reynolds averaged momentum equations are
(
)
i
ij
j
U
UU
tx
r
r
¶¶
+
¶¶
(
)
ijijM
ij
p
uuS
xx
tr
¶¶
=-+-+
¶¶
(2)
The total Reynolds averaged energy equation is
(
)
tot
jtot
j
hp
Uh
ttx
r
r
¶¶¶
-+
¶¶¶
j
jj
T
uh
xx
lr
æö
¶¶
=-
ç÷
ç÷
¶¶
èø
(
)
iijijE
j
UuuS
x
tr
¶
éù
+-+
ëû
¶
(3)
The dynamic viscosity of the lubricating oil decreases as the
oil temperature increases, especially at low oil temperatures. The
relationship between the viscosity and the temperature is generally
expressed as:
(
)
(
)
3
0
0
3
20+
=
20+
T
T
mm
(4)
Where μ0 is the dynamic viscosity of the oil at T0, and T is the
absolute temperature. Since the lubrication oil is assumed to be
incompressible, the density-temperature and density-pressure
effects were ignored in this study.
1.2 Solid region equations:
The thermoelastic finite element matrix equations are detrived
by applying the variational principle to the stress equation of
motion and the heat flow conservation equation coupled by the
thermoelastic constitutive equations as:
[
]
[
]
[
]
[
]
{
}
{
}
[
]
[
]
{
}
{
}
0
0
00
tut
C
uu
M
CC
TT
ìüìü
éù
éù
ïïïï
+
êú
íýíý
êú
éùéù
êú
ëû
ïïïï
ëûëû
îþîþ
ëû
&&&
&&&
[
]
[
]
{
}
{
}
{
}
{
}
0
ut
t
KK
uF
TQ
K
éù
éù
ìüìü
ëû
êú
+=
íýíý
êú
éù
îþîþ
ëû
ëû
(5)
where [M] is the element mass matrix, [C] is the element
structural damping matrix, [K] is the element stiffness matrix, {u}
is the displacement vector, {F} is the sum of the element nodal
force and element pressure vectors, Ct is the element specific heat
matrix, Kt is the element thermal conductivity matrix, {T} is the
temperature vector, {Q} is the sum of the element heat generation
load and element convection surface heat flow vectors, [Kut] is the
element thermoelastic stiffness matrix.
1.3 Fluid-thermal-solid interaction equation
The coupled solution method was used to solve the
fluid-thermal-solid interaction problem using existing
computational fluid dynamics and computational solid mechanics
methods with relatively little memory use. The data exchange method
for the thermal elastic-hydro-dynamic interactions must satisfy the
conservation of pressure, displacement, thermal flux, and
temperature at the interaction interface:
ffss
fs
fs
fs
nn
dd
qq
TT
tt
×=×
ì
ï
=
ï
í
=
ï
ï
=
î
(6)
Where subscript f denotes the fluid and s denotes the solid.
The commercial multi-physics software ANSYSTM was used as the
simulation platform with the MFX feature for the TEHD lubrication
simulation. The algorithm iteratively solved the fluid domain (the
oil film model and surrounding oil) by CFX and solved the solid
domain model (the pad and runner collar) by Mechanical APDL with
the process shown in Fig. 1(a). ANSYS interpolates between the two
domains with all the quantities (temperature, force, displacement,
and heat fluxes) exchanged during the solution process at the
matching surfaces. The oil pressure and heat fluxes generated in
the oil film were transmitted from fluid flow model and then
applied to the solids as boundary condition. In turn, the
temperatures and thermal-elastic deformations of the pad and runner
were obtained from the solid model and used to modify the oil film
geometry. The mapping from one mesh to the other was performed as
presented in Fig. 1(b). Thus, each mesh could be adjusted to the
model requirements usually with a fine mesh for the fluid domain
and a coarse mesh for the solid domain. The two meshes only need to
be geometrically complementary in areas connected internally.
(a) Multi-physics process
(b) Conservative interpolation
Figure 1. Multi physics analysis
2. NUMERICAL MODEL
The technical and operational data for the bidirectional thrust
bearing are presented in Table 1. The bearing consisted of a runner
collar and ten pads with flooded lubrication in which all the pads
were immersed in the oil contained in the bearing housing as shown
in Fig. 2. To simplify the computations, the thrust load was
assumed to be evenly distributed on each pad. The rotational
symmetry system was then used to simplify the model to a single
sector. Figure 3 shows the geometrical model of the single sector
used for the numerical simulations, which consisted of one pad and
one adjacent angular sector of the runner collar and the oil flow
(Fig. 3). The space between the pads was divided into one part
adjacent to the upstream (leading) edge and the other part adjacent
to the downstream (trailing) edge. A rotational periodic boundary
condition was used between these two parts to simulate the entire
bearing, which is a sequence of several pads with the outlet of one
being the inlet to the next. In this way, the model gives a
continuous result for a bearing consisting of several pads with
flow and energy transport from a pad outlet to the next inlet just
as in a real bearing. A grid independence check gave a fluid domain
with about 167,000 8-nodel elements. The gap between the pad and
the collar was divided into 15 layers.
Table 1. Bearing dimensions and operating conditions
Item
Value
Pad outer radius R1/mm
1335
Pad inner radius R2/mm
775
Pad angle θ/deg
31
Pad width, B/mm
560
Pad thickness H1/mm
203
Babbitt layer, H2/mm
3
Number of pads n
10
Total thrust load F/t
260
Rotational speed ω/r·min–1
500
Radial pivot position Or/mm
1065
Circumferential pivot eccentricity Oc/%
50
Collar outer radius R3/mm
1335
Collar inner radius R4/mm
775
Collar thickness H2/mm
660
Inlet flow rate Q/(L﹒s-1)
2.5
Oil supply temperature T/ ºC
25
Table 2. Material properties
Material
Steel
Babbitt
Lubricant
ρ/(k·m-3)
7850
7420
890
E/Pa
2.10×1011
5.30×1010
v
0.33
0.33
--
λ/(W·m-1·K-1)
50
38
0.145
α/ K-1
1.20×10-5
2.20×10-5
3.8×10-4
C/(J·Kg-1·K-1)
465
251
2000
μ40/(Pa·s)
--
--
2.848×10-2
Where ρ, E, v, λ, α, C, μ40 is the density, young's modulus,
poisson's ratio, thermal conductivity, thermal expansivity,
specific heat and dynamic viscosity.
Figure 2. Tilting pad in the bidirectional thrust bearing
The bearing was lubricated with Esso 32 oil which was assumed to
be incompressible, single-phase liquid with a temperature-dependent
viscosity. Fresh oil was supplied to the bearing housing by the
inlet boundary condition with a constant flow rate at the given
inlet temperature. An opening boundary condition was applied at the
housing outlet to allow the oil to flow in or out. A rotational
boundary condition was applied at the collar sliding. The fluid
walls in contact with the pad and the collar were set as
fluid-solid interfaces. In addition, the wall connecting to the FSI
interface wall was set as unspecified motion to allow the pad and
collar to move freely.
(a) Solid model (pad and collar)
(b) Fluid domain (oil film and surrounding oil)
Figure 3. Numerical model of the thrust bearing
The solid domain including the pad and collar was meshed into
3,900 elements with mid-side nodes. The pad and collar were made of
steel with a Babbitt alloy coating for the pad. The pad was
supported on a disk which allow for tilting in both the angular and
radial directions. The disk was modeled as a 3D spar element
link180 with the option of compression-only. The disk was then
locked against rotation around the shaft axis. Machined geometrical
features in the sliding surface, chamfers and hydrostatic jacking
recess were omitted to simplify the computations. The top surface
of the collar was fixed except for its axial motion so that it
could move upwards or downwards due to the pressure in the oil film
and the external load on the collar. Symmetric boundary conditions
were used for the two angular sides of the collar to ensure that
the two sides had the same deformation. The temperatures were
coupled along the angular direction in the collar because the high
rotational speed gives almost the same temperatures in the angular
direction. Equivalent pressures were applied on the collar top
surface to simulate the thrust load. The pad and the collar were
also both supported by damper elements to improve the numerical
stability. Fluid-solid interaction surfaces were imposed at all the
external pad surfaces and the collar sliding walls. The inner and
outer cylindrical surfaces of the collar are set as convention heat
transfer boundary with assumpted heat transfer coefficients and
ambient temperatures.
A uniform thickness oil film was initialized with a given
thickness between the pad and collar which could move to a
positions of static equilibrium. Too thin initial film will lead to
negative grids in the film and stop the computation process, while
too thick initial film will cost too much time to reach the
convergence. In this study, 500 μm is used as the initial thickness
after some attempts. To accelerate the process to the equilibrium,
20 iterative steps in one time step with the convergence criteria
of 1×10-4 and 5 iterative steps in one external coupling step
between the fluid and solid domain with the convergence criteria of
1×10-2 are used.
3. Verification of the model
Since the thrust bearing in this study is a prototype, there are
no enough and abundant measurement data collected during the
operation. Only temperature was monitored at the center of the
cross-section 20 mm below the pad sliding surface in real
operation. Four typical operation states including turbine mode
with 300 MW, turbine mode with 180 MW, turbine mode with no load
and the pump mode were computed. The corresponding thrust load of
the four modes are 260t, 310t, 330t and 150t, respectively. The
numerical temperatures are compared with the measured results as
listed in Table 3. The maximum error between the numerical and
measured results is only 2.06% which this numerical model is
acceptable to some extent. In addition, the temperatures in
different operation condition are almost the same with that of the
pump mode slightly higher than those of the turbine modes.
Table 3. Verification of the temperatures
Numerical (ºC)
Measured (ºC)
Error (%)
Turbine, 300MW
58.17
56
2.06
Turbine, 180MW
59.07
57
1.84
Turbine, no load
59.42
58
2.06
Pump
56.05
58
0.09
4. RESULTS AND DISCUSSIONS
4.1 Oil film pressure
Figure 4(a) and (b) show the pressure distributions n the pad
and collar sliding surfaces. The collar rotates from the right side
(leading edge) to the left side (trailing edge). The two pressure
distributions are quite similar. The pressure has no pressure
gradient through the film thickness. In addition, the collar
sliding surface is larger than the pad. The pressure on the collar
is very small outside the pad region which means that the pressure
in the film is much higher than in the oil tank. Since the support
on the pad is located at the pad center for both rotational
directional, the high pressure is concentrated almost at the center
of the pad sliding surface rather than close to the trailing edge
for the directional thrust bearing with eccentrically support. In
addition, there are no negative pressure region in the film which
suppressed the oil cavitation. Because the type of tilting pads is
used in the bearing, which makes pads tilt freely in both angular
and radial directions and then results in a thinner and thinner
film in the rotational direction.
(a) On the pad sliding surface
(b) On the collar sliding surface
Figure 4. Pressure distribution in the oil film
4.2 TEHD deformation
Figure 5(a) and (b) show the axial TEHD deformation on the pad
and collar sliding surfaces. The thermal deformation makes the pad
form a convex surface with maximum deformation of 176.34 μm near
the intersection of the trailing edge and outer radius. The collar
is slightly lifted up at the trailing edge and significantly pushed
down at the leading edge with almost the same deformation in the
angular direction.
The deformation of the pad and collar has the same order of
magnitude with the film thickness which will greatly change the
geometry of the film. The oil film thickness distribution is shown
in Fig. 6. The thickness decreases gradually with rotational
direction to form a wedge shaped oil film produced a large thrust
to support the rotating part. The thinnest part is located near the
trailing edge with the value of only 142 μm.
(a) On the pad sliding surface
(b) On the collar sliding surface
Figure 5. Deformation of the film
Figure 6. Oil film thickness（unit: μm）
4.3 Temperature
Figure 7(a) and (b) show the temperature distributions on the
pad and collar sliding surfaces. Unlike the pressure, the
temperature contours on the pad are quite different from those on
the collar. The temperature on the pad near the trailing edge is
much higher than the leading edge due to the viscous friction in
the oil film, while the temperatures on the collar are almost the
same in the angular direction, which means that the temperature
gradient in the film is very large in the thickness direction with
significant thermal conduction across the field. The computation
results give the highest temperature on the pad as 68.35 ºC at the
trailing edge with the lowest as 50.36 ºC at the leading edge.
Since the fresh oil is only 25 ºC, the temperature at the leading
edge is the result of mixing of the fresh oil and hot oil from the
trailing edge of the upstream pad. The highest temperature on the
collar is 61.85 ºC which is higher than at the leading edge and
lower than at the trailing edge of the pad, because the collar
continually sweeps over the high and low temperature zone on the
pad so its temperatures are the mixture of the two
temperatures.
(c) On the pad sliding surface
(d) On the collar sliding surface
Figure 5. Deformation of the film
4.4 Heat dissipation in the bearing
Viscous friction torque will act on the collar sliding surface
which causes friction power loss and lots of heat in the film.
There are three main dissipation pathway for the heat generation in
the film: through the pad sliding surface, through the collar
sliding surface and carried to the spaces between the adjacent pads
by the film flow. Figure 8 shows the heat flux distribution on the
pad sliding surface where negative values mean the heat transfers
out of the film and into the pad. The heat flux densities near the
trailing edge and outer radius edge are significantly larger than
other zones on the pad sliding surface with the maximum heat flux
up to 60KW/m2. Integrate the heat flux on the pad and collar
sliding surfaces and get the quantities of the heat through the two
surfaces listed in Tab. 4. The heat dissipations through the pad
and collar sliding surfaces only account for 1.06% and 2.25%
percent of the friction power loss in the film, and the remaining
96.69% of the heat is carried into the space between the two
adjacent pads by the film flow and then transferred into the fresh
oil. Only a very small part of heat is dissipated through the pad
and collar. Therefore, it is reasonable to assume the interfaces
between the film and pad/collar as adiabatic boundaries to
calculate the bearing lubrication using the method iteratively
solving the Reynolds and energy equations.
Figure 8. Heat flux through the pad
Table 4. Heat dissipation in the film
Symbol
Heat（KW）
Proportion（%）
Ploss
109.04
100.00
Qpad
-1.16
1.06
Qcollar
-2.45
2.25
Qflow
-105.43
96.69
Where Ploss, Qpad, Qcollar, Qflow are the Frictional power loss
per pad, heat through the pad sliding surface, heat through the
collar sliding surface and heat carried away to the space by the
film flow
As discussed before, one heat dissipation way is through the pad
where the heat first transfers into the pad sliding surface, then
out of the pad free surfaces and finally dissipated by the
surrounding fresh oil. All the surfaces of the pad are marked in
the Fig.9. Figure 10 shows the wall heat flux density on the pad
free surfaces where positive values mean the heat transfers out of
the pad and into the surrounding fresh oil. The heat flux densities
on the inner radius surface, outer radius surface and trailing
surface are larger than on the bottom surface and leading surface.
The heat quantities on all the surfaces are listed in the Tab.5.
The heat balance error of the pad is only 1.97% which means almost
all the heat transferred into the pad flows out of the pad free
surfaces. The heat out of the inner radius surface, trailing
surface and outer radius surface account for almost 80 percent of
the total heat with 34.10%, 28.14% and 19.16%, respectively. The
heat out of the bottom surface is the least with the proportion of
only 7.15%.
Figure 9. Names of the pad surfaces
Figure 10. Heat flux through the pad free surfaces
Table 5. Heat flux in the pad
Symbol
Heat（KW）
Proportion（%）
Qps
1.16
100.00
Qpl
-0.08
7.15
Qpt
-0.33
28.14
Qpi
-0.22
19.16
Qpo
-0.39
34.10
Qpb
-0.16
13.42
ζ
-0.02
1.97
Where Qps, Qpl, Qpt, Qpi, Qpo, Qpb and ζ are the heat flux
through the sliding surface, through the leading surface, through
the trailing surface, through the inner radius surface, through the
outer radius surface, Through the bottom surface and the heat
balance error in the pad
4.5 Heat transfer coefficients on the pad
Figure 11 shows quite uneven distribution of the heat transfer
coefficients on the sliding surface and the free surface of the
pad. Table 6 lists the range of heat transfer coefficients on each
surfaces of the pad. The coefficients on the sliding surface are
significantly larger than the free surfaces. The free surfaces have
the same order of magnitude of the transfer coefficients where
those on the leading surface are the largest, those on the outer
radius surface are less, those on the trailing surface, inner
radius surface and the bottom surface are almost the same and the
least. The largest coefficients of the three parts are almost
4:2:1. It is interesting that the heat transfer coefficients on the
leading surface are the largest while the heat dissipation through
it is the least.
(a) On the pad sliding surface
(b) On the collar sliding surface
Figure 11. Wall heat transfer coefficients on the pad
surfaces
Table 6. Range of the heat transfer coefficients on the pad
surfaces
Item
Min (W·m2·K-1)
Max (W·m2·K-1)
Hs
4635
13583
Hl
511
7418
Ht
463
1843
Hir
362
1759
Hor
730
3997
Hb
294
1963
Where Hs, Hl, Ht, Hir, Hor and Hb are the heat transfer
coefficients on the sliding surface, the leading surface, the
trailing surface, the inner radius surface, the outer radius
surface and the bottom surface.
Figure 12 shows the velocity distribution on the angular
sections of the space between the adjacent two pads. Due to the
rotating effects of the collar, there exists eddies in the space
with larger velocities close to the leading surface which enhance
the heat transfer capability of the surface. Thus, the heat
transfer coefficients on the leading surface of the pad are larger
than the trailing surface.
Figure 12. Velocity distribution between the two adjacent
pads
5. CONCLUSIONS
This study suggested the FSI technique to analyze the TEHD
lubrication characteristics of a bidirectional thrust bearing in a
pump storage unit. This technique uses a full model of the thrust
bearing including not only the pad and collar but also the film and
the flow in the oil housing without thermal and pressure boundary
condition assumptions for the pad and film, which enables a
lubrication analysis with the boundary conditions moved away from
the lubricating film. Besides the basic lubrication characteristics
like oil film pressure, temperature and thickness, the heat
transfer coefficient on the pad surfaces can be obtained using this
method.
Only a very small part of heat is dissipated through the pad and
collar, while most of the heat is carried away into the fresh oil
by the film flow. Therefore, it is reasonable to assume the
interfaces between the film and pad/collar as adiabatic boundaries
to calculate the bearing lubrication using the method iteratively
solving the Reynolds and energy equations.
The heat out of the inner radius surface, trailing surface and
outer radius surface account for almost 80 percent of the total
heat transferred into the pad. The remaining heat flows out of the
bottom surface and the leading surface.
The heat transfer coefficients on the pad surfaces are quite
uneven. The free surfaces have the same order of magnitude of the
transfer coefficients with the largest on the leading surface and
the least on the trailing surface. The eddies in the space between
the adjacent two pads result in the larger heat transfer
coefficients on the leading surface and less on the trailing
surface.
ACKNOWLEDGEMENTS
The authors thank the National Natural Science Foundation of
China (Grant No. 51439002，Grant No. 51409148), and the Specialized
Research Fund for the Doctoral Program of Higher Education of China
(Grant No. 20120002110011, No. 20130002110072), the State Key
Laboratory of Hydroscience and Engineering (Grant No. 2014-KY-05)
for their financial support.
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