662 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAR 2015, VOLUME 17, ISSUE 2. ISSN 1392-8716
1544. Synchronous and subsynchronous vibration under
the combined effect of bearings and seals: numerical
simulation and its experimental validation
Wanfu Zhang
1
, Jiangang Yang
2
, Chun Li
3
, Ren Dai
4
, Ailing Yang
5
1, 3, 4, 5
School of Energy and Power Engineering, University of Shanghai for Science and Technology,
Shanghai 200093, China
2
National Engineering Research Center of Turbo-Generator Vibration, Southeast University,
Nanjing 210096, Jiangsu Province, China
1
Corresponding author
E-mail:
1
zwf5202006@163.com,
2
jgyang@seu.edu.cn,
3
lichunusst@163.com,
4
dairen@usst.edu.cn,
5
alyang@usst.edu.cn
(Received 20 October 2014; received in revised form 5 December 2014; accepted 3 February 2015)
Abstract. A three-dimensional computational fluid dynamics (CFD) model of a labyrinth seal was
established in order to investigate the influence mechanism of combined effects between bearings
and labyrinth seals on the dynamic characteristics of the rotor-bearing-seal system. The dynamic
coefficients of the labyrinth seal for various rotating speeds were calculated. Results show that the
absolute values of cross-coupled coefficients increase with the increasing rotating speed, while
the absolute values of direct coefficients decrease slightly. The positive preswirl at the inlet tends
to intensify the increase of cross-coupled coefficients and the decrease of direct coefficients. The
negative preswirl shows the opposite effect. A finite element model was further setup. Results
show that the labyrinth seal has a large influence on the synchronous response of rotor in the
resonant region due to its damping effect. For other speeds, it has a minor effect. The labyrinth
seal may promote the instability of the rotor-bearing-seal system. The subsynchronous vibration
increases significantly when the seal force is taken into account. The system stability can be
generally enhanced by introducing the negative preswirl at the inlet. Results also show that the
detrimental influence of the labyrinth seal can be compensated by using suitable bearings. A
proper bearing configuration can be designed to reduce the risks of rotordynamic instabilities due
to seals. An experimental test was finally performed, and it shows good agreements with the
numerical simulation.
Keywords: labyrinth seal, vibration, stability, CFD, rotor-bearing-seal system.
Nomenclature
,
direct and cross damping coefficients of the labyrinth seal (N·s/m)
seal force in the radial direction (N)
steam force in the tangential direction (N)
,
direct and cross stiffness coefficients of the labyrinth seal (N/m)
flow passage length in the axial direction (m)
uneven pressure acting on the rotor (Pa)
rotor radius (m)
rotor eccentricity (m)
Ω
whirling speed (rpm)
the th complex eigenvalue
real part of the th eigenvalue
imaginary part of the th eigenvalue
logarithmic decrement of the th eigenvalue
BC boundary condition
HP/IP high pressure/intermediate pressure cylinder
1544. SYNCHRONOUS AND SUBSYNCHRONOUS VIBRATION UNDER THE COMBINED EFFECT OF BEARINGS AND SEALS: NUMERICAL SIMULATION
AND ITS EXPERIMENTAL VALIDATION. WANFU ZHANG, JIANGANG YANG, CHUN LI, REN DAI, AILING YANG
© JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAR 2015, VOLUME 17, ISSUE 2. ISSN 1392-8716 663
1. Introduction
Many instability faults of rotor, such as fluid-induced vibration, have been observed in
turbomachineries in recent years. With the improvement of the working medium parameters in
modern ultra-supercritical steam turbines, more and more unstable vibration problems were
reported in HP/IP cylinders. These faults not only relate to the bearing but also the seal. The need
exists to investigate the mutual combined effect between the bearing and seal on the synchronous
and subsynchronous vibration.
Thomas and Alford
[1, 2] first studied the rotor whirl instability mechanism. It was indicated
that the major destabilizing force acting on turbomachinery stages is a cross-coupled force. Its
magnitude is proportional to the radial deflection of the rotor and it acts in the direction orthogonal
to the deflection. The cross-coupled force tends to promote the whirling motion of rotor and results
in unstable vibration faults. Kim
[3] studied the stability of a turbine rotor system with Alford
forces. He integrated the structural model of a rotor system with the turbine flow model to examine
effect of Alford forces on the structural stability of the rotor system. This force includes not only
the unsteady aerodynamic blade or impeller forces, but also the forces induced by bearings and
seals. To investigate the influence of dynamics of bearings and seals on the system stability, a lot
of research has been carried out to evaluate the effect of bearings and seals on the rotor dynamics
individually
[4-9].
A HP/IP turbine system generally contains the rotor, bearing and seal. All these factors should
be considered as a whole. However, such theoretical and experimental research based on the
rotor-bearing-seal system has been rare. Hirano
[10] evaluated the unstable vibration induced by
the labyrinth seal of a large scales steam turbine. The stabilities of the steam turbines are confirmed
to be in a stable region. Wang
[11] studied the nonlinear dynamic behaviors of a
rotor-bearing-seal system. Wang showed the destabilization influence of the leakage flow inside
an interlocking seal on the system stability. Li
[12] proposed a novel nonlinear model of a
rotor-bearing-seal system based on the Hamilton principle and conducted numerical analysis. Ma
[13] studied the nonlinear dynamic analysis of a rotor-bearing-seal system under two loading
conditions at high speeds. It was found that the second loading condition (out-of-phase unbalances
of two discs) and the nonlinear seal force can mainly restrain the first mode instability and have
slight effects on the second mode instability. Yan [14] presented a transient CFD procedure to
investigate the nonlinear dynamic performance of the rotor-seal system. The previous studies
emphasized the nonlinear characteristics of the rotor-bearing-seal system. In addition, they mainly
employed the Muszynska model based on a Jeffcot rotor system which made a lot of assumptions
and neglected the gyroscopic effect of the rotor.
This paper further studied the influence mechanism of the combined effect between bearings
and seals on the synchronous and subsynchronous vibration of the rotor-bearing-seal system.
Finite difference method was used to obtain the stiffness and damping coefficients of bearings.
Numerical studies using CFD techniques were presented to analyze the dynamic coefficients of
the labyrinth seal. A finite element model of the rotor-bearing-seal system was further setup to
analyze the rotor dynamics based on a test rig. Finally, an experimental measurement was carried
out, and the combined effects between bearings and labyrinth seals were also observed from the
experimental results in the test rig.
2. Test facility and apparatus
The experimental research was carried out on the fluid-induced vibration test rig. As show in
Fig. 1, the test rotor is supported on two bearings lubricated by ISO VG32 turbine oil. The
dimensions of the two bearings are shown in Table 1. The rotor is driven by a 15 kW
variable-speed motor through a gearbox (4.5:1) via two rigid couplings. The maximal rotating
speed and inlet pressure are 6000 rpm and 0.8 MPa respectively. Two balance disks are mounted
at the two ends of the rotor to change the vibration. Total length and mass of the rotor are about
1544. SYNCHRONOUS AND SUBSYNCHRONOUS VIBRATION UNDER THE COMBINED EFFECT OF BEARINGS AND SEALS: NUMERICAL SIMULATION
AND ITS EXPERIMENTAL VALIDATION. WANFU ZHANG, JIANGANG YANG, CHUN LI, REN DAI, AILING YANG
664 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAR 2015, VOLUME 17, ISSUE 2. ISSN 1392-8716
1.259 m and 70.92 kg respectively. The cylinder is supported by springs in the vertical, horizontal
and axial directions. There are 6 sets of seals in the cylinder. High-pressure air enters the seal
through four inlet holes in the center plane of the cylinder (3 sets of seals for each side). Fig. 2
shows the geometry and dimensions for one set of seals.
As shown in Fig. 3, four eddy current sensors are installed at the two ends of rotor to measure
the rotor vibration in the vertical and horizontal direction. A key phase transducer is used to
measure the rotating speed and phase. The test conditions are shown in Table 2.
Fig. 1. Schematic diagram of the seal test rig
Fig. 2. Dimensions of one set of the seal ring
Table 1. Bearing dimensions
Properties Case 1 Case 2 Case 3 Case 4
Diameter (mm) 50 50 50 50
Width (mm) 50 44 44 44
Radius clearance (mm) 0.025 0.025 0.025 0.025
Pad angle (deg) 130 150 150 150
Preload factor 0 0 0.25 0.45
Table 2. Operating conditions
Properties Data
Fluid Compressed air
Temperature (K) 300
Inlet pressure (MPa) 0.55
Outlet pressure (MPa) 0.10
Fig. 3. Displacement of sensors
3. Numerical model of the rotor-bearing-seal system
The main purpose by setting up the present model is to analyze the influence of the labyrinth
seal on the system dynamics and the combined effect between the bearings and seals. The
computation models in this section are built based on the test rig in order to ensure the reliability
and validity of the numerical simulation.
3.1. General model for the bearing
The general formula for the film force in bearings can be defined by the following linearized
1544. SYNCHRONOUS AND SUBSYNCHRONOUS VIBRATION UNDER THE COMBINED EFFECT OF BEARINGS AND SEALS: NUMERICAL SIMULATION
AND ITS EXPERIMENTAL VALIDATION. WANFU ZHANG, JIANGANG YANG, CHUN LI, REN DAI, AILING YANG
© JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAR 2015, VOLUME 17, ISSUE 2. ISSN 1392-8716 665
force-displacement model:
−
=
+
,
(1)
where (,) define the motion of the rotor relative to its stator, (
,
) are the components of the
reaction force acting on the rotor. (
,
,
,
) and (
,
,
,
) are the stiffness
and damping coefficients, respectively.
For the bearing dynamics, much previous work has been done. This paper adopts the solution
model based on Reynolds equation and perturbation method
[15].
3.2. Solution model for the labyrinth seal
For small motion of the rotor about a centered position, a simpler model for the seal dynamics
can be reduced from Eq. (1) as:
−
=
−
+
−
.
(2)
Previous studies
[16-20] using CFD techniques show that the details of the flow field inside
seals prove to be simulated close to the fact. In this paper, Reynolds-Averaged-Navier-Stokes
(RANS) CFD analysis of the flow field inside the labyrinth seal is performed in ANSYS FLUENT.
An advantage of using CFD is its capacity to analyze a large number of complex design
configurations and parameters, and it makes no fundamental assumptions on geometry, shear
stress at the wall, as well as internal flow structure. The equations can be written in Cartesian
tensor form as:
∂
∂
+
∂
∂
(
)=0,
∂
∂
(
)
+
∂
∂
=−
∂
∂
+
∂
∂
∂
∂
+
∂
∂
−
2
3
∂
∂
+
∂
∂
−
,
(3)
where
denotes a scalar such as velocity, pressure, energy, or species concentration
(= 1, 2, 3).
ANSYS FLUENT is a finite-volume-based code, and it solves the equations for conservation
of mass, momentum, and energy in terms of the dependent variables, velocity and pressure.
Three-dimensional computational grid is generated using Gambit 2.2 software. A completed
360 degrees model with eccentric rotor is established to obtain the dynamic coefficients of the
labyrinth seal. The eccentricity of the rotor is 0.05 mm, which equals 10 % of the tip clearance.
Fig. 4 shows the calculation model and boundary conditions defined for the labyrinth seal. The
calculation assumes the fluid to be an ideal gas at constant temperature and the entire flow to be
turbulent. The RNG - model and standard wall functions are used for the calculation. The
momentum equations, the continuity equation, and the turbulence model equations are solved
using the SIMPLE pressure-velocity coupling algorithm. Second-order upwind discretization is
employed for the continuity, momentum and energy equations, and the pressure is discretized with
standard scheme. Pressure is specified at the inlet and outlet boundary. In addition, the inlet
boundary is set with zero preswirl, positive (+30 deg) preswirl and negative (–30 deg) preswirl
individually. A no slip and an adiabatic boundary condition are imposed for the stator wall and
rotor surface.
Since the dynamic coefficients of the labyrinth seal are of interest in present study, the rotating
frame of reference is adopted for the calculation at a specific rotating speed
[16, 17]. In the rotating
1544. SYNCHRONOUS AND SUBSYNCHRONOUS VIBRATION UNDER THE COMBINED EFFECT OF BEARINGS AND SEALS: NUMERICAL SIMULATION
AND ITS EXPERIMENTAL VALIDATION. WANFU ZHANG, JIANGANG YANG, CHUN LI, REN DAI, AILING YANG
666 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAR 2015, VOLUME 17, ISSUE 2. ISSN 1392-8716
frame of reference, the fluid rotates at the same speed with the rotor around the eccentric axis. A
solution is obtained at multiple whirl frequency ratio values. After solutions with various whirl
speeds are obtained, the seal force of each case is calculated by integrating the pressure on the
rotor surface as follows:
−
=
sin,
−
=
cos,
(4)
where is the rotor radius; is the flow passage length in the axial direction;
is the uneven
pressure acting on the rotor surface.
Inlet
Outlet
Rotor
Stato
r
BC:
0.55 MPa,
300 K,
Air
BC:
0.1 MPa
A-A
z
y
x
Fig. 4. Calculation model and boundary conditions
Assuming the effect of inertia is negligible, the dynamic coefficients can be derived from the
following equations:
=−−Ω
,
=−Ω
,
(5)
where , are direct and cross-coupled stiffness coefficients, , are direct and cross-coupled
damping coefficients,
,
are seal forces in radial and tangential directions, Ω
is whirling
speed, and e is eccentricity.
The mesh density study is firstly performed to investigate the effect of mesh density and to
know how fine of a mesh is required to capture the important flow physics. The grid elements are
clustered along the walls in order to capture the details of the boundary layers. The ratio is set to
1.05. This process includes incremental adjustments to grid size. As a result of this analysis, the
total number of nodes for the current calculation model is about 5.5 million. The final mesh in the
meridional plane (A-A Section in Fig. 4) is shown in Fig. 5. Near wall values are also checked to
ensure that the mesh is appropriate for application of the wall functions. Values of the
dimensionless wall distance (+) lie between 37 and 243, which is acceptable for the wall
functions in the model.