A Theory for Fluidelastic Instability of CONF-910602--2
Tube-Support-Plate InactiveModes DE91 006001
by
Y. Cai, S. S. Chen
Argonne National Laboratory
Argonne, Illinois
S. Chandra
Northeast Utilities Service Company
Hartford, Connecticut
ABSTRACT
Fluidelastic instability of loosely supported tubes, vibrating in a tube support
plate (TSP)-inactive mode, is suspected to be one of the main causes of tube
failures in some operating steam generators and heat exchangers. A
mathematical model, which incorporates all motion-dependent fluid-forces based
on the unsteady flow theory, is presented here for fluidelastic instability of loosely
supported tubes exposed to nonuniform crossflow. In the unstable region
associated with a TSP-inactive mode, the tube motion can be described by two
linear models: TSP-inactive mode when tubes do not contact with the TSP and
TSP-active mode when tubes contact the TSP. A bilinear model, consisting of
these linear models, is presented in this paper to simulate the characteristics of
fluidelastic instability of loosely supported tubes in the stable and unstable region
associated with TSP-inactive modes. The analytical results are compared with
published experimental data; they agree reasonably well. The prediction
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procedure presented for fluidelastic instability response of loosely supported tubes
is applicable in the stable and unstable regions of TSP-inactive mode.
NOMENCLATURE
Ajn Coefficients
ajn,_jn Generalized tube displacement and velocity in the x direction
Bjn Coefficients
bin, t_jn Generalized tube displacement and velocity in the y direction
D ' Tube diameter
E Young's modulus
e Wall thickness of a tube
el, e2 Tube-support gaps
Fi Impact force
f Oscillation frequency
fn Natural frequency of the n-th mode
I Moment of inertia of tube cross section
Kc Equivalent stiffness
knl, kn2 Eigenvalues for tube vibration
Tube length
mj Mass per unit length of the j-th tube
N Number of tubes in an array or row
n Number of modes
P Pitch
R Radius of a tube
Tc Contact time of tube/support
t Time
ts The time when tube strikes TSP
td The time when tube leaves TSP
U Flow velocity
Um Mean flow velocity
Uv Reduced flow velocity (= Urn/lD or U/fD)
uj, flj Displacement and velocity of the j-rh tube in the x direction
Ul, ii I Displacement and velocity of the tube for TSP-inactive mode
u2, 6 2 Displacement and velocity of the tube for TSP-active mode
vi, _j Displacement and velocity of the j-th tube in the y direction
x, y, z Cartesian coordinates
Wr Wear work rate parameters
O_jk, _jk, C_jk, _jk Added mass coefficients
' t !
O_ik,_jk,O'jk,1;jk Fluid damping coefficients
O_jk,_jk,(_jk,_jk Fluid stiffness coefficients
_, _1, _2 Dimensionless coordinates
_j, _j Damping ratio in vacuum
7j Mass ratio (= p_R2/mj) of the j-th tube
_, v Dimensionless distances
co Circular frequency
6Oi, f_j Natural frequency in radian of the j-rh tube in vacuum
_n(Z) Orthonormal function of the n-th mode
_(z) Flow velocity distribution function
t. &
Subscripts
j,k Tube number j,k (j,k = 1 to N)
N Uumber of tubes
n Number of modes
1 For model 1 (TSP-inactive mode)
2 For model 2 (TSP-active mode)
1. INTRODUCTION
Fluidelastic instability associated with a TSP (tube support plate)-inactive
mode for loosely held tubes has been demonstrated in laboratory tests and
observed in a few heat exchangers [1-6]. It is suspected to be one of the main
causes of tube failures in some operating steam generators and heat exchangers.
The phenomenon occurs as a result of design inherent clearances between the
tubes and their' supports, such as baffle plates and anti-vibration bars. When
these tubes are subjected to a cross flow, flow-induced vibration can cause tubes to
impact and rub against their supports and result in tube wear. If tube vibration is
excessive in duration and amplitude, wear can result in sufficient tube wall
. material loss to cause fatigue cracking and/or tube leaks.
In recent years, extensive experimental and analytical studies have been
performed for fluidelastic instability of loosely held tubes and wear. Chen et al. [1]
investigated the fluidelastic behavior of loosely held tubes in laboratory. They
' observed that as the flow velocity is increased to a threshold value, instability in a
TSP-inactive mode may occur. Then for a range of flow velocities larger than the
threshold flow velocity, the tube vibrates predominantly in a TSP-inactive mode
with the response amplitude limited by the clearance between the tube and the
TSP. With a further increase of flow velocity a second threshold, or critical, flow
velocity is reached at which instability in a TSP-active mode begins. In this case,
large amplitude oscillations occur and, in many cases, tubes may impact with
one another.
Additional studies have been published recently to determine the response of
loosely supported tubes under some specific flow conditions [7-16]. References [7]
through [11] described experiments on tubes vibrating in a TSP-inactive mode
including dynamic contact forces between tubes and supports and fretting wear
mechanism. Other studies [12-19] considered computer simulations of the
fluidelastic instability of loosely held tubes using both analytical and numerical
methods. For example, Fisher [13] and Rao [12,15] developed finite element
computer codes to simulate tube vibration and fretting wear, and compared these
results with experimental measurements. Nonlinear analytical methods for
analyzing the fluidelastic instability and impacting behavior of loosely held tubes
were presented by Fricker [14] and Axisa [20]. They used quasi-static or quasi-
steady flow theories, which are applicable in specific parameter ranges. A study
[23] on the basis of the unsteady flow model has also been developed for fluidelastic
instability of tubes in nonuniform flow.
This paper presents an analytical mode] to predict tube response for loosely
held tube arrays in crossflow. An analytical/numerical procedure is described to
predict the critical flow velocity and tube response in rbe instability zone
associated with a TSP-inactive mode. First, fluid coupling effects among tubes
including fluid inertia, fluid damping, and fluid stiffness are described by the
unsteady flow theory. At present, there are very limited data available for these
fluid-force coefficients. In this paper, we use the coefficients presented in Ref. 23
based on the experimental data by Tanaka and his colleagues. Second, as the first